http://www.physicsbook.gatech.edu/api.php?action=feedcontributions&feedformat=atom&user=Ilukowiecki3Physics Book - User contributions [en]2026-04-27T10:37:11ZUser contributionsMediaWiki 1.42.7http://www.physicsbook.gatech.edu/index.php?title=Iterative_Prediction&diff=38641Iterative Prediction2020-03-28T19:20:07Z<p>Ilukowiecki3: </p>
<hr />
<div><br />
<br />
<br />
This page describes iterative prediction, a technique used to predict the motion of particles over a period of time often used in simulations.<br />
<br />
==The Main Idea==<br />
<br />
Iterative prediction is a mathematical technique for approximating the behavior of one or more particles over an interval of time. An iteration is a repeated procedure, so iterative prediction uses repeated procedures to predict the motion of a system. Specifically, 'iterate' means to repeat; in physics, it means to perform the same calculation repeatedly using information produced by the previous calculation. To perform iterative prediction, the initial position, initial momentum, and mass of each particle must be known, and the forces acting on each particle must be known. This information is used to update the position of each particle periodically.<br />
<br />
To apply the momentum principle:<br />
<br />
<br />
'''1.''' Chose your system<br />
<br />
'''2.''' Draw a force diagram (no internal forces, everything depends on interactions with the environment)<br />
<br />
'''3.''' Chose your time interval<br />
<br />
'''4.''' Substitute known values and solve for unknowns<br />
<br />
'''5.''' Check your units and reasonableness of your answer<br />
<br />
<br />
To perform iterative prediction, the time interval of interest must be divided into small sub-intervals of duration <math>\Delta t</math> called time steps. For each time step, the following steps should be performed:<br />
<br />
<ol><br />
<li>The [[Net Force]] acting on each particle should be calculated,</li><br />
<li>The [[Linear Momentum]] of each particle should be updated from the last time step using the newly calculated net force, and</li><br />
<li>The position of each particle should be updated from the last time step using its newly calculated [[Velocity]]. Repeat for the total number of time steps.</li><br />
</ol><br />
<br />
[[File:Iterative_Model.png|center]]<br />
<br />
<center>This is a visual representation of iterative prediction being used to repeatedly update a particle's momentum. It was found on [https://p3server.pa.msu.edu/coursewiki/doku.php?id=183_notes:iterativepredict this page] of the Projects & Practices in Physics website, where it is provided under a [https://creativecommons.org/licenses/by-nc-sa/3.0/ CC Attribution-Noncommercial-Share Alike 3.0 Unported] license. It is shown here under the same license.</center><br />
<br />
<br />
<br />
The size of the time steps used during iterative prediction is referred to as the resolution of the model. Dividing a time interval into a few large time steps is called low resolution while dividing the same time interval into many small time steps is called high resolution. Performing iterative prediction with a higher resolution requires more computations but produces more accurate results. This is because steps 2 and 3 assume constant force and velocity respectively during the duration of each time step (see the "mathematical model" section). In reality, force and velocity change continually, so this is where inaccuracy is introduced. However, if the time steps are small enough that force and velocity do not significantly change during any time step, the iterative prediction is a sufficiently accurate model. In the limit where the time interval is divided into an infinite number of infinitely small time steps, iterative prediction becomes a perfectly accurate model. However, this would require an infinite number of computations, and is therefore impossible to do, although, in some simple situations, analytic approaches can be used to this end instead. Deciding what time step to use requires consideration of both the computational resources available and the resolution required to accurately approximate the situation. When performing iterative prediction by hand, it is impractical to perform more than a few time steps, which cannot be used to accurately model much. If computers are available, it becomes feasible to divide the time interval into thousands of time steps, which can accurately simulate most day-to-day situations. Some of the most sensitive simulations, however, require incredibly small time steps. For example, some physicists perform atom-by-atom simulations of molecular structures interacting. A slight change in the position of an atom can cause a huge change in the forces acting on it, so tiny time steps must be used in order to ensure that force and velocity do not significantly change during any one time step. These simulations are run on the world's most powerful supercomputers and often take weeks to complete. They can simulate time intervals of several nanoseconds (<math>10^{-9}</math> s) using time steps of only a few femtoseconds. (<math>10^{-15}</math> s) [https://www.ncbi.nlm.nih.gov/pmc/articles/PMC6286305/ (Click here for an example of a paper about such a simulation.)] Most of the VPython simulations on this wiki use time steps of about .01s.<br />
<br />
Iterative prediction is only an approximation, but it is a very powerful tool because it can model complex systems that are impossible to model using known analytical approaches such as calculus and kinematics due to unsolvable differential equations and other obstacles.<br />
<br />
===Relation between Newton's 2nd Law and Momentum Principle===<br />
<br />
The rate of change of momentum of an object is directly proportional to the net force applied and is in the direction of the net force; the net force is equal to the rate of change of momentum. <br />
<br />
[[File:Fma_.jpg]]<br />
<br />
- F = net force in newtons (N)<br />
<br />
- m = mass in kilograms (kg)<br />
<br />
- a = acceleration in meters per second per second (ms-2)<br />
<br />
[[File:rate-of-change-of-momentum-2.jpg]]<br />
<br />
- F = net force in newtons (N)<br />
<br />
- <math>\Delta mv</math> = change in momentum in kilogram metres per second (kg ms-1)<br />
<br />
- t = change in time in seconds (s)<br />
<br />
===A Mathematical Model===<br />
<br />
Below are the mathematical formulas required for each of the three steps outlined above.<br />
<br />
<ol><br />
<br />
<li>The net force acting on each particle should be calculated.</li><br />
<br />
This step varies widely from program to program, depending on what system is being modeled. Sometimes, the forces acting on each particle are constant, such as when modeling the motion of a ball with a specific weight thrown near the surface of the earth. Other times, the forces are functions of properties such as the particles' positions, such as when modeling the motion of celestial bodies whose gravitational attraction is greater when they are closer together. Sometimes, there is only one force acting on each particle, and other times, the net force must be found by adding together the individual force vectors.<br />
<br />
<li>The momentum of each particle should be updated from the last time step using the newly calculated net force.</li><br />
<br />
This step uses the [[Impulse and Momentum|impulse-momentum theorem]]. The impulse-momentum theorem states that <math>\vec{p}_f = \vec{p}_i + \vec{J} = \vec{p}_i + \vec{F}_{net, avg} * \Delta t</math>. In iterative prediction, the initial momentum is always known; it is simply the momentum the particle had during the previous time step (or if there is no previous time step, the initial momentum of the particle). <math>\vec{F}_{net, avg}</math> represents the average net force that has been acting on the particle since the previous time step, resulting in a new momentum this time step. To calculate it, the net force found in step 1 should be used. Note that in reality, the net force acting on the particle is constantly changing, and the net force found in step 1 is the final net force, which is not truly the average net force over the course of the previous time step (except in a scenario with constant force). As described in the introductory section, this introduces inaccuracy that can be reduced by using shorter time steps; if short time steps are used, the net force does not have much time to change over the course of the time step, meaning the average net force is very similar to the final net force.<br />
<br />
<li>The position of each particle should be updated from the last time step using its newly calculated velocity.</li><br />
<br />
This step uses the kinematic equation <math>\vec{r}_f = \vec{r}_i + \vec{v}_{avg} * \Delta t</math>. In iterative prediction, the initial position is always known; it is simply the position the particle had during the previous time step (or if there is no previous time step, the initial position of the particle). <math>\vec{v}_{avg}</math> represents the average velocity with which the particle has been moving since the previous time step, resulting in a new position this time step. To calculate it, the momentum found in step 2 should simply be divided by the mass of the particle: <math>\vec{v}_f = \frac{\vec{p}_f}{m}</math>. Note that in reality, the velocity of the particle is constantly changing, and the velocity found above is the final velocity, which is not truly the average velocity over the course of the previous time step (even in a scenario with constant force). As described in the introductory section, this introduces inaccuracy that can be reduced by using shorter time steps; if short time steps are used, the velocity does not have much time to change over the course of the time step, meaning the average velocity is very similar to the final velocity.<br />
<br />
</ol><br />
<br />
===A Computational Model===<br />
<br />
Iterative prediction is typically done on computers because many computations are necessary in order to perform it at a meaningful resolution. VPython is a useful program to perform iterative prediction because it can graphically display the positions of the particles each time step. Below is an example of a VPython program that uses iterative prediction to simulate a ball thrown near the surface of the earth. Be sure to read and understand the source code by clicking on "view this program" in the top left corner; there are comments for each line designed to introduce readers to their first iterative prediction program.<br />
<br />
[https://www.glowscript.org/#/user/YorickAndeweg/folder/PhysicsBookFolder/program/IterativePrediction1 high-resolution ball trajectory simulation]<br />
<br />
To demonstrate differing resolutions, here is a lower resolution version of the same simulation, which uses larger time steps:<br />
<br />
[https://www.glowscript.org/#/user/YorickAndeweg/folder/PhysicsBookFolder/program/IterativePrediction2 low-resolution ball trajectory simulation]<br />
<br />
Note that the first program is more realistic; the more frequent position updates result in a smoother curve that more closely approximates the true trajectory of a ball. On the other hand, the second program has fewer calculations to do. Both programs are artificially slowed using the "rate" command in order to simulate a realistic travel time, but if both programs were allowed to run as quickly as possible, the second program would finish first. For larger, more complicated simulations, time can be a significant factor.<br />
<br />
Here is another more complicated simulation that uses iterative prediction to simulate the motion of a mass swinging on a spring. This is an example of iterative prediction with a varying force; during each time step, the forces acting on the mass must be calculated depending on its position using [[Hooke's Law]].<br />
<br />
[https://www.glowscript.org/#/user/YorickAndeweg/folder/PhysicsBookFolder/program/SpringMass spring mass simulation]<br />
<br />
Note: both projectile motion and spring-mass systems can be analytically modeled; that is, in both of the above systems, analytical techniques such as calculus and kinematics can be used to find the position of the particles as a function of time. However, for some other, more complicated systems, analytical approaches may not exist, while iterative prediction often still works. For example, it would be easy to modify the projectile motion simulation to include the effects of [[Air Resistance]], which is difficult to take into account analytically.<br />
<br />
==Examples==<br />
<br />
In addition to the example below, know how to create V-Python simulations using iterative prediction with both constant and varying forces.<br />
<br />
===1. (Middling)===<br />
<br />
A 2kg particle is released from rest at time t=0. A constant force of 6N is applied to it. How far has the particle traveled after 4 seconds?<br />
<br />
Solve this question<br />
<br />
A.) analytically (using kinematic equations)<br />
<br />
B.) using iterative prediction with 12 equal time steps<br />
<br />
C.) using iterative prediction with 4 equal time steps<br />
<br />
D.) using iterative prediction with a single time step<br />
<br />
Pay attention to how the resolution of the iterative prediction affects the accuracy of the answer. The analytical answer is completely accurate.<br />
<br />
<br />
A.) To solve this analytically, the following kinematic equation should be used:<br />
<br />
<math>\Delta x = \frac{1}{2} a t^2 + v_0 t</math><br />
<br />
Using [[Newton's Second Law: the Momentum Principle]], we know that <math>a = \frac{f}{m} = 3</math>m/s^2. We are also given that <math>v_0 = 0</math>, and we are interested in the time t=4. Substituting these values into the kinematic equation yields<br />
<br />
<math>\Delta x = \frac{1}{2} (3) (4)^2 = 24</math>m.<br />
<br />
B.) To solve this using iterative prediction, a computational tool should be used. Since animation isn't necessary for this problem, I used Microsoft Excel; each row populates itself based on the information in the previous row using formula functions. Google Spreadsheets can also be used, as can V-Python or any other programming language.<br />
<br />
[[File:12steps.PNG]]<br />
<br />
Using 12-time steps of 1/3 seconds each results in a simulated displacement of 26m, which is fairly close to the actual amount of 24m.<br />
<br />
C.)<br />
<br />
[[File:4steps.PNG]]<br />
<br />
using 4-time steps of 1 second each results in a simulated displacement of 30m, which is fairly different from the actual amount of 24m.<br />
<br />
D.)<br />
<br />
[[File:1step.PNG]]<br />
<br />
using a 1-time step of 4 seconds results in a simulated displacement of 48m, which is twice the actual amount!<br />
<br />
This demonstrates that higher-resolution simulations with smaller time steps are able to predict the motion of particles more accurately than lower-resolution simulations with larger time steps. It also demonstrates that as the length of each time step approaches 0, the predicted motion of the particles approaches their actual motion. It is important to choose a resolution that keeps error within an acceptable margin.<br />
<br />
==Connectedness==<br />
<br />
===Application: video game industry===<br />
<br />
One popular application of iterative prediction is in physics-based video games. These are often little more than physics simulations with player input. Iterative prediction is used in most major video games to predict the motion of in-game objects. It can be used even if the laws of physics within the video game are different from those of the real world.<br />
<br />
==History==<br />
<br />
The impulse-momentum theorem, which is used in iterative prediction to update the momenta of particles, is derived from Newton's Second Law, which Isaac Newton (1643-1727) publish in his 1687 book Principia Mathematica. From this point onward, all of the math necessary for iterative prediction was known, but it was not useful to perform iterative prediction until significant computational advances in the twentieth century allowed for high resolutions.<br />
<br />
== See also ==<br />
<br />
*[[Linear Momentum]]<br />
*[[Impulse and Momentum]]<br />
*[[Analytical Prediction]]<br />
*[[Iterative Prediction of Spring-Mass System]]<br />
*[[Determinism]]<br />
<br />
===External links===<br />
<br />
*[http://p3server.pa.msu.edu/coursewiki/doku.php?id=183_notes:modeling_with_vpython Iterative Prediction with VPython]<br />
<br />
==References==<br />
<br />
*Georgia Institute of Technology. Physics Department. PHYS 2211. Fall 2016. '''Wednesday, Week 2 Lecture Slides. Fenton, Flavio H'''<br />
*Georgia Institute of Technology. Physics Department. PHYS 2211. Fall 2016. '''Monday, Week 3 Lecture Slides. Fenton, Flavio H'''<br />
*Georgia Institute of Technology. Physics Department. PHYS 2211. Fall 2016. Lab 07 Fancart Energy & Spring-Mass Instructions, Greco, Edwin<br />
*http://p3server.pa.msu.edu/coursewiki/doku.php?id=183_notes:iterativepredict<br />
*http://physicsnet.co.uk/a-level-physics-as-a2/mechanics/newtons-laws-of-motion/<br />
*Bañó-Polo, M., Baeza-Delgado, C., Tamborero, S., Hazel, A., Grau, B., Nilsson, I., … Mingarro, I. (2018). Transmembrane but not soluble helices fold inside the ribosome tunnel. Nature communications, 9(1), 5246. doi:10.1038/s41467-018-07554-7</div>Ilukowiecki3http://www.physicsbook.gatech.edu/index.php?title=Iterative_Prediction&diff=38615Iterative Prediction2020-03-24T18:49:47Z<p>Ilukowiecki3: </p>
<hr />
<div>'''Claimed by ilukowiecki3 Spring 2020'''<br />
<br />
<br />
This page describes iterative prediction, a technique used to predict the motion of particles over a period of time often used in simulations.<br />
<br />
==The Main Idea==<br />
<br />
Iterative prediction is a mathematical technique for approximating the behavior of one or more particles over an interval of time. An iteration is a repeated procedure, so iterative prediction uses repeated procedures to predict the motion of a system. Specifically, 'iterate' means to repeat; in physics, it means to perform the same calculation repeatedly using information produced by the previous calculation. To perform iterative prediction, the initial position, initial momentum, and mass of each particle must be known, and the forces acting on each particle must be known. This information is used to update the position of each particle periodically.<br />
<br />
To apply the momentum principle:<br />
<br />
<br />
'''1.''' Chose your system<br />
<br />
'''2.''' Draw a force diagram (no internal forces, everything depends on interactions with the environment)<br />
<br />
'''3.''' Chose your time interval<br />
<br />
'''4.''' Substitute known values and solve for unknowns<br />
<br />
'''5.''' Check your units and reasonableness of your answer<br />
<br />
<br />
To perform iterative prediction, the time interval of interest must be divided into small sub-intervals of duration <math>\Delta t</math> called time steps. For each time step, the following steps should be performed:<br />
<br />
<ol><br />
<li>The [[Net Force]] acting on each particle should be calculated,</li><br />
<li>The [[Linear Momentum]] of each particle should be updated from the last time step using the newly calculated net force, and</li><br />
<li>The position of each particle should be updated from the last time step using its newly calculated [[Velocity]]. Repeat for the total number of time steps.</li><br />
</ol><br />
<br />
[[File:Iterative_Model.png|center]]<br />
<br />
<center>This is a visual representation of iterative prediction being used to repeatedly update a particle's momentum. It was found on [https://p3server.pa.msu.edu/coursewiki/doku.php?id=183_notes:iterativepredict this page] of the Projects & Practices in Physics website, where it is provided under a [https://creativecommons.org/licenses/by-nc-sa/3.0/ CC Attribution-Noncommercial-Share Alike 3.0 Unported] license. It is shown here under the same license.</center><br />
<br />
<br />
<br />
The size of the time steps used during iterative prediction is referred to as the resolution of the model. Dividing a time interval into a few large time steps is called low resolution while dividing the same time interval into many small time steps is called high resolution. Performing iterative prediction with a higher resolution requires more computations but produces more accurate results. This is because steps 2 and 3 assume constant force and velocity respectively during the duration of each time step (see the "mathematical model" section). In reality, force and velocity change continually, so this is where inaccuracy is introduced. However, if the time steps are small enough that force and velocity do not significantly change during any time step, the iterative prediction is a sufficiently accurate model. In the limit where the time interval is divided into an infinite number of infinitely small time steps, iterative prediction becomes a perfectly accurate model. However, this would require an infinite number of computations, and is therefore impossible to do, although, in some simple situations, analytic approaches can be used to this end instead. Deciding what time step to use requires consideration of both the computational resources available and the resolution required to accurately approximate the situation. When performing iterative prediction by hand, it is impractical to perform more than a few time steps, which cannot be used to accurately model much. If computers are available, it becomes feasible to divide the time interval into thousands of time steps, which can accurately simulate most day-to-day situations. Some of the most sensitive simulations, however, require incredibly small time steps. For example, some physicists perform atom-by-atom simulations of molecular structures interacting. A slight change in the position of an atom can cause a huge change in the forces acting on it, so tiny time steps must be used in order to ensure that force and velocity do not significantly change during any one time step. These simulations are run on the world's most powerful supercomputers and often take weeks to complete. They can simulate time intervals of several nanoseconds (<math>10^{-9}</math> s) using time steps of only a few femtoseconds. (<math>10^{-15}</math> s) [https://www.ncbi.nlm.nih.gov/pmc/articles/PMC6286305/ (Click here for an example of a paper about such a simulation.)] Most of the VPython simulations on this wiki use time steps of about .01s.<br />
<br />
Iterative prediction is only an approximation, but it is a very powerful tool because it can model complex systems that are impossible to model using known analytical approaches such as calculus and kinematics due to unsolvable differential equations and other obstacles.<br />
<br />
===Relation between Newton's 2nd Law and Momentum Principle===<br />
<br />
The rate of change of momentum of an object is directly proportional to the net force applied and is in the direction of the net force; the net force is equal to the rate of change of momentum. <br />
<br />
[[File:Fma_.jpg]]<br />
<br />
- F = net force in newtons (N)<br />
<br />
- m = mass in kilograms (kg)<br />
<br />
- a = acceleration in meters per second per second (ms-2)<br />
<br />
[[File:rate-of-change-of-momentum-2.jpg]]<br />
<br />
- F = net force in newtons (N)<br />
<br />
- <math>\Delta mv</math> = change in momentum in kilogram metres per second (kg ms-1)<br />
<br />
- t = change in time in seconds (s)<br />
<br />
===A Mathematical Model===<br />
<br />
Below are the mathematical formulas required for each of the three steps outlined above.<br />
<br />
<ol><br />
<br />
<li>The net force acting on each particle should be calculated.</li><br />
<br />
This step varies widely from program to program, depending on what system is being modeled. Sometimes, the forces acting on each particle are constant, such as when modeling the motion of a ball with a specific weight thrown near the surface of the earth. Other times, the forces are functions of properties such as the particles' positions, such as when modeling the motion of celestial bodies whose gravitational attraction is greater when they are closer together. Sometimes, there is only one force acting on each particle, and other times, the net force must be found by adding together the individual force vectors.<br />
<br />
<li>The momentum of each particle should be updated from the last time step using the newly calculated net force.</li><br />
<br />
This step uses the [[Impulse and Momentum|impulse-momentum theorem]]. The impulse-momentum theorem states that <math>\vec{p}_f = \vec{p}_i + \vec{J} = \vec{p}_i + \vec{F}_{net, avg} * \Delta t</math>. In iterative prediction, the initial momentum is always known; it is simply the momentum the particle had during the previous time step (or if there is no previous time step, the initial momentum of the particle). <math>\vec{F}_{net, avg}</math> represents the average net force that has been acting on the particle since the previous time step, resulting in a new momentum this time step. To calculate it, the net force found in step 1 should be used. Note that in reality, the net force acting on the particle is constantly changing, and the net force found in step 1 is the final net force, which is not truly the average net force over the course of the previous time step (except in a scenario with constant force). As described in the introductory section, this introduces inaccuracy that can be reduced by using shorter time steps; if short time steps are used, the net force does not have much time to change over the course of the time step, meaning the average net force is very similar to the final net force.<br />
<br />
<li>The position of each particle should be updated from the last time step using its newly calculated velocity.</li><br />
<br />
This step uses the kinematic equation <math>\vec{r}_f = \vec{r}_i + \vec{v}_{avg} * \Delta t</math>. In iterative prediction, the initial position is always known; it is simply the position the particle had during the previous time step (or if there is no previous time step, the initial position of the particle). <math>\vec{v}_{avg}</math> represents the average velocity with which the particle has been moving since the previous time step, resulting in a new position this time step. To calculate it, the momentum found in step 2 should simply be divided by the mass of the particle: <math>\vec{v}_f = \frac{\vec{p}_f}{m}</math>. Note that in reality, the velocity of the particle is constantly changing, and the velocity found above is the final velocity, which is not truly the average velocity over the course of the previous time step (even in a scenario with constant force). As described in the introductory section, this introduces inaccuracy that can be reduced by using shorter time steps; if short time steps are used, the velocity does not have much time to change over the course of the time step, meaning the average velocity is very similar to the final velocity.<br />
<br />
</ol><br />
<br />
===A Computational Model===<br />
<br />
Iterative prediction is typically done on computers because many computations are necessary in order to perform it at a meaningful resolution. VPython is a useful program to perform iterative prediction because it can graphically display the positions of the particles each time step. Below is an example of a VPython program that uses iterative prediction to simulate a ball thrown near the surface of the earth. Be sure to read and understand the source code by clicking on "view this program" in the top left corner; there are comments for each line designed to introduce readers to their first iterative prediction program.<br />
<br />
[https://www.glowscript.org/#/user/YorickAndeweg/folder/PhysicsBookFolder/program/IterativePrediction1 high-resolution ball trajectory simulation]<br />
<br />
To demonstrate differing resolutions, here is a lower resolution version of the same simulation, which uses larger time steps:<br />
<br />
[https://www.glowscript.org/#/user/YorickAndeweg/folder/PhysicsBookFolder/program/IterativePrediction2 low-resolution ball trajectory simulation]<br />
<br />
Note that the first program is more realistic; the more frequent position updates result in a smoother curve that more closely approximates the true trajectory of a ball. On the other hand, the second program has fewer calculations to do. Both programs are artificially slowed using the "rate" command in order to simulate a realistic travel time, but if both programs were allowed to run as quickly as possible, the second program would finish first. For larger, more complicated simulations, time can be a significant factor.<br />
<br />
Here is another more complicated simulation that uses iterative prediction to simulate the motion of a mass swinging on a spring. This is an example of iterative prediction with a varying force; during each time step, the forces acting on the mass must be calculated depending on its position using [[Hooke's Law]].<br />
<br />
[https://www.glowscript.org/#/user/YorickAndeweg/folder/PhysicsBookFolder/program/SpringMass spring mass simulation]<br />
<br />
Note: both projectile motion and spring-mass systems can be analytically modeled; that is, in both of the above systems, analytical techniques such as calculus and kinematics can be used to find the position of the particles as a function of time. However, for some other, more complicated systems, analytical approaches may not exist, while iterative prediction often still works. For example, it would be easy to modify the projectile motion simulation to include the effects of [[Air Resistance]], which is difficult to take into account analytically.<br />
<br />
==Examples==<br />
<br />
In addition to the example below, know how to create V-Python simulations using iterative prediction with both constant and varying forces.<br />
<br />
===1. (Middling)===<br />
<br />
A 2kg particle is released from rest at time t=0. A constant force of 6N is applied to it. How far has the particle traveled after 4 seconds?<br />
<br />
Solve this question<br />
<br />
A.) analytically (using kinematic equations)<br />
<br />
B.) using iterative prediction with 12 equal time steps<br />
<br />
C.) using iterative prediction with 4 equal time steps<br />
<br />
D.) using iterative prediction with a single time step<br />
<br />
Pay attention to how the resolution of the iterative prediction affects the accuracy of the answer. The analytical answer is completely accurate.<br />
<br />
<br />
A.) To solve this analytically, the following kinematic equation should be used:<br />
<br />
<math>\Delta x = \frac{1}{2} a t^2 + v_0 t</math><br />
<br />
Using [[Newton's Second Law: the Momentum Principle]], we know that <math>a = \frac{f}{m} = 3</math>m/s^2. We are also given that <math>v_0 = 0</math>, and we are interested in the time t=4. Substituting these values into the kinematic equation yields<br />
<br />
<math>\Delta x = \frac{1}{2} (3) (4)^2 = 24</math>m.<br />
<br />
B.) To solve this using iterative prediction, a computational tool should be used. Since animation isn't necessary for this problem, I used Microsoft Excel; each row populates itself based on the information in the previous row using formula functions. Google Spreadsheets can also be used, as can V-Python or any other programming language.<br />
<br />
[[File:12steps.PNG]]<br />
<br />
Using 12-time steps of 1/3 seconds each results in a simulated displacement of 26m, which is fairly close to the actual amount of 24m.<br />
<br />
C.)<br />
<br />
[[File:4steps.PNG]]<br />
<br />
using 4-time steps of 1 second each results in a simulated displacement of 30m, which is fairly different from the actual amount of 24m.<br />
<br />
D.)<br />
<br />
[[File:1step.PNG]]<br />
<br />
using a 1-time step of 4 seconds results in a simulated displacement of 48m, which is twice the actual amount!<br />
<br />
This demonstrates that higher-resolution simulations with smaller time steps are able to predict the motion of particles more accurately than lower-resolution simulations with larger time steps. It also demonstrates that as the length of each time step approaches 0, the predicted motion of the particles approaches their actual motion. It is important to choose a resolution that keeps error within an acceptable margin.<br />
<br />
==Connectedness==<br />
<br />
===Application: video game industry===<br />
<br />
One popular application of iterative prediction is in physics-based video games. These are often little more than physics simulations with player input. Iterative prediction is used in most major video games to predict the motion of in-game objects. It can be used even if the laws of physics within the video game are different from those of the real world.<br />
<br />
==History==<br />
<br />
The impulse-momentum theorem, which is used in iterative prediction to update the momenta of particles, is derived from Newton's Second Law, which Isaac Newton (1643-1727) publish in his 1687 book Principia Mathematica. From this point onward, all of the math necessary for iterative prediction was known, but it was not useful to perform iterative prediction until significant computational advances in the twentieth century allowed for high resolutions.<br />
<br />
== See also ==<br />
<br />
*[[Linear Momentum]]<br />
*[[Impulse and Momentum]]<br />
*[[Analytical Prediction]]<br />
*[[Iterative Prediction of Spring-Mass System]]<br />
*[[Determinism]]<br />
<br />
===External links===<br />
<br />
*[http://p3server.pa.msu.edu/coursewiki/doku.php?id=183_notes:modeling_with_vpython Iterative Prediction with VPython]<br />
<br />
==References==<br />
<br />
*Georgia Institute of Technology. Physics Department. PHYS 2211. Fall 2016. '''Wednesday, Week 2 Lecture Slides. Fenton, Flavio H'''<br />
*Georgia Institute of Technology. Physics Department. PHYS 2211. Fall 2016. '''Monday, Week 3 Lecture Slides. Fenton, Flavio H'''<br />
*Georgia Institute of Technology. Physics Department. PHYS 2211. Fall 2016. Lab 07 Fancart Energy & Spring-Mass Instructions, Greco, Edwin<br />
*http://p3server.pa.msu.edu/coursewiki/doku.php?id=183_notes:iterativepredict<br />
*http://physicsnet.co.uk/a-level-physics-as-a2/mechanics/newtons-laws-of-motion/<br />
*Bañó-Polo, M., Baeza-Delgado, C., Tamborero, S., Hazel, A., Grau, B., Nilsson, I., … Mingarro, I. (2018). Transmembrane but not soluble helices fold inside the ribosome tunnel. Nature communications, 9(1), 5246. doi:10.1038/s41467-018-07554-7</div>Ilukowiecki3http://www.physicsbook.gatech.edu/index.php?title=Iterative_Prediction&diff=38612Iterative Prediction2020-03-24T18:26:47Z<p>Ilukowiecki3: /* The Main Idea */</p>
<hr />
<div>'''Claimed by ilukowiecki3 Spring 2020'''<br />
<br />
<br />
This page describes iterative prediction, a technique used to predict the motion of particles over a period of time often used in simulations.<br />
<br />
==The Main Idea==<br />
<br />
Iterative prediction is a mathematical technique for approximating the behavior of one or more particles over an interval of time. An iteration is a repeated procedure, so iterative prediction uses repeated procedures to predict the motion of a system. Specifically, 'iterate' means to repeat; in physics, it means to perform the same calculation repeatedly using information produced by the previous calculation. To perform iterative prediction, the initial position, initial momentum, and mass of each particle must be known, and the forces acting on each particle must be known. This information is used to update the position of each particle periodically.<br />
<br />
To apply the momentum principle:<br />
<br />
<br />
'''1.''' Chose your system<br />
<br />
'''2.''' Draw a force diagram (no internal forces, everything depends on interactions with the environment)<br />
<br />
'''3.''' Chose your time interval<br />
<br />
'''4.''' Substitute known values and solve for unknowns<br />
<br />
'''5.''' Check your units and reasonableness of your answer<br />
<br />
<br />
To perform iterative prediction, the time interval of interest must be divided into small sub-intervals of duration <math>\Delta t</math> called time steps. For each time step, the following steps should be performed:<br />
<br />
<ol><br />
<li>The [[Net Force]] acting on each particle should be calculated,</li><br />
<li>The [[Linear Momentum]] of each particle should be updated from the last time step using the newly calculated net force, and</li><br />
<li>The position of each particle should be updated from the last time step using its newly calculated [[Velocity]]. Repeat for the total number of time steps.</li><br />
</ol><br />
<br />
[[File:Iterative_Model.png|center]]<br />
<br />
<center>This is a visual representation of iterative prediction being used to repeatedly update a particle's momentum. It was found on [https://p3server.pa.msu.edu/coursewiki/doku.php?id=183_notes:iterativepredict this page] of the Projects & Practices in Physics website, where it is provided under a [https://creativecommons.org/licenses/by-nc-sa/3.0/ CC Attribution-Noncommercial-Share Alike 3.0 Unported] license. It is shown here under the same license.</center><br />
<br />
<br />
<br />
The size of the time steps used during iterative prediction is referred to as the resolution of the model. Dividing a time interval into a few large time steps is called low resolution while dividing the same time interval into many small time steps is called high resolution. Performing iterative prediction with a higher resolution requires more computations but produces more accurate results. This is because steps 2 and 3 assume constant force and velocity respectively during the duration of each time step (see the "mathematical model" section). In reality, force and velocity change continually, so this is where inaccuracy is introduced. However, if the time steps are small enough that force and velocity do not significantly change during any time step, iterative prediction is a sufficiently accurate model. In the limit where the time interval is divided into an infinite number of infinitely small time steps, iterative prediction becomes a perfectly accurate model. However, this would require an infinite number of computations, and is therefore impossible to do, although in some simple situations, analytic approaches can be used to this end instead. Deciding what time step to use requires consideration of both the computational resources available and the resolution required to accurately approximate the situation. When performing iterative prediction by hand, it is impractical to perform more than a few time steps, which cannot be used to accurately model much. If computers are available, it becomes feasible to divide the time interval into thousands of time steps, which can accurately simulate most day-to-day situations. Some of the most sensitive simulations, however, require incredibly small time steps. For example, some physicists perform atom-by-atom simulations of molecular structures interacting. A slight change in the position of an atom can cause a huge change in the forces acting on it, so tiny time steps must be used in order to ensure that force and velocity do not significantly change during any one time step. These simulations are run on the world's most powerful supercomputers and often take weeks to complete. They can simulate time intervals of several nanoseconds (<math>10^{-9}</math> s) using time steps of only a few femtoseconds. (<math>10^{-15}</math> s) [https://www.ncbi.nlm.nih.gov/pmc/articles/PMC6286305/ (Click here for an example of a paper about such a simulation.)] Most of the VPython simulations on this wiki use time steps of about .01s.<br />
<br />
Iterative prediction is only an approximation, but it is a very powerful tool because it can model complex systems that are impossible to model using known analytical approaches such as calculus and kinematics due to unsolvable differential equations and other obstacles.<br />
<br />
===Relation between Newton's 2nd Law and Momentum Principle===<br />
<br />
The rate of change of momentum of an object is directly proportional to the net force applied and is in the direction of the net force; the net force is equal to the rate of change of momentum. <br />
<br />
[[File:Fma_.jpg]]<br />
<br />
- F = net force in newtons (N)<br />
<br />
- m = mass in kilograms (kg)<br />
<br />
- a = acceleration in meters per second per second (ms-2)<br />
<br />
[[File:rate-of-change-of-momentum-2.jpg]]<br />
<br />
- F = net force in newtons (N)<br />
<br />
- <math>\Delta mv</math> = change in momentum in kilogram metres per second (kg ms-1)<br />
<br />
- t = change in time in seconds (s)<br />
<br />
===A Mathematical Model===<br />
<br />
Below are the mathematical formulas required for each of the three steps outlined above.<br />
<br />
<ol><br />
<br />
<li>The net force acting on each particle should be calculated.</li><br />
<br />
This step varies widely from program to program, depending on what system is being modeled. Sometimes, the forces acting on each particle are constant, such as when modeling the motion of a ball with a specific weight thrown near the surface of the earth. Other times, the forces are functions of properties such as the particles' positions, such as when modeling the motion of celestial bodies whose gravitational attraction is greater when they are closer together. Sometimes, there is only one force acting on each particle, and other times, the net force must be found by adding together the individual force vectors.<br />
<br />
<li>The momentum of each particle should be updated from the last time step using the newly calculated net force.</li><br />
<br />
This step uses the [[Impulse and Momentum|impulse-momentum theorem]]. The impulse-momentum theorem states that <math>\vec{p}_f = \vec{p}_i + \vec{J} = \vec{p}_i + \vec{F}_{net, avg} * \Delta t</math>. In iterative prediction, the initial momentum is always known; it is simply the momentum the particle had during the previous time step (or, if there is no previous time step, the initial momentum of the particle). <math>\vec{F}_{net, avg}</math> represents the average net force that has been acting on the particle since the previous time step, resulting in a new momentum this time step. To calculate it, the net force found in step 1 should be used. Note that in reality, the net force acting on the particle is constantly changing, and the net force found in step 1 is the final net force, which is not truly the average net force over the course of the previous time step (except in a scenario with constant force). As described in the introductory section, this introduces inaccuracy that can be reduced by using shorter time steps; if short time steps are used, the net force does not have much time to change over the course of the time step, meaning the average net force is very similar to the final net force.<br />
<br />
<li>The position of each particle should be updated from the last time step using its newly calculated velocity.</li><br />
<br />
This step uses the kinematic equation <math>\vec{r}_f = \vec{r}_i + \vec{v}_{avg} * \Delta t</math>. In iterative prediction, the initial position is always known; it is simply the position the particle had during the previous time step (or, if there is no previous time step, the initial position of the particle). <math>\vec{v}_{avg}</math> represents the average velocity with which the particle has been moving since the previous time step, resulting in a new position this time step. To calculate it, the momentum found in step 2 should simply be divided by the mass of the particle: <math>\vec{v}_f = \frac{\vec{p}_f}{m}</math>. Note that in reality, the velocity of the particle is constantly changing, and the velocity found above is the final velocity, which is not truly the average velocity over the course of the previous time step (even in a scenario with constant force). As described in the introductory section, this introduces inaccuracy that can be reduced by using shorter time steps; if short time steps are used, the velocity does not have much time to change over the course of the time step, meaning the average velocity is very similar to the final velocity.<br />
<br />
</ol><br />
<br />
===A Computational Model===<br />
<br />
Iterative prediction is typically done on computers because many computations are necessary in order to perform it at a meaningful resolution. VPython is a useful program to perform iterative prediction because of it can graphically display the positions of the particles each time step. Below is an example of a VPython program that uses iterative prediction to simulate a ball thrown near the surface of the earth. Be sure to read and understand the source code by clicking on "view this program" in the top left corner; there are comments for each line designed to introduce readers to their first iterative prediction program.<br />
<br />
[https://www.glowscript.org/#/user/YorickAndeweg/folder/PhysicsBookFolder/program/IterativePrediction1 high-resolution ball trajectory simulation]<br />
<br />
To demonstrate differing resolutions, here is a lower resolution version of the same simulation, which uses larger time steps:<br />
<br />
[https://www.glowscript.org/#/user/YorickAndeweg/folder/PhysicsBookFolder/program/IterativePrediction2 low-resolution ball trajectory simulation]<br />
<br />
Note that the first program is more realistic; the more frequent position updates result in a smoother curve that more closely approximates the true trajectory of a ball. On the other hand, the second program has fewer calculations to do. Both programs are artificially slowed using the "rate" command in order to simulate a realistic travel time, but if both programs were allowed to run as quickly as possible, the second program would finish first. For larger, more complicated simulations, time can be a significant factor.<br />
<br />
Here is another more complicated simulation that uses iterative prediction to simulate the motion of a mass swinging on a spring. This is an example of iterative prediction with a varying force; during each time step, the forces acting on the mass must be calculated depending on its position using [[Hooke's Law]].<br />
<br />
[https://www.glowscript.org/#/user/YorickAndeweg/folder/PhysicsBookFolder/program/SpringMass spring mass simulation]<br />
<br />
Note: both projectile motion and spring-mass systems can be analytically modeled; that is, in both of the above systems, analytical techniques such as calculus and kinematics can be used to find the position of the particles as a function of time. However, for some other, more complicated systems, analytical approaches may not exist, while iterative prediction often still works. For example, it would be easy to modify the projectile motion simulation to include the effects of [[Air Resistance]], which is difficult to take into account analytically.<br />
<br />
==Examples==<br />
<br />
In addition to the example below, know how to create vPython simulations using iterative prediction with both constant and varying forces.<br />
<br />
===1. (Middling)===<br />
<br />
A 2kg particle is released from rest at time t=0. A constant force of 6N is applied to it. How far has the particle traveled after 4 seconds?<br />
<br />
Solve this question<br />
<br />
A.) analytically (using kinematic equations)<br />
<br />
B.) using iterative prediction with 12 equal time steps<br />
<br />
C.) using iterative prediction with 4 equal time steps<br />
<br />
D.) using iterative prediction with a single time step<br />
<br />
Pay attention to how the resolution of the iterative prediction affects the accuracy of the answer. The analytical answer is completely accurate.<br />
<br />
<br />
A.) To solve this analytically, the following kinematic equation should be used:<br />
<br />
<math>\Delta x = \frac{1}{2} a t^2 + v_0 t</math><br />
<br />
Using [[Newton's Second Law: the Momentum Principle]], we know that <math>a = \frac{f}{m} = 3</math>m/s^2. We are also given that <math>v_0 = 0</math>, and we are interested in the time t=4. Substituting these values into the kinematic equation yields<br />
<br />
<math>\Delta x = \frac{1}{2} (3) (4)^2 = 24</math>m.<br />
<br />
B.) To solve this using iterative prediction, a computational tool should be used. Since an animation isn't necessary for this problem, I used Microsoft Excel; each row populates itself based on the information in the previous row using formula functions. Google Spreadsheets can also be used, as can vPython or any other programming language.<br />
<br />
[[File:12steps.PNG]]<br />
<br />
Using 12 time steps of 1/3 seconds each results in a simulated displacement of 26m, which is fairly close to the actual amount of 24m.<br />
<br />
C.)<br />
<br />
[[File:4steps.PNG]]<br />
<br />
using 4 time steps of 1 second each results in a simulated displacement of 30m, which is fairly different from the actual amount of 24m.<br />
<br />
D.)<br />
<br />
[[File:1step.PNG]]<br />
<br />
using 1 time step of 4 seconds results in a simulated displacement of 48m, which is twice the actual amount!<br />
<br />
This demonstrates that higher-resolution simulations with smaller time steps are able to predict the motion of particles more accurately than lower-resolution simulations with larger time steps. It also demonstrates that as the length of each time step approaches 0, the predicted motion of the particles approaches their actual motion. It is important to choose a resolution that keeps error within an acceptable margin.<br />
<br />
==Connectedness==<br />
<br />
===Application: video game industry===<br />
<br />
One popular application of iterative prediction is in physics-based video games. These are often little more than physics simulations with player input. Iterative prediction is used in most major video games to predict the motion of in-game objects. It can be used even if the laws of physics within the video game are different from those of the real world.<br />
<br />
==History==<br />
<br />
The impulse-momentum theorem, which is used in iterative prediction to update the momenta of particles, is derived from Newton's Second Law, which Isaac Newton (1643-1727) publish in his 1687 book Principia Mathematica. From this point onward, all of the math necessary for iterative prediction was known, but it was not useful to perform iterative prediction until significant computational advances in the twentieth century allowed for high resolutions.<br />
<br />
== See also ==<br />
<br />
*[[Linear Momentum]]<br />
*[[Impulse and Momentum]]<br />
*[[Analytical Prediction]]<br />
*[[Iterative Prediction of Spring-Mass System]]<br />
*[[Determinism]]<br />
<br />
===External links===<br />
<br />
*[http://p3server.pa.msu.edu/coursewiki/doku.php?id=183_notes:modeling_with_vpython Iterative Prediction with VPython]<br />
<br />
==References==<br />
<br />
*Georgia Institute of Technology. Physics Department. PHYS 2211. Fall 2016. '''Wednesday, Week 2 Lecture Slides. Fenton, Flavio H'''<br />
*Georgia Institute of Technology. Physics Department. PHYS 2211. Fall 2016. '''Monday, Week 3 Lecture Slides. Fenton, Flavio H'''<br />
*Georgia Institute of Technology. Physics Department. PHYS 2211. Fall 2016. Lab 07 Fancart Energy & Spring-Mass Instructions, Greco, Edwin<br />
*http://p3server.pa.msu.edu/coursewiki/doku.php?id=183_notes:iterativepredict<br />
*http://physicsnet.co.uk/a-level-physics-as-a2/mechanics/newtons-laws-of-motion/<br />
*Bañó-Polo, M., Baeza-Delgado, C., Tamborero, S., Hazel, A., Grau, B., Nilsson, I., … Mingarro, I. (2018). Transmembrane but not soluble helices fold inside the ribosome tunnel. Nature communications, 9(1), 5246. doi:10.1038/s41467-018-07554-7</div>Ilukowiecki3http://www.physicsbook.gatech.edu/index.php?title=Iterative_Prediction&diff=38611Iterative Prediction2020-03-24T18:26:04Z<p>Ilukowiecki3: /* The Main Idea */</p>
<hr />
<div>'''Claimed by ilukowiecki3 Spring 2020'''<br />
<br />
<br />
This page describes iterative prediction, a technique used to predict the motion of particles over a period of time often used in simulations.<br />
<br />
==The Main Idea==<br />
<br />
Iterative prediction is a mathematical technique for approximating the behavior of one or more particles over an interval of time. An iteration is a repeated procedure, so iterative prediction uses repeated procedures to predict the motion of a system. Specifically, 'iterate' means to repeat; in physics, it means to perform the same calculation repeatedly using information produced by the previous calculation. To perform iterative prediction, the initial position, initial momentum, and mass of each particle must be known, and the forces acting on each particle must be known. This information is used to update the position of each particle periodically.<br />
<br />
To apply the momentum principle:<br />
<br />
<br />
<li>1. Chose your system<li><br />
<br />
2. Draw a force diagram (no internal forces, everything depends on interactions with the environment)<br />
<br />
3. Chose your time interval<br />
<br />
4. Substitute known values and solve for unknowns<br />
<br />
5. Check your units and reasonableness of your answer<br />
<br />
<br />
To perform iterative prediction, the time interval of interest must be divided into small sub-intervals of duration <math>\Delta t</math> called time steps. For each time step, the following steps should be performed:<br />
<br />
<ol><br />
<li>The [[Net Force]] acting on each particle should be calculated,</li><br />
<li>The [[Linear Momentum]] of each particle should be updated from the last time step using the newly calculated net force, and</li><br />
<li>The position of each particle should be updated from the last time step using its newly calculated [[Velocity]]. Repeat for the total number of time steps.</li><br />
</ol><br />
<br />
[[File:Iterative_Model.png|center]]<br />
<br />
<center>This is a visual representation of iterative prediction being used to repeatedly update a particle's momentum. It was found on [https://p3server.pa.msu.edu/coursewiki/doku.php?id=183_notes:iterativepredict this page] of the Projects & Practices in Physics website, where it is provided under a [https://creativecommons.org/licenses/by-nc-sa/3.0/ CC Attribution-Noncommercial-Share Alike 3.0 Unported] license. It is shown here under the same license.</center><br />
<br />
<br />
<br />
The size of the time steps used during iterative prediction is referred to as the resolution of the model. Dividing a time interval into a few large time steps is called low resolution while dividing the same time interval into many small time steps is called high resolution. Performing iterative prediction with a higher resolution requires more computations but produces more accurate results. This is because steps 2 and 3 assume constant force and velocity respectively during the duration of each time step (see the "mathematical model" section). In reality, force and velocity change continually, so this is where inaccuracy is introduced. However, if the time steps are small enough that force and velocity do not significantly change during any time step, iterative prediction is a sufficiently accurate model. In the limit where the time interval is divided into an infinite number of infinitely small time steps, iterative prediction becomes a perfectly accurate model. However, this would require an infinite number of computations, and is therefore impossible to do, although in some simple situations, analytic approaches can be used to this end instead. Deciding what time step to use requires consideration of both the computational resources available and the resolution required to accurately approximate the situation. When performing iterative prediction by hand, it is impractical to perform more than a few time steps, which cannot be used to accurately model much. If computers are available, it becomes feasible to divide the time interval into thousands of time steps, which can accurately simulate most day-to-day situations. Some of the most sensitive simulations, however, require incredibly small time steps. For example, some physicists perform atom-by-atom simulations of molecular structures interacting. A slight change in the position of an atom can cause a huge change in the forces acting on it, so tiny time steps must be used in order to ensure that force and velocity do not significantly change during any one time step. These simulations are run on the world's most powerful supercomputers and often take weeks to complete. They can simulate time intervals of several nanoseconds (<math>10^{-9}</math> s) using time steps of only a few femtoseconds. (<math>10^{-15}</math> s) [https://www.ncbi.nlm.nih.gov/pmc/articles/PMC6286305/ (Click here for an example of a paper about such a simulation.)] Most of the VPython simulations on this wiki use time steps of about .01s.<br />
<br />
Iterative prediction is only an approximation, but it is a very powerful tool because it can model complex systems that are impossible to model using known analytical approaches such as calculus and kinematics due to unsolvable differential equations and other obstacles.<br />
<br />
===Relation between Newton's 2nd Law and Momentum Principle===<br />
<br />
The rate of change of momentum of an object is directly proportional to the net force applied and is in the direction of the net force; the net force is equal to the rate of change of momentum. <br />
<br />
[[File:Fma_.jpg]]<br />
<br />
- F = net force in newtons (N)<br />
<br />
- m = mass in kilograms (kg)<br />
<br />
- a = acceleration in meters per second per second (ms-2)<br />
<br />
[[File:rate-of-change-of-momentum-2.jpg]]<br />
<br />
- F = net force in newtons (N)<br />
<br />
- <math>\Delta mv</math> = change in momentum in kilogram metres per second (kg ms-1)<br />
<br />
- t = change in time in seconds (s)<br />
<br />
===A Mathematical Model===<br />
<br />
Below are the mathematical formulas required for each of the three steps outlined above.<br />
<br />
<ol><br />
<br />
<li>The net force acting on each particle should be calculated.</li><br />
<br />
This step varies widely from program to program, depending on what system is being modeled. Sometimes, the forces acting on each particle are constant, such as when modeling the motion of a ball with a specific weight thrown near the surface of the earth. Other times, the forces are functions of properties such as the particles' positions, such as when modeling the motion of celestial bodies whose gravitational attraction is greater when they are closer together. Sometimes, there is only one force acting on each particle, and other times, the net force must be found by adding together the individual force vectors.<br />
<br />
<li>The momentum of each particle should be updated from the last time step using the newly calculated net force.</li><br />
<br />
This step uses the [[Impulse and Momentum|impulse-momentum theorem]]. The impulse-momentum theorem states that <math>\vec{p}_f = \vec{p}_i + \vec{J} = \vec{p}_i + \vec{F}_{net, avg} * \Delta t</math>. In iterative prediction, the initial momentum is always known; it is simply the momentum the particle had during the previous time step (or, if there is no previous time step, the initial momentum of the particle). <math>\vec{F}_{net, avg}</math> represents the average net force that has been acting on the particle since the previous time step, resulting in a new momentum this time step. To calculate it, the net force found in step 1 should be used. Note that in reality, the net force acting on the particle is constantly changing, and the net force found in step 1 is the final net force, which is not truly the average net force over the course of the previous time step (except in a scenario with constant force). As described in the introductory section, this introduces inaccuracy that can be reduced by using shorter time steps; if short time steps are used, the net force does not have much time to change over the course of the time step, meaning the average net force is very similar to the final net force.<br />
<br />
<li>The position of each particle should be updated from the last time step using its newly calculated velocity.</li><br />
<br />
This step uses the kinematic equation <math>\vec{r}_f = \vec{r}_i + \vec{v}_{avg} * \Delta t</math>. In iterative prediction, the initial position is always known; it is simply the position the particle had during the previous time step (or, if there is no previous time step, the initial position of the particle). <math>\vec{v}_{avg}</math> represents the average velocity with which the particle has been moving since the previous time step, resulting in a new position this time step. To calculate it, the momentum found in step 2 should simply be divided by the mass of the particle: <math>\vec{v}_f = \frac{\vec{p}_f}{m}</math>. Note that in reality, the velocity of the particle is constantly changing, and the velocity found above is the final velocity, which is not truly the average velocity over the course of the previous time step (even in a scenario with constant force). As described in the introductory section, this introduces inaccuracy that can be reduced by using shorter time steps; if short time steps are used, the velocity does not have much time to change over the course of the time step, meaning the average velocity is very similar to the final velocity.<br />
<br />
</ol><br />
<br />
===A Computational Model===<br />
<br />
Iterative prediction is typically done on computers because many computations are necessary in order to perform it at a meaningful resolution. VPython is a useful program to perform iterative prediction because of it can graphically display the positions of the particles each time step. Below is an example of a VPython program that uses iterative prediction to simulate a ball thrown near the surface of the earth. Be sure to read and understand the source code by clicking on "view this program" in the top left corner; there are comments for each line designed to introduce readers to their first iterative prediction program.<br />
<br />
[https://www.glowscript.org/#/user/YorickAndeweg/folder/PhysicsBookFolder/program/IterativePrediction1 high-resolution ball trajectory simulation]<br />
<br />
To demonstrate differing resolutions, here is a lower resolution version of the same simulation, which uses larger time steps:<br />
<br />
[https://www.glowscript.org/#/user/YorickAndeweg/folder/PhysicsBookFolder/program/IterativePrediction2 low-resolution ball trajectory simulation]<br />
<br />
Note that the first program is more realistic; the more frequent position updates result in a smoother curve that more closely approximates the true trajectory of a ball. On the other hand, the second program has fewer calculations to do. Both programs are artificially slowed using the "rate" command in order to simulate a realistic travel time, but if both programs were allowed to run as quickly as possible, the second program would finish first. For larger, more complicated simulations, time can be a significant factor.<br />
<br />
Here is another more complicated simulation that uses iterative prediction to simulate the motion of a mass swinging on a spring. This is an example of iterative prediction with a varying force; during each time step, the forces acting on the mass must be calculated depending on its position using [[Hooke's Law]].<br />
<br />
[https://www.glowscript.org/#/user/YorickAndeweg/folder/PhysicsBookFolder/program/SpringMass spring mass simulation]<br />
<br />
Note: both projectile motion and spring-mass systems can be analytically modeled; that is, in both of the above systems, analytical techniques such as calculus and kinematics can be used to find the position of the particles as a function of time. However, for some other, more complicated systems, analytical approaches may not exist, while iterative prediction often still works. For example, it would be easy to modify the projectile motion simulation to include the effects of [[Air Resistance]], which is difficult to take into account analytically.<br />
<br />
==Examples==<br />
<br />
In addition to the example below, know how to create vPython simulations using iterative prediction with both constant and varying forces.<br />
<br />
===1. (Middling)===<br />
<br />
A 2kg particle is released from rest at time t=0. A constant force of 6N is applied to it. How far has the particle traveled after 4 seconds?<br />
<br />
Solve this question<br />
<br />
A.) analytically (using kinematic equations)<br />
<br />
B.) using iterative prediction with 12 equal time steps<br />
<br />
C.) using iterative prediction with 4 equal time steps<br />
<br />
D.) using iterative prediction with a single time step<br />
<br />
Pay attention to how the resolution of the iterative prediction affects the accuracy of the answer. The analytical answer is completely accurate.<br />
<br />
<br />
A.) To solve this analytically, the following kinematic equation should be used:<br />
<br />
<math>\Delta x = \frac{1}{2} a t^2 + v_0 t</math><br />
<br />
Using [[Newton's Second Law: the Momentum Principle]], we know that <math>a = \frac{f}{m} = 3</math>m/s^2. We are also given that <math>v_0 = 0</math>, and we are interested in the time t=4. Substituting these values into the kinematic equation yields<br />
<br />
<math>\Delta x = \frac{1}{2} (3) (4)^2 = 24</math>m.<br />
<br />
B.) To solve this using iterative prediction, a computational tool should be used. Since an animation isn't necessary for this problem, I used Microsoft Excel; each row populates itself based on the information in the previous row using formula functions. Google Spreadsheets can also be used, as can vPython or any other programming language.<br />
<br />
[[File:12steps.PNG]]<br />
<br />
Using 12 time steps of 1/3 seconds each results in a simulated displacement of 26m, which is fairly close to the actual amount of 24m.<br />
<br />
C.)<br />
<br />
[[File:4steps.PNG]]<br />
<br />
using 4 time steps of 1 second each results in a simulated displacement of 30m, which is fairly different from the actual amount of 24m.<br />
<br />
D.)<br />
<br />
[[File:1step.PNG]]<br />
<br />
using 1 time step of 4 seconds results in a simulated displacement of 48m, which is twice the actual amount!<br />
<br />
This demonstrates that higher-resolution simulations with smaller time steps are able to predict the motion of particles more accurately than lower-resolution simulations with larger time steps. It also demonstrates that as the length of each time step approaches 0, the predicted motion of the particles approaches their actual motion. It is important to choose a resolution that keeps error within an acceptable margin.<br />
<br />
==Connectedness==<br />
<br />
===Application: video game industry===<br />
<br />
One popular application of iterative prediction is in physics-based video games. These are often little more than physics simulations with player input. Iterative prediction is used in most major video games to predict the motion of in-game objects. It can be used even if the laws of physics within the video game are different from those of the real world.<br />
<br />
==History==<br />
<br />
The impulse-momentum theorem, which is used in iterative prediction to update the momenta of particles, is derived from Newton's Second Law, which Isaac Newton (1643-1727) publish in his 1687 book Principia Mathematica. From this point onward, all of the math necessary for iterative prediction was known, but it was not useful to perform iterative prediction until significant computational advances in the twentieth century allowed for high resolutions.<br />
<br />
== See also ==<br />
<br />
*[[Linear Momentum]]<br />
*[[Impulse and Momentum]]<br />
*[[Analytical Prediction]]<br />
*[[Iterative Prediction of Spring-Mass System]]<br />
*[[Determinism]]<br />
<br />
===External links===<br />
<br />
*[http://p3server.pa.msu.edu/coursewiki/doku.php?id=183_notes:modeling_with_vpython Iterative Prediction with VPython]<br />
<br />
==References==<br />
<br />
*Georgia Institute of Technology. Physics Department. PHYS 2211. Fall 2016. '''Wednesday, Week 2 Lecture Slides. Fenton, Flavio H'''<br />
*Georgia Institute of Technology. Physics Department. PHYS 2211. Fall 2016. '''Monday, Week 3 Lecture Slides. Fenton, Flavio H'''<br />
*Georgia Institute of Technology. Physics Department. PHYS 2211. Fall 2016. Lab 07 Fancart Energy & Spring-Mass Instructions, Greco, Edwin<br />
*http://p3server.pa.msu.edu/coursewiki/doku.php?id=183_notes:iterativepredict<br />
*http://physicsnet.co.uk/a-level-physics-as-a2/mechanics/newtons-laws-of-motion/<br />
*Bañó-Polo, M., Baeza-Delgado, C., Tamborero, S., Hazel, A., Grau, B., Nilsson, I., … Mingarro, I. (2018). Transmembrane but not soluble helices fold inside the ribosome tunnel. Nature communications, 9(1), 5246. doi:10.1038/s41467-018-07554-7</div>Ilukowiecki3http://www.physicsbook.gatech.edu/index.php?title=Iterative_Prediction&diff=38610Iterative Prediction2020-03-24T18:25:28Z<p>Ilukowiecki3: /* The Main Idea */</p>
<hr />
<div>'''Claimed by ilukowiecki3 Spring 2020'''<br />
<br />
<br />
This page describes iterative prediction, a technique used to predict the motion of particles over a period of time often used in simulations.<br />
<br />
==The Main Idea==<br />
<br />
Iterative prediction is a mathematical technique for approximating the behavior of one or more particles over an interval of time. An iteration is a repeated procedure, so iterative prediction uses repeated procedures to predict the motion of a system. Specifically, 'iterate' means to repeat; in physics, it means to perform the same calculation repeatedly using information produced by the previous calculation. To perform iterative prediction, the initial position, initial momentum, and mass of each particle must be known, and the forces acting on each particle must be known. This information is used to update the position of each particle periodically.<br />
<br />
To apply the momentum principle:<br />
<br />
<br />
1. Chose your system<br />
<br />
2. Draw a force diagram (no internal forces, everything depends on interactions with the environment)<br />
<br />
3. Chose your time interval<br />
<br />
4. Substitute known values and solve for unknowns<br />
<br />
5. Check your units and reasonableness of your answer<br />
<br />
<br />
To perform iterative prediction, the time interval of interest must be divided into small sub-intervals of duration <math>\Delta t</math> called time steps. For each time step, the following steps should be performed:<br />
<br />
<ol><br />
<li>The [[Net Force]] acting on each particle should be calculated,</li><br />
<li>The [[Linear Momentum]] of each particle should be updated from the last time step using the newly calculated net force, and</li><br />
<li>The position of each particle should be updated from the last time step using its newly calculated [[Velocity]]. Repeat for the total number of time steps.</li><br />
</ol><br />
<br />
[[File:Iterative_Model.png|center]]<br />
<br />
<center>This is a visual representation of iterative prediction being used to repeatedly update a particle's momentum. It was found on [https://p3server.pa.msu.edu/coursewiki/doku.php?id=183_notes:iterativepredict this page] of the Projects & Practices in Physics website, where it is provided under a [https://creativecommons.org/licenses/by-nc-sa/3.0/ CC Attribution-Noncommercial-Share Alike 3.0 Unported] license. It is shown here under the same license.</center><br />
<br />
<br />
<br />
The size of the time steps used during iterative prediction is referred to as the resolution of the model. Dividing a time interval into a few large time steps is called low resolution while dividing the same time interval into many small time steps is called high resolution. Performing iterative prediction with a higher resolution requires more computations but produces more accurate results. This is because steps 2 and 3 assume constant force and velocity respectively during the duration of each time step (see the "mathematical model" section). In reality, force and velocity change continually, so this is where inaccuracy is introduced. However, if the time steps are small enough that force and velocity do not significantly change during any time step, iterative prediction is a sufficiently accurate model. In the limit where the time interval is divided into an infinite number of infinitely small time steps, iterative prediction becomes a perfectly accurate model. However, this would require an infinite number of computations, and is therefore impossible to do, although in some simple situations, analytic approaches can be used to this end instead. Deciding what time step to use requires consideration of both the computational resources available and the resolution required to accurately approximate the situation. When performing iterative prediction by hand, it is impractical to perform more than a few time steps, which cannot be used to accurately model much. If computers are available, it becomes feasible to divide the time interval into thousands of time steps, which can accurately simulate most day-to-day situations. Some of the most sensitive simulations, however, require incredibly small time steps. For example, some physicists perform atom-by-atom simulations of molecular structures interacting. A slight change in the position of an atom can cause a huge change in the forces acting on it, so tiny time steps must be used in order to ensure that force and velocity do not significantly change during any one time step. These simulations are run on the world's most powerful supercomputers and often take weeks to complete. They can simulate time intervals of several nanoseconds (<math>10^{-9}</math> s) using time steps of only a few femtoseconds. (<math>10^{-15}</math> s) [https://www.ncbi.nlm.nih.gov/pmc/articles/PMC6286305/ (Click here for an example of a paper about such a simulation.)] Most of the VPython simulations on this wiki use time steps of about .01s.<br />
<br />
Iterative prediction is only an approximation, but it is a very powerful tool because it can model complex systems that are impossible to model using known analytical approaches such as calculus and kinematics due to unsolvable differential equations and other obstacles.<br />
<br />
===Relation between Newton's 2nd Law and Momentum Principle===<br />
<br />
The rate of change of momentum of an object is directly proportional to the net force applied and is in the direction of the net force; the net force is equal to the rate of change of momentum. <br />
<br />
[[File:Fma_.jpg]]<br />
<br />
- F = net force in newtons (N)<br />
<br />
- m = mass in kilograms (kg)<br />
<br />
- a = acceleration in meters per second per second (ms-2)<br />
<br />
[[File:rate-of-change-of-momentum-2.jpg]]<br />
<br />
- F = net force in newtons (N)<br />
<br />
- <math>\Delta mv</math> = change in momentum in kilogram metres per second (kg ms-1)<br />
<br />
- t = change in time in seconds (s)<br />
<br />
===A Mathematical Model===<br />
<br />
Below are the mathematical formulas required for each of the three steps outlined above.<br />
<br />
<ol><br />
<br />
<li>The net force acting on each particle should be calculated.</li><br />
<br />
This step varies widely from program to program, depending on what system is being modeled. Sometimes, the forces acting on each particle are constant, such as when modeling the motion of a ball with a specific weight thrown near the surface of the earth. Other times, the forces are functions of properties such as the particles' positions, such as when modeling the motion of celestial bodies whose gravitational attraction is greater when they are closer together. Sometimes, there is only one force acting on each particle, and other times, the net force must be found by adding together the individual force vectors.<br />
<br />
<li>The momentum of each particle should be updated from the last time step using the newly calculated net force.</li><br />
<br />
This step uses the [[Impulse and Momentum|impulse-momentum theorem]]. The impulse-momentum theorem states that <math>\vec{p}_f = \vec{p}_i + \vec{J} = \vec{p}_i + \vec{F}_{net, avg} * \Delta t</math>. In iterative prediction, the initial momentum is always known; it is simply the momentum the particle had during the previous time step (or, if there is no previous time step, the initial momentum of the particle). <math>\vec{F}_{net, avg}</math> represents the average net force that has been acting on the particle since the previous time step, resulting in a new momentum this time step. To calculate it, the net force found in step 1 should be used. Note that in reality, the net force acting on the particle is constantly changing, and the net force found in step 1 is the final net force, which is not truly the average net force over the course of the previous time step (except in a scenario with constant force). As described in the introductory section, this introduces inaccuracy that can be reduced by using shorter time steps; if short time steps are used, the net force does not have much time to change over the course of the time step, meaning the average net force is very similar to the final net force.<br />
<br />
<li>The position of each particle should be updated from the last time step using its newly calculated velocity.</li><br />
<br />
This step uses the kinematic equation <math>\vec{r}_f = \vec{r}_i + \vec{v}_{avg} * \Delta t</math>. In iterative prediction, the initial position is always known; it is simply the position the particle had during the previous time step (or, if there is no previous time step, the initial position of the particle). <math>\vec{v}_{avg}</math> represents the average velocity with which the particle has been moving since the previous time step, resulting in a new position this time step. To calculate it, the momentum found in step 2 should simply be divided by the mass of the particle: <math>\vec{v}_f = \frac{\vec{p}_f}{m}</math>. Note that in reality, the velocity of the particle is constantly changing, and the velocity found above is the final velocity, which is not truly the average velocity over the course of the previous time step (even in a scenario with constant force). As described in the introductory section, this introduces inaccuracy that can be reduced by using shorter time steps; if short time steps are used, the velocity does not have much time to change over the course of the time step, meaning the average velocity is very similar to the final velocity.<br />
<br />
</ol><br />
<br />
===A Computational Model===<br />
<br />
Iterative prediction is typically done on computers because many computations are necessary in order to perform it at a meaningful resolution. VPython is a useful program to perform iterative prediction because of it can graphically display the positions of the particles each time step. Below is an example of a VPython program that uses iterative prediction to simulate a ball thrown near the surface of the earth. Be sure to read and understand the source code by clicking on "view this program" in the top left corner; there are comments for each line designed to introduce readers to their first iterative prediction program.<br />
<br />
[https://www.glowscript.org/#/user/YorickAndeweg/folder/PhysicsBookFolder/program/IterativePrediction1 high-resolution ball trajectory simulation]<br />
<br />
To demonstrate differing resolutions, here is a lower resolution version of the same simulation, which uses larger time steps:<br />
<br />
[https://www.glowscript.org/#/user/YorickAndeweg/folder/PhysicsBookFolder/program/IterativePrediction2 low-resolution ball trajectory simulation]<br />
<br />
Note that the first program is more realistic; the more frequent position updates result in a smoother curve that more closely approximates the true trajectory of a ball. On the other hand, the second program has fewer calculations to do. Both programs are artificially slowed using the "rate" command in order to simulate a realistic travel time, but if both programs were allowed to run as quickly as possible, the second program would finish first. For larger, more complicated simulations, time can be a significant factor.<br />
<br />
Here is another more complicated simulation that uses iterative prediction to simulate the motion of a mass swinging on a spring. This is an example of iterative prediction with a varying force; during each time step, the forces acting on the mass must be calculated depending on its position using [[Hooke's Law]].<br />
<br />
[https://www.glowscript.org/#/user/YorickAndeweg/folder/PhysicsBookFolder/program/SpringMass spring mass simulation]<br />
<br />
Note: both projectile motion and spring-mass systems can be analytically modeled; that is, in both of the above systems, analytical techniques such as calculus and kinematics can be used to find the position of the particles as a function of time. However, for some other, more complicated systems, analytical approaches may not exist, while iterative prediction often still works. For example, it would be easy to modify the projectile motion simulation to include the effects of [[Air Resistance]], which is difficult to take into account analytically.<br />
<br />
==Examples==<br />
<br />
In addition to the example below, know how to create vPython simulations using iterative prediction with both constant and varying forces.<br />
<br />
===1. (Middling)===<br />
<br />
A 2kg particle is released from rest at time t=0. A constant force of 6N is applied to it. How far has the particle traveled after 4 seconds?<br />
<br />
Solve this question<br />
<br />
A.) analytically (using kinematic equations)<br />
<br />
B.) using iterative prediction with 12 equal time steps<br />
<br />
C.) using iterative prediction with 4 equal time steps<br />
<br />
D.) using iterative prediction with a single time step<br />
<br />
Pay attention to how the resolution of the iterative prediction affects the accuracy of the answer. The analytical answer is completely accurate.<br />
<br />
<br />
A.) To solve this analytically, the following kinematic equation should be used:<br />
<br />
<math>\Delta x = \frac{1}{2} a t^2 + v_0 t</math><br />
<br />
Using [[Newton's Second Law: the Momentum Principle]], we know that <math>a = \frac{f}{m} = 3</math>m/s^2. We are also given that <math>v_0 = 0</math>, and we are interested in the time t=4. Substituting these values into the kinematic equation yields<br />
<br />
<math>\Delta x = \frac{1}{2} (3) (4)^2 = 24</math>m.<br />
<br />
B.) To solve this using iterative prediction, a computational tool should be used. Since an animation isn't necessary for this problem, I used Microsoft Excel; each row populates itself based on the information in the previous row using formula functions. Google Spreadsheets can also be used, as can vPython or any other programming language.<br />
<br />
[[File:12steps.PNG]]<br />
<br />
Using 12 time steps of 1/3 seconds each results in a simulated displacement of 26m, which is fairly close to the actual amount of 24m.<br />
<br />
C.)<br />
<br />
[[File:4steps.PNG]]<br />
<br />
using 4 time steps of 1 second each results in a simulated displacement of 30m, which is fairly different from the actual amount of 24m.<br />
<br />
D.)<br />
<br />
[[File:1step.PNG]]<br />
<br />
using 1 time step of 4 seconds results in a simulated displacement of 48m, which is twice the actual amount!<br />
<br />
This demonstrates that higher-resolution simulations with smaller time steps are able to predict the motion of particles more accurately than lower-resolution simulations with larger time steps. It also demonstrates that as the length of each time step approaches 0, the predicted motion of the particles approaches their actual motion. It is important to choose a resolution that keeps error within an acceptable margin.<br />
<br />
==Connectedness==<br />
<br />
===Application: video game industry===<br />
<br />
One popular application of iterative prediction is in physics-based video games. These are often little more than physics simulations with player input. Iterative prediction is used in most major video games to predict the motion of in-game objects. It can be used even if the laws of physics within the video game are different from those of the real world.<br />
<br />
==History==<br />
<br />
The impulse-momentum theorem, which is used in iterative prediction to update the momenta of particles, is derived from Newton's Second Law, which Isaac Newton (1643-1727) publish in his 1687 book Principia Mathematica. From this point onward, all of the math necessary for iterative prediction was known, but it was not useful to perform iterative prediction until significant computational advances in the twentieth century allowed for high resolutions.<br />
<br />
== See also ==<br />
<br />
*[[Linear Momentum]]<br />
*[[Impulse and Momentum]]<br />
*[[Analytical Prediction]]<br />
*[[Iterative Prediction of Spring-Mass System]]<br />
*[[Determinism]]<br />
<br />
===External links===<br />
<br />
*[http://p3server.pa.msu.edu/coursewiki/doku.php?id=183_notes:modeling_with_vpython Iterative Prediction with VPython]<br />
<br />
==References==<br />
<br />
*Georgia Institute of Technology. Physics Department. PHYS 2211. Fall 2016. '''Wednesday, Week 2 Lecture Slides. Fenton, Flavio H'''<br />
*Georgia Institute of Technology. Physics Department. PHYS 2211. Fall 2016. '''Monday, Week 3 Lecture Slides. Fenton, Flavio H'''<br />
*Georgia Institute of Technology. Physics Department. PHYS 2211. Fall 2016. Lab 07 Fancart Energy & Spring-Mass Instructions, Greco, Edwin<br />
*http://p3server.pa.msu.edu/coursewiki/doku.php?id=183_notes:iterativepredict<br />
*http://physicsnet.co.uk/a-level-physics-as-a2/mechanics/newtons-laws-of-motion/<br />
*Bañó-Polo, M., Baeza-Delgado, C., Tamborero, S., Hazel, A., Grau, B., Nilsson, I., … Mingarro, I. (2018). Transmembrane but not soluble helices fold inside the ribosome tunnel. Nature communications, 9(1), 5246. doi:10.1038/s41467-018-07554-7</div>Ilukowiecki3http://www.physicsbook.gatech.edu/index.php?title=Iterative_Prediction&diff=38609Iterative Prediction2020-03-24T18:25:13Z<p>Ilukowiecki3: /* The Main Idea */</p>
<hr />
<div>'''Claimed by ilukowiecki3 Spring 2020'''<br />
<br />
<br />
This page describes iterative prediction, a technique used to predict the motion of particles over a period of time often used in simulations.<br />
<br />
==The Main Idea==<br />
<br />
Iterative prediction is a mathematical technique for approximating the behavior of one or more particles over an interval of time. An iteration is a repeated procedure, so iterative prediction uses repeated procedures to predict the motion of a system. Specifically, 'iterate' means to repeat; in physics, it means to perform the same calculation repeatedly using information produced by the previous calculation. To perform iterative prediction, the initial position, initial momentum, and mass of each particle must be known, and the forces acting on each particle must be known. This information is used to update the position of each particle periodically.<br />
<br />
To apply the momentum principle:<br />
<br />
1. Chose your system<br />
<br />
2. Draw a force diagram (no internal forces, everything depends on interactions with the environment)<br />
<br />
3. Chose your time interval<br />
<br />
4. Substitute known values and solve for unknowns<br />
<br />
5. Check your units and reasonableness of your answer<br />
<br />
<br />
To perform iterative prediction, the time interval of interest must be divided into small sub-intervals of duration <math>\Delta t</math> called time steps. For each time step, the following steps should be performed:<br />
<br />
<ol><br />
<li>The [[Net Force]] acting on each particle should be calculated,</li><br />
<li>The [[Linear Momentum]] of each particle should be updated from the last time step using the newly calculated net force, and</li><br />
<li>The position of each particle should be updated from the last time step using its newly calculated [[Velocity]]. Repeat for the total number of time steps.</li><br />
</ol><br />
<br />
[[File:Iterative_Model.png|center]]<br />
<br />
<center>This is a visual representation of iterative prediction being used to repeatedly update a particle's momentum. It was found on [https://p3server.pa.msu.edu/coursewiki/doku.php?id=183_notes:iterativepredict this page] of the Projects & Practices in Physics website, where it is provided under a [https://creativecommons.org/licenses/by-nc-sa/3.0/ CC Attribution-Noncommercial-Share Alike 3.0 Unported] license. It is shown here under the same license.</center><br />
<br />
<br />
<br />
The size of the time steps used during iterative prediction is referred to as the resolution of the model. Dividing a time interval into a few large time steps is called low resolution while dividing the same time interval into many small time steps is called high resolution. Performing iterative prediction with a higher resolution requires more computations but produces more accurate results. This is because steps 2 and 3 assume constant force and velocity respectively during the duration of each time step (see the "mathematical model" section). In reality, force and velocity change continually, so this is where inaccuracy is introduced. However, if the time steps are small enough that force and velocity do not significantly change during any time step, iterative prediction is a sufficiently accurate model. In the limit where the time interval is divided into an infinite number of infinitely small time steps, iterative prediction becomes a perfectly accurate model. However, this would require an infinite number of computations, and is therefore impossible to do, although in some simple situations, analytic approaches can be used to this end instead. Deciding what time step to use requires consideration of both the computational resources available and the resolution required to accurately approximate the situation. When performing iterative prediction by hand, it is impractical to perform more than a few time steps, which cannot be used to accurately model much. If computers are available, it becomes feasible to divide the time interval into thousands of time steps, which can accurately simulate most day-to-day situations. Some of the most sensitive simulations, however, require incredibly small time steps. For example, some physicists perform atom-by-atom simulations of molecular structures interacting. A slight change in the position of an atom can cause a huge change in the forces acting on it, so tiny time steps must be used in order to ensure that force and velocity do not significantly change during any one time step. These simulations are run on the world's most powerful supercomputers and often take weeks to complete. They can simulate time intervals of several nanoseconds (<math>10^{-9}</math> s) using time steps of only a few femtoseconds. (<math>10^{-15}</math> s) [https://www.ncbi.nlm.nih.gov/pmc/articles/PMC6286305/ (Click here for an example of a paper about such a simulation.)] Most of the VPython simulations on this wiki use time steps of about .01s.<br />
<br />
Iterative prediction is only an approximation, but it is a very powerful tool because it can model complex systems that are impossible to model using known analytical approaches such as calculus and kinematics due to unsolvable differential equations and other obstacles.<br />
<br />
===Relation between Newton's 2nd Law and Momentum Principle===<br />
<br />
The rate of change of momentum of an object is directly proportional to the net force applied and is in the direction of the net force; the net force is equal to the rate of change of momentum. <br />
<br />
[[File:Fma_.jpg]]<br />
<br />
- F = net force in newtons (N)<br />
<br />
- m = mass in kilograms (kg)<br />
<br />
- a = acceleration in meters per second per second (ms-2)<br />
<br />
[[File:rate-of-change-of-momentum-2.jpg]]<br />
<br />
- F = net force in newtons (N)<br />
<br />
- <math>\Delta mv</math> = change in momentum in kilogram metres per second (kg ms-1)<br />
<br />
- t = change in time in seconds (s)<br />
<br />
===A Mathematical Model===<br />
<br />
Below are the mathematical formulas required for each of the three steps outlined above.<br />
<br />
<ol><br />
<br />
<li>The net force acting on each particle should be calculated.</li><br />
<br />
This step varies widely from program to program, depending on what system is being modeled. Sometimes, the forces acting on each particle are constant, such as when modeling the motion of a ball with a specific weight thrown near the surface of the earth. Other times, the forces are functions of properties such as the particles' positions, such as when modeling the motion of celestial bodies whose gravitational attraction is greater when they are closer together. Sometimes, there is only one force acting on each particle, and other times, the net force must be found by adding together the individual force vectors.<br />
<br />
<li>The momentum of each particle should be updated from the last time step using the newly calculated net force.</li><br />
<br />
This step uses the [[Impulse and Momentum|impulse-momentum theorem]]. The impulse-momentum theorem states that <math>\vec{p}_f = \vec{p}_i + \vec{J} = \vec{p}_i + \vec{F}_{net, avg} * \Delta t</math>. In iterative prediction, the initial momentum is always known; it is simply the momentum the particle had during the previous time step (or, if there is no previous time step, the initial momentum of the particle). <math>\vec{F}_{net, avg}</math> represents the average net force that has been acting on the particle since the previous time step, resulting in a new momentum this time step. To calculate it, the net force found in step 1 should be used. Note that in reality, the net force acting on the particle is constantly changing, and the net force found in step 1 is the final net force, which is not truly the average net force over the course of the previous time step (except in a scenario with constant force). As described in the introductory section, this introduces inaccuracy that can be reduced by using shorter time steps; if short time steps are used, the net force does not have much time to change over the course of the time step, meaning the average net force is very similar to the final net force.<br />
<br />
<li>The position of each particle should be updated from the last time step using its newly calculated velocity.</li><br />
<br />
This step uses the kinematic equation <math>\vec{r}_f = \vec{r}_i + \vec{v}_{avg} * \Delta t</math>. In iterative prediction, the initial position is always known; it is simply the position the particle had during the previous time step (or, if there is no previous time step, the initial position of the particle). <math>\vec{v}_{avg}</math> represents the average velocity with which the particle has been moving since the previous time step, resulting in a new position this time step. To calculate it, the momentum found in step 2 should simply be divided by the mass of the particle: <math>\vec{v}_f = \frac{\vec{p}_f}{m}</math>. Note that in reality, the velocity of the particle is constantly changing, and the velocity found above is the final velocity, which is not truly the average velocity over the course of the previous time step (even in a scenario with constant force). As described in the introductory section, this introduces inaccuracy that can be reduced by using shorter time steps; if short time steps are used, the velocity does not have much time to change over the course of the time step, meaning the average velocity is very similar to the final velocity.<br />
<br />
</ol><br />
<br />
===A Computational Model===<br />
<br />
Iterative prediction is typically done on computers because many computations are necessary in order to perform it at a meaningful resolution. VPython is a useful program to perform iterative prediction because of it can graphically display the positions of the particles each time step. Below is an example of a VPython program that uses iterative prediction to simulate a ball thrown near the surface of the earth. Be sure to read and understand the source code by clicking on "view this program" in the top left corner; there are comments for each line designed to introduce readers to their first iterative prediction program.<br />
<br />
[https://www.glowscript.org/#/user/YorickAndeweg/folder/PhysicsBookFolder/program/IterativePrediction1 high-resolution ball trajectory simulation]<br />
<br />
To demonstrate differing resolutions, here is a lower resolution version of the same simulation, which uses larger time steps:<br />
<br />
[https://www.glowscript.org/#/user/YorickAndeweg/folder/PhysicsBookFolder/program/IterativePrediction2 low-resolution ball trajectory simulation]<br />
<br />
Note that the first program is more realistic; the more frequent position updates result in a smoother curve that more closely approximates the true trajectory of a ball. On the other hand, the second program has fewer calculations to do. Both programs are artificially slowed using the "rate" command in order to simulate a realistic travel time, but if both programs were allowed to run as quickly as possible, the second program would finish first. For larger, more complicated simulations, time can be a significant factor.<br />
<br />
Here is another more complicated simulation that uses iterative prediction to simulate the motion of a mass swinging on a spring. This is an example of iterative prediction with a varying force; during each time step, the forces acting on the mass must be calculated depending on its position using [[Hooke's Law]].<br />
<br />
[https://www.glowscript.org/#/user/YorickAndeweg/folder/PhysicsBookFolder/program/SpringMass spring mass simulation]<br />
<br />
Note: both projectile motion and spring-mass systems can be analytically modeled; that is, in both of the above systems, analytical techniques such as calculus and kinematics can be used to find the position of the particles as a function of time. However, for some other, more complicated systems, analytical approaches may not exist, while iterative prediction often still works. For example, it would be easy to modify the projectile motion simulation to include the effects of [[Air Resistance]], which is difficult to take into account analytically.<br />
<br />
==Examples==<br />
<br />
In addition to the example below, know how to create vPython simulations using iterative prediction with both constant and varying forces.<br />
<br />
===1. (Middling)===<br />
<br />
A 2kg particle is released from rest at time t=0. A constant force of 6N is applied to it. How far has the particle traveled after 4 seconds?<br />
<br />
Solve this question<br />
<br />
A.) analytically (using kinematic equations)<br />
<br />
B.) using iterative prediction with 12 equal time steps<br />
<br />
C.) using iterative prediction with 4 equal time steps<br />
<br />
D.) using iterative prediction with a single time step<br />
<br />
Pay attention to how the resolution of the iterative prediction affects the accuracy of the answer. The analytical answer is completely accurate.<br />
<br />
<br />
A.) To solve this analytically, the following kinematic equation should be used:<br />
<br />
<math>\Delta x = \frac{1}{2} a t^2 + v_0 t</math><br />
<br />
Using [[Newton's Second Law: the Momentum Principle]], we know that <math>a = \frac{f}{m} = 3</math>m/s^2. We are also given that <math>v_0 = 0</math>, and we are interested in the time t=4. Substituting these values into the kinematic equation yields<br />
<br />
<math>\Delta x = \frac{1}{2} (3) (4)^2 = 24</math>m.<br />
<br />
B.) To solve this using iterative prediction, a computational tool should be used. Since an animation isn't necessary for this problem, I used Microsoft Excel; each row populates itself based on the information in the previous row using formula functions. Google Spreadsheets can also be used, as can vPython or any other programming language.<br />
<br />
[[File:12steps.PNG]]<br />
<br />
Using 12 time steps of 1/3 seconds each results in a simulated displacement of 26m, which is fairly close to the actual amount of 24m.<br />
<br />
C.)<br />
<br />
[[File:4steps.PNG]]<br />
<br />
using 4 time steps of 1 second each results in a simulated displacement of 30m, which is fairly different from the actual amount of 24m.<br />
<br />
D.)<br />
<br />
[[File:1step.PNG]]<br />
<br />
using 1 time step of 4 seconds results in a simulated displacement of 48m, which is twice the actual amount!<br />
<br />
This demonstrates that higher-resolution simulations with smaller time steps are able to predict the motion of particles more accurately than lower-resolution simulations with larger time steps. It also demonstrates that as the length of each time step approaches 0, the predicted motion of the particles approaches their actual motion. It is important to choose a resolution that keeps error within an acceptable margin.<br />
<br />
==Connectedness==<br />
<br />
===Application: video game industry===<br />
<br />
One popular application of iterative prediction is in physics-based video games. These are often little more than physics simulations with player input. Iterative prediction is used in most major video games to predict the motion of in-game objects. It can be used even if the laws of physics within the video game are different from those of the real world.<br />
<br />
==History==<br />
<br />
The impulse-momentum theorem, which is used in iterative prediction to update the momenta of particles, is derived from Newton's Second Law, which Isaac Newton (1643-1727) publish in his 1687 book Principia Mathematica. From this point onward, all of the math necessary for iterative prediction was known, but it was not useful to perform iterative prediction until significant computational advances in the twentieth century allowed for high resolutions.<br />
<br />
== See also ==<br />
<br />
*[[Linear Momentum]]<br />
*[[Impulse and Momentum]]<br />
*[[Analytical Prediction]]<br />
*[[Iterative Prediction of Spring-Mass System]]<br />
*[[Determinism]]<br />
<br />
===External links===<br />
<br />
*[http://p3server.pa.msu.edu/coursewiki/doku.php?id=183_notes:modeling_with_vpython Iterative Prediction with VPython]<br />
<br />
==References==<br />
<br />
*Georgia Institute of Technology. Physics Department. PHYS 2211. Fall 2016. '''Wednesday, Week 2 Lecture Slides. Fenton, Flavio H'''<br />
*Georgia Institute of Technology. Physics Department. PHYS 2211. Fall 2016. '''Monday, Week 3 Lecture Slides. Fenton, Flavio H'''<br />
*Georgia Institute of Technology. Physics Department. PHYS 2211. Fall 2016. Lab 07 Fancart Energy & Spring-Mass Instructions, Greco, Edwin<br />
*http://p3server.pa.msu.edu/coursewiki/doku.php?id=183_notes:iterativepredict<br />
*http://physicsnet.co.uk/a-level-physics-as-a2/mechanics/newtons-laws-of-motion/<br />
*Bañó-Polo, M., Baeza-Delgado, C., Tamborero, S., Hazel, A., Grau, B., Nilsson, I., … Mingarro, I. (2018). Transmembrane but not soluble helices fold inside the ribosome tunnel. Nature communications, 9(1), 5246. doi:10.1038/s41467-018-07554-7</div>Ilukowiecki3http://www.physicsbook.gatech.edu/index.php?title=Iterative_Prediction&diff=38608Iterative Prediction2020-03-24T18:21:41Z<p>Ilukowiecki3: </p>
<hr />
<div>'''Claimed by ilukowiecki3 Spring 2020'''<br />
<br />
<br />
This page describes iterative prediction, a technique used to predict the motion of particles over a period of time often used in simulations.<br />
<br />
==The Main Idea==<br />
<br />
Iterative prediction is a mathematical technique for approximating the behavior of one or more particles over an interval of time. An iteration is a repeated procedure, so iterative prediction uses repeated procedures to predict the motion of a system. Specifically, 'iterate' means to repeat; in physics, it means to perform the same calculation repeatedly using information produced by the previous calculation. To perform iterative prediction, the initial position, initial momentum, and mass of each particle must be known, and the forces acting on each particle must be known. This information is used to update the position of each particle periodically.<br />
<br />
To perform iterative prediction, the time interval of interest must be divided into small sub-intervals of duration <math>\Delta t</math> called time steps. For each time step, the following steps should be performed:<br />
<br />
<ol><br />
<li>The [[Net Force]] acting on each particle should be calculated,</li><br />
<li>The [[Linear Momentum]] of each particle should be updated from the last time step using the newly calculated net force, and</li><br />
<li>The position of each particle should be updated from the last time step using its newly calculated [[Velocity]]. Repeat for the total number of time steps.</li><br />
</ol><br />
<br />
[[File:Iterative_Model.png|center]]<br />
<br />
<center>This is a visual representation of iterative prediction being used to repeatedly update a particle's momentum. It was found on [https://p3server.pa.msu.edu/coursewiki/doku.php?id=183_notes:iterativepredict this page] of the Projects & Practices in Physics website, where it is provided under a [https://creativecommons.org/licenses/by-nc-sa/3.0/ CC Attribution-Noncommercial-Share Alike 3.0 Unported] license. It is shown here under the same license.</center><br />
<br />
<br />
<br />
The size of the time steps used during iterative prediction is referred to as the resolution of the model. Dividing a time interval into a few large time steps is called low resolution while dividing the same time interval into many small time steps is called high resolution. Performing iterative prediction with a higher resolution requires more computations but produces more accurate results. This is because steps 2 and 3 assume constant force and velocity respectively during the duration of each time step (see the "mathematical model" section). In reality, force and velocity change continually, so this is where inaccuracy is introduced. However, if the time steps are small enough that force and velocity do not significantly change during any time step, iterative prediction is a sufficiently accurate model. In the limit where the time interval is divided into an infinite number of infinitely small time steps, iterative prediction becomes a perfectly accurate model. However, this would require an infinite number of computations, and is therefore impossible to do, although in some simple situations, analytic approaches can be used to this end instead. Deciding what time step to use requires consideration of both the computational resources available and the resolution required to accurately approximate the situation. When performing iterative prediction by hand, it is impractical to perform more than a few time steps, which cannot be used to accurately model much. If computers are available, it becomes feasible to divide the time interval into thousands of time steps, which can accurately simulate most day-to-day situations. Some of the most sensitive simulations, however, require incredibly small time steps. For example, some physicists perform atom-by-atom simulations of molecular structures interacting. A slight change in the position of an atom can cause a huge change in the forces acting on it, so tiny time steps must be used in order to ensure that force and velocity do not significantly change during any one time step. These simulations are run on the world's most powerful supercomputers and often take weeks to complete. They can simulate time intervals of several nanoseconds (<math>10^{-9}</math> s) using time steps of only a few femtoseconds. (<math>10^{-15}</math> s) [https://www.ncbi.nlm.nih.gov/pmc/articles/PMC6286305/ (Click here for an example of a paper about such a simulation.)] Most of the VPython simulations on this wiki use time steps of about .01s.<br />
<br />
Iterative prediction is only an approximation, but it is a very powerful tool because it can model complex systems that are impossible to model using known analytical approaches such as calculus and kinematics due to unsolvable differential equations and other obstacles.<br />
<br />
===Relation between Newton's 2nd Law and Momentum Principle===<br />
<br />
The rate of change of momentum of an object is directly proportional to the net force applied and is in the direction of the net force; the net force is equal to the rate of change of momentum. <br />
<br />
[[File:Fma_.jpg]]<br />
<br />
- F = net force in newtons (N)<br />
<br />
- m = mass in kilograms (kg)<br />
<br />
- a = acceleration in meters per second per second (ms-2)<br />
<br />
[[File:rate-of-change-of-momentum-2.jpg]]<br />
<br />
- F = net force in newtons (N)<br />
<br />
- <math>\Delta mv</math> = change in momentum in kilogram metres per second (kg ms-1)<br />
<br />
- t = change in time in seconds (s)<br />
<br />
===A Mathematical Model===<br />
<br />
Below are the mathematical formulas required for each of the three steps outlined above.<br />
<br />
<ol><br />
<br />
<li>The net force acting on each particle should be calculated.</li><br />
<br />
This step varies widely from program to program, depending on what system is being modeled. Sometimes, the forces acting on each particle are constant, such as when modeling the motion of a ball with a specific weight thrown near the surface of the earth. Other times, the forces are functions of properties such as the particles' positions, such as when modeling the motion of celestial bodies whose gravitational attraction is greater when they are closer together. Sometimes, there is only one force acting on each particle, and other times, the net force must be found by adding together the individual force vectors.<br />
<br />
<li>The momentum of each particle should be updated from the last time step using the newly calculated net force.</li><br />
<br />
This step uses the [[Impulse and Momentum|impulse-momentum theorem]]. The impulse-momentum theorem states that <math>\vec{p}_f = \vec{p}_i + \vec{J} = \vec{p}_i + \vec{F}_{net, avg} * \Delta t</math>. In iterative prediction, the initial momentum is always known; it is simply the momentum the particle had during the previous time step (or, if there is no previous time step, the initial momentum of the particle). <math>\vec{F}_{net, avg}</math> represents the average net force that has been acting on the particle since the previous time step, resulting in a new momentum this time step. To calculate it, the net force found in step 1 should be used. Note that in reality, the net force acting on the particle is constantly changing, and the net force found in step 1 is the final net force, which is not truly the average net force over the course of the previous time step (except in a scenario with constant force). As described in the introductory section, this introduces inaccuracy that can be reduced by using shorter time steps; if short time steps are used, the net force does not have much time to change over the course of the time step, meaning the average net force is very similar to the final net force.<br />
<br />
<li>The position of each particle should be updated from the last time step using its newly calculated velocity.</li><br />
<br />
This step uses the kinematic equation <math>\vec{r}_f = \vec{r}_i + \vec{v}_{avg} * \Delta t</math>. In iterative prediction, the initial position is always known; it is simply the position the particle had during the previous time step (or, if there is no previous time step, the initial position of the particle). <math>\vec{v}_{avg}</math> represents the average velocity with which the particle has been moving since the previous time step, resulting in a new position this time step. To calculate it, the momentum found in step 2 should simply be divided by the mass of the particle: <math>\vec{v}_f = \frac{\vec{p}_f}{m}</math>. Note that in reality, the velocity of the particle is constantly changing, and the velocity found above is the final velocity, which is not truly the average velocity over the course of the previous time step (even in a scenario with constant force). As described in the introductory section, this introduces inaccuracy that can be reduced by using shorter time steps; if short time steps are used, the velocity does not have much time to change over the course of the time step, meaning the average velocity is very similar to the final velocity.<br />
<br />
</ol><br />
<br />
===A Computational Model===<br />
<br />
Iterative prediction is typically done on computers because many computations are necessary in order to perform it at a meaningful resolution. VPython is a useful program to perform iterative prediction because of it can graphically display the positions of the particles each time step. Below is an example of a VPython program that uses iterative prediction to simulate a ball thrown near the surface of the earth. Be sure to read and understand the source code by clicking on "view this program" in the top left corner; there are comments for each line designed to introduce readers to their first iterative prediction program.<br />
<br />
[https://www.glowscript.org/#/user/YorickAndeweg/folder/PhysicsBookFolder/program/IterativePrediction1 high-resolution ball trajectory simulation]<br />
<br />
To demonstrate differing resolutions, here is a lower resolution version of the same simulation, which uses larger time steps:<br />
<br />
[https://www.glowscript.org/#/user/YorickAndeweg/folder/PhysicsBookFolder/program/IterativePrediction2 low-resolution ball trajectory simulation]<br />
<br />
Note that the first program is more realistic; the more frequent position updates result in a smoother curve that more closely approximates the true trajectory of a ball. On the other hand, the second program has fewer calculations to do. Both programs are artificially slowed using the "rate" command in order to simulate a realistic travel time, but if both programs were allowed to run as quickly as possible, the second program would finish first. For larger, more complicated simulations, time can be a significant factor.<br />
<br />
Here is another more complicated simulation that uses iterative prediction to simulate the motion of a mass swinging on a spring. This is an example of iterative prediction with a varying force; during each time step, the forces acting on the mass must be calculated depending on its position using [[Hooke's Law]].<br />
<br />
[https://www.glowscript.org/#/user/YorickAndeweg/folder/PhysicsBookFolder/program/SpringMass spring mass simulation]<br />
<br />
Note: both projectile motion and spring-mass systems can be analytically modeled; that is, in both of the above systems, analytical techniques such as calculus and kinematics can be used to find the position of the particles as a function of time. However, for some other, more complicated systems, analytical approaches may not exist, while iterative prediction often still works. For example, it would be easy to modify the projectile motion simulation to include the effects of [[Air Resistance]], which is difficult to take into account analytically.<br />
<br />
==Examples==<br />
<br />
In addition to the example below, know how to create vPython simulations using iterative prediction with both constant and varying forces.<br />
<br />
===1. (Middling)===<br />
<br />
A 2kg particle is released from rest at time t=0. A constant force of 6N is applied to it. How far has the particle traveled after 4 seconds?<br />
<br />
Solve this question<br />
<br />
A.) analytically (using kinematic equations)<br />
<br />
B.) using iterative prediction with 12 equal time steps<br />
<br />
C.) using iterative prediction with 4 equal time steps<br />
<br />
D.) using iterative prediction with a single time step<br />
<br />
Pay attention to how the resolution of the iterative prediction affects the accuracy of the answer. The analytical answer is completely accurate.<br />
<br />
<br />
A.) To solve this analytically, the following kinematic equation should be used:<br />
<br />
<math>\Delta x = \frac{1}{2} a t^2 + v_0 t</math><br />
<br />
Using [[Newton's Second Law: the Momentum Principle]], we know that <math>a = \frac{f}{m} = 3</math>m/s^2. We are also given that <math>v_0 = 0</math>, and we are interested in the time t=4. Substituting these values into the kinematic equation yields<br />
<br />
<math>\Delta x = \frac{1}{2} (3) (4)^2 = 24</math>m.<br />
<br />
B.) To solve this using iterative prediction, a computational tool should be used. Since an animation isn't necessary for this problem, I used Microsoft Excel; each row populates itself based on the information in the previous row using formula functions. Google Spreadsheets can also be used, as can vPython or any other programming language.<br />
<br />
[[File:12steps.PNG]]<br />
<br />
Using 12 time steps of 1/3 seconds each results in a simulated displacement of 26m, which is fairly close to the actual amount of 24m.<br />
<br />
C.)<br />
<br />
[[File:4steps.PNG]]<br />
<br />
using 4 time steps of 1 second each results in a simulated displacement of 30m, which is fairly different from the actual amount of 24m.<br />
<br />
D.)<br />
<br />
[[File:1step.PNG]]<br />
<br />
using 1 time step of 4 seconds results in a simulated displacement of 48m, which is twice the actual amount!<br />
<br />
This demonstrates that higher-resolution simulations with smaller time steps are able to predict the motion of particles more accurately than lower-resolution simulations with larger time steps. It also demonstrates that as the length of each time step approaches 0, the predicted motion of the particles approaches their actual motion. It is important to choose a resolution that keeps error within an acceptable margin.<br />
<br />
==Connectedness==<br />
<br />
===Application: video game industry===<br />
<br />
One popular application of iterative prediction is in physics-based video games. These are often little more than physics simulations with player input. Iterative prediction is used in most major video games to predict the motion of in-game objects. It can be used even if the laws of physics within the video game are different from those of the real world.<br />
<br />
==History==<br />
<br />
The impulse-momentum theorem, which is used in iterative prediction to update the momenta of particles, is derived from Newton's Second Law, which Isaac Newton (1643-1727) publish in his 1687 book Principia Mathematica. From this point onward, all of the math necessary for iterative prediction was known, but it was not useful to perform iterative prediction until significant computational advances in the twentieth century allowed for high resolutions.<br />
<br />
== See also ==<br />
<br />
*[[Linear Momentum]]<br />
*[[Impulse and Momentum]]<br />
*[[Analytical Prediction]]<br />
*[[Iterative Prediction of Spring-Mass System]]<br />
*[[Determinism]]<br />
<br />
===External links===<br />
<br />
*[http://p3server.pa.msu.edu/coursewiki/doku.php?id=183_notes:modeling_with_vpython Iterative Prediction with VPython]<br />
<br />
==References==<br />
<br />
*Georgia Institute of Technology. Physics Department. PHYS 2211. Fall 2016. '''Wednesday, Week 2 Lecture Slides. Fenton, Flavio H'''<br />
*Georgia Institute of Technology. Physics Department. PHYS 2211. Fall 2016. '''Monday, Week 3 Lecture Slides. Fenton, Flavio H'''<br />
*Georgia Institute of Technology. Physics Department. PHYS 2211. Fall 2016. Lab 07 Fancart Energy & Spring-Mass Instructions, Greco, Edwin<br />
*http://p3server.pa.msu.edu/coursewiki/doku.php?id=183_notes:iterativepredict<br />
*http://physicsnet.co.uk/a-level-physics-as-a2/mechanics/newtons-laws-of-motion/<br />
*Bañó-Polo, M., Baeza-Delgado, C., Tamborero, S., Hazel, A., Grau, B., Nilsson, I., … Mingarro, I. (2018). Transmembrane but not soluble helices fold inside the ribosome tunnel. Nature communications, 9(1), 5246. doi:10.1038/s41467-018-07554-7</div>Ilukowiecki3http://www.physicsbook.gatech.edu/index.php?title=Kinetic_Energy&diff=38601Kinetic Energy2020-03-24T17:43:07Z<p>Ilukowiecki3: </p>
<hr />
<div>'''Claimed by ilukowiecki3 Spring 2020'''<br />
== Introduction==<br />
Objects in motion have energy associated with them. This energy of motion is called kinetic energy. Kinetic energy, often abbreviated as KE, is usually given in the standard S.I. units of Joules (J). KE is also given in units of kilo Joules (kJ). <math>1 kJ = 1000 J</math>. <math>1 J = 1 kg*(m²/s²)</math>. Other types of energy include [[Rest Mass Energy]] and [[Potential Energy]]. Kinetic energy is also known as the work needed to accelerate a mass from rest to a final velocity. Once the object reaches this speed, the kinetic energy remains constant unless the speed is changed. <br />
<br />
Kinetic energy can take many forms, such as vibrational kinetic energy, rotational kinetic energy, and translational kinetic energy. Vibrational kinetic energy is the energy due to vibrational motion, while rotational kinetic energy is the energy due to rotational motion. Translational kinetic energy, the focus of this wiki page, is energy due to motion from place to place. More information on these three kinds of kinetic motions can be found at [[Translational, Rotational, and Vibrational Energy]]. Kinetic energy does not have a direction and is a scalar value. It is not a vector quantity but is rather a magnitude. Kinetic energy can be also be transferred into other types of energy, such as potential energy or chemical energy. In order to gain kinetic energy, a force must be applied, which causes work to be done on an object. This work causes the object to move, which is referred to as kinetic energy. The value of kinetic energy depends on the relationship between the object in question and the point of observation/the related frame of reference.<br />
<br />
==Rest Energy==<br />
Rest energy can be defined as the energy of an object at rest. To find the rest energy of an object/particle, multiply the mass by its velocity squared '''(E = mv²)'''. This is the correct form of Einstein’s most famous equation, which shows that energy is related to the mass of an object at rest. For example, if energy is stored in the object, its rest mass increases. This also implies that mass can be destroyed to release energy. <br />
<br />
Does this mean that only objects with mass have energy? NO! there is an equation that helps us understand this topic, Einstein's Relativistic Mass Formula. E² = (pc)² + (mc²)². When the mass = 0, E = pc<br />
<br />
==Mathematical Model==<br />
The relativistic equation for kinetic energy according to Einstein's Theory of Relativity is <math>KE=mc²(\frac{1}{\sqrt{1-\frac{v²}{c²}}} -1)</math>. However, for cases where an object's velocity is far less than the speed of light (<math>3X10^8 m/s</math>), one can use the simplified kinetic energy formula as an approximation: <math>KE=\frac{1}{2}mv²</math>. In most cases, the simplified kinetic energy formula gives a result with only minimal error. However, for near light-speed calculations, such as those involving subatomic particles such as electrons, protons, or photons, the relativistic equation must be used. Usually, the simplified kinetic energy formula is used to calculate the kinetic energy of an average object and will be used in almost all cases in introductory physics. <br />
<br />
==A Computational Model==<br />
By [[Conservation of Energy]], energy can be converted, but it cannot be created nor destroyed. Hence, in an isolated system, energy can continuously be converted back and forth between potential and kinetic energy without any loss. The following link is an excellent visualization of energy that can be demonstrated with v-python. The spring will oscillate up and down constantly converting between [[Elastic Potential Energy]] and Kinetic Energy. https://trinket.io/glowscript/87a35d5778<br />
<br />
<br />
<br />
[[File:springkinetic.gif|frame|center]]<br />
<br />
==Examples==<br />
===Simple===<br />
[[File:KEproblem.jpg|400px|center]]<br />
A ball is rolling along a frictionless surface at a constant <math>19 m/s</math>. The ball has mass <math>12 kg</math>. What is the kinetic energy of the ball in Joules?<br />
<br>Solution:<br><br />
#Because the ball's velocity is far less than the speed of light, we can use the simplified kinetic energy formula.<br> <br />
#<math>KE=\frac{1}{2}mv²</math><br> <br />
#<math>KE=\frac{1}{2}(12 kg)*(19 m/s)²</math><br> <br />
#Hence, KE = <math>2166 J</math> or <math>2.166 kJ</math><br><br />
<br />
===Middling===<br />
An electron is moving through space at a constant <math>2.9X10^8 m/s</math>. The electron has mass <math>9.1X10^-31 kg</math>. What is the kinetic energy of the ball in Joules?<br><br />
<br>Solution:<br><br />
#Because the electron's velocity is close to the speed of light, we must use the relativistic kinetic energy formula.<br> <br />
#<math>KE=mc²(\frac{1}{\sqrt{1-\frac{v²}{c²}}} -1)</math><br><br />
#<math>KE=(9.1X10^-31 kg)(3X10^8 m/s)²(\frac{1}{\sqrt{1-\frac{(2.9X10^8 m/s)²}{(3X10^8 m/s)²}}} -1)</math><br><br />
#Hence, KE = <math>2.38X10^-13 J</math> or <math>2.38X10^-16 kJ</math><br><br />
===Difficult===<br />
An proton is moving through space at constant velocity. It is found to have kinetic energy <math>2.38X10^-15 J</math>. The proton has mass <math>1.67X10^-27 kg</math>. What is the proton's velocity?<br><br />
<br>Solution:<br><br />
#Because we are dealing with a subatomic particle, we should probably use the relativistic kinetic energy formula as the approximate kinetic energy formula may be very inaccurate if the particle is moving at a velocity near the speed of sound.<br> <br />
#<math>KE=mc²(\frac{1}{\sqrt{1-\frac{v²}{c²}}} -1)</math><br><br />
#<math>\frac{KE}{mc²}=(\frac{1}{\sqrt{1-\frac{v²}{c²}}} -1)</math><br><br />
#<math>\frac{KE}{mc²} + 1=\frac{1}{\sqrt{1-\frac{v²}{c²}}}</math><br><br />
#<math>\frac{1}{\frac{KE}{mc²} + 1}=\sqrt{1-\frac{v²}{c²}}</math><br><br />
#<math>1-\frac{v²}{c²}=(\frac{1}{\frac{KE}{mc²} + 1})²</math><br><br />
#<math>\frac{v²}{c²}=1 - (\frac{1}{\frac{KE}{mc²} + 1})²</math><br><br />
#<math>v²=c²*(1 - (\frac{1}{\frac{KE}{mc²} + 1})²)</math><br><br />
#<math>v=\sqrt{c²*(1 - (\frac{1}{\frac{KE}{mc²} + 1})²)}</math><br><br />
#<math>v=c*\sqrt{1 - (\frac{1}{\frac{KE}{mc²} + 1})²}</math><br><br />
#<math>v=3X10^8 m/s*\sqrt{1 - (\frac{1}{\frac{(2.38X10^-15 J)}{(1.67X10^-27 kg)(3X10^8 m/s)²} + 1})²}</math><br><br />
#v = <math>2.999X10^8 m/s</math>. This means that the proton is moving very close to the speed of light and hence our choice to use the relativistic kinetic energy equation was a good one.<br><br />
<br />
==Connectedness==<br />
Kinetic energy is very important in baseball. When a batter hits a ball, kinetic energy from the bat is transferred into the ball. leading to the ball flying out into the field. Assuming solid contact between the ball and the bat, the amount of kinetic energy transferred to the ball is directly proportional to the kinetic energy of the bat. As this is a non-relativistic application, there are only two variables to consider when calculating the kinetic energy of the bad: m and v. While a heaver bat would have greater kinetic energy than a bat of smaller mass traveling at the same velocity, heavier bats are more difficult to swing quickly and generally have lower velocity. Hence, batters must try to choose the a bat that they can swing very fast that still has reasonable mass to it to try to maximize kinetic energy (<math>\frac{1}{2}mv²</math>).<br />
https://www.youtube.com/watch?v=zDcf7eEaP0M<br />
<br />
Kinetic energy also plays a role in horse-back riding. For instance, a horse's jump happens in three parts.<br />
<br />
<br />
First, at the beginning of the jump, the horse tenses its muscles and kicks off the ground to lift into the air, converting chemical energy into horizontal and vertical kinetic energy. While in the air, the horse's horizontal kinetic energy doesn't change, but its vertical kinetic energy is converted into gravitational potential energy until it reaches the apex of the jump. As the horse falls, this gravitational potential is converted back into kinetic energy. Finally, when the horse lands, the horse's legs absorb the impact and slow down slightly by converting the vertical and some of the horizontal kinetic energy into heat.<br />
<br />
Kinetic energy is also important to biomedical engineering. Biomedical engineering, such as the design of prosthetics, must focus on how humans move and produce kinetic energy in order to make the best prosthetic for a person. <br />
<br />
<br />
The biomedical engineer must consider how much energy the person will need to move the prosthetics, how heavy the mass of the prosthetic must be to be efficient, and how much energy must continuously be applied to keep the prosthetic moving.<br />
<br />
==History==<br />
<br />
Kinetic energy can be traced all the way back to Aristotle (384 BCE to 322 BCE) who first proposed the concept of actuality and potentiality (actuality being kinetic energy and potentiality being [[Potential Energy]]). The connection between energy and mv² was first developed in the 1600-1700s by Gottfried Leibniz and Johann Bernoulli, who described it as the "living force." William Gravesande tested this by dropping weights from different heights into a block of clay, discovering a proportionality between penetration depth and impact velocity squared. However, William Thomson is credited for devising the term "kinetic energy" in the mid-1800s.<br />
<br />
== See also ==<br />
<br />
[[Potential Energy]]<br> <br />
[[Rest Mass Energy]]<br><br />
[[Translational, Rotational and Vibrational Energy]]<br />
<br />
===Further reading===<br />
<br />
Matter and Interactions By Ruth W. Chabay, Bruce A. Sherwood - Chapter 9 <br />
<br />
===External links===<br />
<br />
http://www.physicsclassroom.com/class/energy/Lesson-1/Kinetic-Energy<br> <br />
https://en.wikipedia.org/wiki/Kinetic_energy<br />
http://hyperphysics.phy-astr.gsu.edu/hbase/ke.html<br />
http://www.livescience.com/46278-kinetic-energy.html<br />
https://www.khanacademy.org/science/physics/work-and-energy/work-and-energy-tutorial/a/what-is-kinetic-energy<br />
<br />
==References==<br />
Matter and Interactions By Ruth W. Chabay, Bruce A. Sherwood - Chapter 9<br> <br />
http://www.physicsclassroom.com/class/energy/Lesson-1/Kinetic-Energy<br> <br />
https://en.wikipedia.org/wiki/Kinetic_energy<br> <br />
https://courses.lumenlearning.com/physics/chapter/28-6-relativistic-energy/<br> <br />
<br />
[[Category:Energy]]</div>Ilukowiecki3http://www.physicsbook.gatech.edu/index.php?title=Kinetic_Energy&diff=38600Kinetic Energy2020-03-24T17:42:21Z<p>Ilukowiecki3: </p>
<hr />
<div>'''Claimed by ilukowiecki3 Spring 2020'''<br />
== Introduction==<br />
Objects in motion have energy associated with them. This energy of motion is called kinetic energy. Kinetic energy, often abbreviated as KE, is usually given in the standard S.I. units of Joules (J). KE is also given in units of kilo Joules (kJ). <math>1 kJ = 1000 J</math>. <math>1 J = 1 kg*(m²/s²)</math>. Other types of energy include [[Rest Mass Energy]] and [[Potential Energy]]. Kinetic energy is also known as the work needed to accelerate a mass from rest to a final velocity. Once the object reaches this speed, the kinetic energy remains constant unless the speed is changed. <br />
<br />
Kinetic energy can take many forms, such as vibrational kinetic energy, rotational kinetic energy, and translational kinetic energy. Vibrational kinetic energy is the energy due to vibrational motion, while rotational kinetic energy is the energy due to rotational motion. Translational kinetic energy, the focus of this wiki page, is energy due to motion from place to place. More information on these three kinds of kinetic motions can be found at [[Translational, Rotational, and Vibrational Energy]]. Kinetic energy does not have a direction and is a scalar value. It is not a vector quantity but is rather a magnitude. Kinetic energy can be also be transferred into other types of energy, such as potential energy or chemical energy. In order to gain kinetic energy, a force must be applied, which causes work to be done on an object. This work causes the object to move, which is referred to as kinetic energy. The value of kinetic energy depends on the relationship between the object in question and the point of observation/the related frame of reference.<br />
<br />
==Rest Energy==<br />
Rest energy can be defined as the energy of an object at rest. To find the rest energy of an object/particle, multiply the mass by its velocity squared '''(E = mv²)'''. This is the correct form of Einstein’s most famous equation, which shows that energy is related to the mass of an object at rest. For example, if energy is stored in the object, its rest mass increases. This also implies that mass can be destroyed to release energy. <br />
<br />
Does this mean that only objects with mass have energy? NO! there is an equation that helps us understand this topic, Einstein's Relativistic Mass Formula. E² = (pc)² + (mc²)². When the mass = 0, E = pc<br />
[[File:restenergy.jpg]]<br />
<br />
==Mathematical Model==<br />
The relativistic equation for kinetic energy according to Einstein's Theory of Relativity is <math>KE=mc²(\frac{1}{\sqrt{1-\frac{v²}{c²}}} -1)</math>. However, for cases where an object's velocity is far less than the speed of light (<math>3X10^8 m/s</math>), one can use the simplified kinetic energy formula as an approximation: <math>KE=\frac{1}{2}mv²</math>. In most cases, the simplified kinetic energy formula gives a result with only minimal error. However, for near light-speed calculations, such as those involving subatomic particles such as electrons, protons, or photons, the relativistic equation must be used. Usually, the simplified kinetic energy formula is used to calculate the kinetic energy of an average object and will be used in almost all cases in introductory physics. <br />
<br />
==A Computational Model==<br />
By [[Conservation of Energy]], energy can be converted, but it cannot be created nor destroyed. Hence, in an isolated system, energy can continuously be converted back and forth between potential and kinetic energy without any loss. The following link is an excellent visualization of energy that can be demonstrated with v-python. The spring will oscillate up and down constantly converting between [[Elastic Potential Energy]] and Kinetic Energy. https://trinket.io/glowscript/87a35d5778<br />
<br />
<br />
<br />
[[File:springkinetic.gif|frame|center]]<br />
<br />
==Examples==<br />
===Simple===<br />
[[File:KEproblem.jpg|400px|center]]<br />
A ball is rolling along a frictionless surface at a constant <math>19 m/s</math>. The ball has mass <math>12 kg</math>. What is the kinetic energy of the ball in Joules?<br />
<br>Solution:<br><br />
#Because the ball's velocity is far less than the speed of light, we can use the simplified kinetic energy formula.<br> <br />
#<math>KE=\frac{1}{2}mv²</math><br> <br />
#<math>KE=\frac{1}{2}(12 kg)*(19 m/s)²</math><br> <br />
#Hence, KE = <math>2166 J</math> or <math>2.166 kJ</math><br><br />
<br />
===Middling===<br />
An electron is moving through space at a constant <math>2.9X10^8 m/s</math>. The electron has mass <math>9.1X10^-31 kg</math>. What is the kinetic energy of the ball in Joules?<br><br />
<br>Solution:<br><br />
#Because the electron's velocity is close to the speed of light, we must use the relativistic kinetic energy formula.<br> <br />
#<math>KE=mc²(\frac{1}{\sqrt{1-\frac{v²}{c²}}} -1)</math><br><br />
#<math>KE=(9.1X10^-31 kg)(3X10^8 m/s)²(\frac{1}{\sqrt{1-\frac{(2.9X10^8 m/s)²}{(3X10^8 m/s)²}}} -1)</math><br><br />
#Hence, KE = <math>2.38X10^-13 J</math> or <math>2.38X10^-16 kJ</math><br><br />
===Difficult===<br />
An proton is moving through space at constant velocity. It is found to have kinetic energy <math>2.38X10^-15 J</math>. The proton has mass <math>1.67X10^-27 kg</math>. What is the proton's velocity?<br><br />
<br>Solution:<br><br />
#Because we are dealing with a subatomic particle, we should probably use the relativistic kinetic energy formula as the approximate kinetic energy formula may be very inaccurate if the particle is moving at a velocity near the speed of sound.<br> <br />
#<math>KE=mc²(\frac{1}{\sqrt{1-\frac{v²}{c²}}} -1)</math><br><br />
#<math>\frac{KE}{mc²}=(\frac{1}{\sqrt{1-\frac{v²}{c²}}} -1)</math><br><br />
#<math>\frac{KE}{mc²} + 1=\frac{1}{\sqrt{1-\frac{v²}{c²}}}</math><br><br />
#<math>\frac{1}{\frac{KE}{mc²} + 1}=\sqrt{1-\frac{v²}{c²}}</math><br><br />
#<math>1-\frac{v²}{c²}=(\frac{1}{\frac{KE}{mc²} + 1})²</math><br><br />
#<math>\frac{v²}{c²}=1 - (\frac{1}{\frac{KE}{mc²} + 1})²</math><br><br />
#<math>v²=c²*(1 - (\frac{1}{\frac{KE}{mc²} + 1})²)</math><br><br />
#<math>v=\sqrt{c²*(1 - (\frac{1}{\frac{KE}{mc²} + 1})²)}</math><br><br />
#<math>v=c*\sqrt{1 - (\frac{1}{\frac{KE}{mc²} + 1})²}</math><br><br />
#<math>v=3X10^8 m/s*\sqrt{1 - (\frac{1}{\frac{(2.38X10^-15 J)}{(1.67X10^-27 kg)(3X10^8 m/s)²} + 1})²}</math><br><br />
#v = <math>2.999X10^8 m/s</math>. This means that the proton is moving very close to the speed of light and hence our choice to use the relativistic kinetic energy equation was a good one.<br><br />
<br />
==Connectedness==<br />
Kinetic energy is very important in baseball. When a batter hits a ball, kinetic energy from the bat is transferred into the ball. leading to the ball flying out into the field. Assuming solid contact between the ball and the bat, the amount of kinetic energy transferred to the ball is directly proportional to the kinetic energy of the bat. As this is a non-relativistic application, there are only two variables to consider when calculating the kinetic energy of the bad: m and v. While a heaver bat would have greater kinetic energy than a bat of smaller mass traveling at the same velocity, heavier bats are more difficult to swing quickly and generally have lower velocity. Hence, batters must try to choose the a bat that they can swing very fast that still has reasonable mass to it to try to maximize kinetic energy (<math>\frac{1}{2}mv²</math>).<br />
https://www.youtube.com/watch?v=zDcf7eEaP0M<br />
<br />
Kinetic energy also plays a role in horse-back riding. For instance, a horse's jump happens in three parts.<br />
<br />
<br />
First, at the beginning of the jump, the horse tenses its muscles and kicks off the ground to lift into the air, converting chemical energy into horizontal and vertical kinetic energy. While in the air, the horse's horizontal kinetic energy doesn't change, but its vertical kinetic energy is converted into gravitational potential energy until it reaches the apex of the jump. As the horse falls, this gravitational potential is converted back into kinetic energy. Finally, when the horse lands, the horse's legs absorb the impact and slow down slightly by converting the vertical and some of the horizontal kinetic energy into heat.<br />
<br />
Kinetic energy is also important to biomedical engineering. Biomedical engineering, such as the design of prosthetics, must focus on how humans move and produce kinetic energy in order to make the best prosthetic for a person. <br />
<br />
<br />
The biomedical engineer must consider how much energy the person will need to move the prosthetics, how heavy the mass of the prosthetic must be to be efficient, and how much energy must continuously be applied to keep the prosthetic moving.<br />
<br />
==History==<br />
<br />
Kinetic energy can be traced all the way back to Aristotle (384 BCE to 322 BCE) who first proposed the concept of actuality and potentiality (actuality being kinetic energy and potentiality being [[Potential Energy]]). The connection between energy and mv² was first developed in the 1600-1700s by Gottfried Leibniz and Johann Bernoulli, who described it as the "living force." William Gravesande tested this by dropping weights from different heights into a block of clay, discovering a proportionality between penetration depth and impact velocity squared. However, William Thomson is credited for devising the term "kinetic energy" in the mid-1800s.<br />
<br />
== See also ==<br />
<br />
[[Potential Energy]]<br> <br />
[[Rest Mass Energy]]<br><br />
[[Translational, Rotational and Vibrational Energy]]<br />
<br />
===Further reading===<br />
<br />
Matter and Interactions By Ruth W. Chabay, Bruce A. Sherwood - Chapter 9 <br />
<br />
===External links===<br />
<br />
http://www.physicsclassroom.com/class/energy/Lesson-1/Kinetic-Energy<br> <br />
https://en.wikipedia.org/wiki/Kinetic_energy<br />
http://hyperphysics.phy-astr.gsu.edu/hbase/ke.html<br />
http://www.livescience.com/46278-kinetic-energy.html<br />
https://www.khanacademy.org/science/physics/work-and-energy/work-and-energy-tutorial/a/what-is-kinetic-energy<br />
<br />
==References==<br />
Matter and Interactions By Ruth W. Chabay, Bruce A. Sherwood - Chapter 9<br> <br />
http://www.physicsclassroom.com/class/energy/Lesson-1/Kinetic-Energy<br> <br />
https://en.wikipedia.org/wiki/Kinetic_energy<br> <br />
https://courses.lumenlearning.com/physics/chapter/28-6-relativistic-energy/<br> <br />
<br />
[[Category:Energy]]</div>Ilukowiecki3http://www.physicsbook.gatech.edu/index.php?title=Kinetic_Energy&diff=38599Kinetic Energy2020-03-24T17:38:28Z<p>Ilukowiecki3: </p>
<hr />
<div>'''Claimed by ilukowiecki3 Spring 2020'''<br />
== Introduction==<br />
Objects in motion have energy associated with them. This energy of motion is called kinetic energy. Kinetic energy, often abbreviated as KE, is usually given in the standard S.I. units of Joules (J). KE is also given in units of kilo Joules (kJ). <math>1 kJ = 1000 J</math>. <math>1 J = 1 kg*(m²/s²)</math>. Other types of energy include [[Rest Mass Energy]] and [[Potential Energy]]. Kinetic energy is also known as the work needed to accelerate a mass from rest to a final velocity. Once the object reaches this speed, the kinetic energy remains constant unless the speed is changed. <br />
<br />
Kinetic energy can take many forms, such as vibrational kinetic energy, rotational kinetic energy, and translational kinetic energy. Vibrational kinetic energy is the energy due to vibrational motion, while rotational kinetic energy is the energy due to rotational motion. Translational kinetic energy, the focus of this wiki page, is energy due to motion from place to place. More information on these three kinds of kinetic motions can be found at [[Translational, Rotational, and Vibrational Energy]]. Kinetic energy does not have a direction and is a scalar value. It is not a vector quantity but is rather a magnitude. Kinetic energy can be also be transferred into other types of energy, such as potential energy or chemical energy. In order to gain kinetic energy, a force must be applied, which causes work to be done on an object. This work causes the object to move, which is referred to as kinetic energy. The value of kinetic energy depends on the relationship between the object in question and the point of observation/the related frame of reference.<br />
<br />
==Rest Energy==<br />
Rest energy can be defined as the energy of an object at rest. To find the rest energy of an object/particle, multiply the mass by its velocity squared '''(E = mv²)'''. This is the correct form of Einstein’s most famous equation, which shows that energy is related to the mass of an object at rest. For example, if energy is stored in the object, its rest mass increases. This also implies that mass can be destroyed to release energy. <br />
<br />
Does this mean that only objects with mass have energy? NO! there is an equation that helps us understand this topic, Einstein's Relativistic Mass Formula. E² = (pc)² + (mc²)². When the mass = 0, E = pc<br />
[[File:restenergy.jpg|400px|center]]<br />
<br />
==Mathematical Model==<br />
The relativistic equation for kinetic energy according to Einstein's Theory of Relativity is <math>KE=mc²(\frac{1}{\sqrt{1-\frac{v²}{c²}}} -1)</math>. However, for cases where an object's velocity is far less than the speed of light (<math>3X10^8 m/s</math>), one can use the simplified kinetic energy formula as an approximation: <math>KE=\frac{1}{2}mv²</math>. In most cases, the simplified kinetic energy formula gives a result with only minimal error. However, for near light-speed calculations, such as those involving subatomic particles such as electrons, protons, or photons, the relativistic equation must be used. Usually, the simplified kinetic energy formula is used to calculate the kinetic energy of an average object and will be used in almost all cases in introductory physics. <br />
<br />
==A Computational Model==<br />
By [[Conservation of Energy]], energy can be converted, but it cannot be created nor destroyed. Hence, in an isolated system, energy can continuously be converted back and forth between potential and kinetic energy without any loss. The following link is an excellent visualization of energy that can be demonstrated with v-python. The spring will oscillate up and down constantly converting between [[Elastic Potential Energy]] and Kinetic Energy. https://trinket.io/glowscript/87a35d5778<br />
<br />
<br />
<br />
[[File:springkinetic.gif|frame|center]]<br />
<br />
==Examples==<br />
===Simple===<br />
[[File:KEproblem.jpg|400px|center]]<br />
A ball is rolling along a frictionless surface at a constant <math>19 m/s</math>. The ball has mass <math>12 kg</math>. What is the kinetic energy of the ball in Joules?<br />
<br>Solution:<br><br />
#Because the ball's velocity is far less than the speed of light, we can use the simplified kinetic energy formula.<br> <br />
#<math>KE=\frac{1}{2}mv²</math><br> <br />
#<math>KE=\frac{1}{2}(12 kg)*(19 m/s)²</math><br> <br />
#Hence, KE = <math>2166 J</math> or <math>2.166 kJ</math><br><br />
<br />
===Middling===<br />
An electron is moving through space at a constant <math>2.9X10^8 m/s</math>. The electron has mass <math>9.1X10^-31 kg</math>. What is the kinetic energy of the ball in Joules?<br><br />
<br>Solution:<br><br />
#Because the electron's velocity is close to the speed of light, we must use the relativistic kinetic energy formula.<br> <br />
#<math>KE=mc²(\frac{1}{\sqrt{1-\frac{v²}{c²}}} -1)</math><br><br />
#<math>KE=(9.1X10^-31 kg)(3X10^8 m/s)²(\frac{1}{\sqrt{1-\frac{(2.9X10^8 m/s)²}{(3X10^8 m/s)²}}} -1)</math><br><br />
#Hence, KE = <math>2.38X10^-13 J</math> or <math>2.38X10^-16 kJ</math><br><br />
===Difficult===<br />
An proton is moving through space at constant velocity. It is found to have kinetic energy <math>2.38X10^-15 J</math>. The proton has mass <math>1.67X10^-27 kg</math>. What is the proton's velocity?<br><br />
<br>Solution:<br><br />
#Because we are dealing with a subatomic particle, we should probably use the relativistic kinetic energy formula as the approximate kinetic energy formula may be very inaccurate if the particle is moving at a velocity near the speed of sound.<br> <br />
#<math>KE=mc²(\frac{1}{\sqrt{1-\frac{v²}{c²}}} -1)</math><br><br />
#<math>\frac{KE}{mc²}=(\frac{1}{\sqrt{1-\frac{v²}{c²}}} -1)</math><br><br />
#<math>\frac{KE}{mc²} + 1=\frac{1}{\sqrt{1-\frac{v²}{c²}}}</math><br><br />
#<math>\frac{1}{\frac{KE}{mc²} + 1}=\sqrt{1-\frac{v²}{c²}}</math><br><br />
#<math>1-\frac{v²}{c²}=(\frac{1}{\frac{KE}{mc²} + 1})²</math><br><br />
#<math>\frac{v²}{c²}=1 - (\frac{1}{\frac{KE}{mc²} + 1})²</math><br><br />
#<math>v²=c²*(1 - (\frac{1}{\frac{KE}{mc²} + 1})²)</math><br><br />
#<math>v=\sqrt{c²*(1 - (\frac{1}{\frac{KE}{mc²} + 1})²)}</math><br><br />
#<math>v=c*\sqrt{1 - (\frac{1}{\frac{KE}{mc²} + 1})²}</math><br><br />
#<math>v=3X10^8 m/s*\sqrt{1 - (\frac{1}{\frac{(2.38X10^-15 J)}{(1.67X10^-27 kg)(3X10^8 m/s)²} + 1})²}</math><br><br />
#v = <math>2.999X10^8 m/s</math>. This means that the proton is moving very close to the speed of light and hence our choice to use the relativistic kinetic energy equation was a good one.<br><br />
<br />
==Connectedness==<br />
Kinetic energy is very important in baseball. When a batter hits a ball, kinetic energy from the bat is transferred into the ball. leading to the ball flying out into the field. Assuming solid contact between the ball and the bat, the amount of kinetic energy transferred to the ball is directly proportional to the kinetic energy of the bat. As this is a non-relativistic application, there are only two variables to consider when calculating the kinetic energy of the bad: m and v. While a heaver bat would have greater kinetic energy than a bat of smaller mass traveling at the same velocity, heavier bats are more difficult to swing quickly and generally have lower velocity. Hence, batters must try to choose the a bat that they can swing very fast that still has reasonable mass to it to try to maximize kinetic energy (<math>\frac{1}{2}mv²</math>).<br />
https://www.youtube.com/watch?v=zDcf7eEaP0M<br />
<br />
Kinetic energy also plays a role in horse-back riding. For instance, a horse's jump happens in three parts.<br />
<br />
<br />
First, at the beginning of the jump, the horse tenses its muscles and kicks off the ground to lift into the air, converting chemical energy into horizontal and vertical kinetic energy. While in the air, the horse's horizontal kinetic energy doesn't change, but its vertical kinetic energy is converted into gravitational potential energy until it reaches the apex of the jump. As the horse falls, this gravitational potential is converted back into kinetic energy. Finally, when the horse lands, the horse's legs absorb the impact and slow down slightly by converting the vertical and some of the horizontal kinetic energy into heat.<br />
<br />
Kinetic energy is also important to biomedical engineering. Biomedical engineering, such as the design of prosthetics, must focus on how humans move and produce kinetic energy in order to make the best prosthetic for a person. <br />
<br />
<br />
The biomedical engineer must consider how much energy the person will need to move the prosthetics, how heavy the mass of the prosthetic must be to be efficient, and how much energy must continuously be applied to keep the prosthetic moving.<br />
<br />
==History==<br />
<br />
Kinetic energy can be traced all the way back to Aristotle (384 BCE to 322 BCE) who first proposed the concept of actuality and potentiality (actuality being kinetic energy and potentiality being [[Potential Energy]]). The connection between energy and mv² was first developed in the 1600-1700s by Gottfried Leibniz and Johann Bernoulli, who described it as the "living force." William Gravesande tested this by dropping weights from different heights into a block of clay, discovering a proportionality between penetration depth and impact velocity squared. However, William Thomson is credited for devising the term "kinetic energy" in the mid-1800s.<br />
<br />
== See also ==<br />
<br />
[[Potential Energy]]<br> <br />
[[Rest Mass Energy]]<br><br />
[[Translational, Rotational and Vibrational Energy]]<br />
<br />
===Further reading===<br />
<br />
Matter and Interactions By Ruth W. Chabay, Bruce A. Sherwood - Chapter 9 <br />
<br />
===External links===<br />
<br />
http://www.physicsclassroom.com/class/energy/Lesson-1/Kinetic-Energy<br> <br />
https://en.wikipedia.org/wiki/Kinetic_energy<br />
http://hyperphysics.phy-astr.gsu.edu/hbase/ke.html<br />
http://www.livescience.com/46278-kinetic-energy.html<br />
https://www.khanacademy.org/science/physics/work-and-energy/work-and-energy-tutorial/a/what-is-kinetic-energy<br />
<br />
==References==<br />
Matter and Interactions By Ruth W. Chabay, Bruce A. Sherwood - Chapter 9<br> <br />
http://www.physicsclassroom.com/class/energy/Lesson-1/Kinetic-Energy<br> <br />
https://en.wikipedia.org/wiki/Kinetic_energy<br> <br />
https://courses.lumenlearning.com/physics/chapter/28-6-relativistic-energy/<br> <br />
<br />
[[Category:Energy]]</div>Ilukowiecki3http://www.physicsbook.gatech.edu/index.php?title=Kinetic_Energy&diff=38598Kinetic Energy2020-03-24T17:33:14Z<p>Ilukowiecki3: </p>
<hr />
<div>'''Claimed by ilukowiecki3 Spring 2020'''<br />
== Introduction==<br />
Objects in motion have energy associated with them. This energy of motion is called kinetic energy. Kinetic energy, often abbreviated as KE, is usually given in the standard S.I. units of Joules (J). KE is also given in units of kilo Joules (kJ). <math>1 kJ = 1000 J</math>. <math>1 J = 1 kg*(m²/s²)</math>. Other types of energy include [[Rest Mass Energy]] and [[Potential Energy]]. Kinetic energy is also known as the work needed to accelerate a mass from rest to a final velocity. Once the object reaches this speed, the kinetic energy remains constant unless the speed is changed. <br />
<br />
Kinetic energy can take many forms, such as vibrational kinetic energy, rotational kinetic energy, and translational kinetic energy. Vibrational kinetic energy is the energy due to vibrational motion, while rotational kinetic energy is the energy due to rotational motion. Translational kinetic energy, the focus of this wiki page, is energy due to motion from place to place. More information on these three kinds of kinetic motions can be found at [[Translational, Rotational, and Vibrational Energy]]. Kinetic energy does not have a direction and is a scalar value. It is not a vector quantity but is rather a magnitude. Kinetic energy can be also be transferred into other types of energy, such as potential energy or chemical energy. In order to gain kinetic energy, a force must be applied, which causes work to be done on an object. This work causes the object to move, which is referred to as kinetic energy. The value of kinetic energy depends on the relationship between the object in question and the point of observation/the related frame of reference.<br />
<br />
==Rest Energy==<br />
Rest energy can be defined as the energy of an object at rest. To find the rest energy of an object/particle, multiply the mass by its velocity squared '''(E = mv²)'''. This is the correct form of Einstein’s most famous equation, which shows that energy is related to the mass of an object at rest. For example, if energy is stored in the object, its rest mass increases. This also implies that mass can be destroyed to release energy. <br />
<br />
Does this mean that only objects with mass have energy? NO! there is an equation that helps us understand this topic, Einstein's Relativistic Mass Formula. E² = (pc)² + (mc²)². When the mass = 0, E = pc<br />
<br />
==Mathematical Model==<br />
The relativistic equation for kinetic energy according to Einstein's Theory of Relativity is <math>KE=mc²(\frac{1}{\sqrt{1-\frac{v²}{c²}}} -1)</math>. However, for cases where an object's velocity is far less than the speed of light (<math>3X10^8 m/s</math>), one can use the simplified kinetic energy formula as an approximation: <math>KE=\frac{1}{2}mv²</math>. In most cases, the simplified kinetic energy formula gives a result with only minimal error. However, for near light-speed calculations, such as those involving subatomic particles such as electrons, protons, or photons, the relativistic equation must be used. Usually, the simplified kinetic energy formula is used to calculate the kinetic energy of an average object and will be used in almost all cases in introductory physics. <br />
<br />
==A Computational Model==<br />
By [[Conservation of Energy]], energy can be converted, but it cannot be created nor destroyed. Hence, in an isolated system, energy can continuously be converted back and forth between potential and kinetic energy without any loss. The following link is an excellent visualization of energy that can be demonstrated with v-python. The spring will oscillate up and down constantly converting between [[Elastic Potential Energy]] and Kinetic Energy. https://trinket.io/glowscript/87a35d5778<br />
<br />
<br />
<br />
[[File:springkinetic.gif|frame|center]]<br />
<br />
==Examples==<br />
===Simple===<br />
[[File:KEproblem.jpg|400px|center]]<br />
A ball is rolling along a frictionless surface at a constant <math>19 m/s</math>. The ball has mass <math>12 kg</math>. What is the kinetic energy of the ball in Joules?<br />
<br>Solution:<br><br />
#Because the ball's velocity is far less than the speed of light, we can use the simplified kinetic energy formula.<br> <br />
#<math>KE=\frac{1}{2}mv²</math><br> <br />
#<math>KE=\frac{1}{2}(12 kg)*(19 m/s)²</math><br> <br />
#Hence, KE = <math>2166 J</math> or <math>2.166 kJ</math><br><br />
<br />
===Middling===<br />
An electron is moving through space at a constant <math>2.9X10^8 m/s</math>. The electron has mass <math>9.1X10^-31 kg</math>. What is the kinetic energy of the ball in Joules?<br><br />
<br>Solution:<br><br />
#Because the electron's velocity is close to the speed of light, we must use the relativistic kinetic energy formula.<br> <br />
#<math>KE=mc²(\frac{1}{\sqrt{1-\frac{v²}{c²}}} -1)</math><br><br />
#<math>KE=(9.1X10^-31 kg)(3X10^8 m/s)²(\frac{1}{\sqrt{1-\frac{(2.9X10^8 m/s)²}{(3X10^8 m/s)²}}} -1)</math><br><br />
#Hence, KE = <math>2.38X10^-13 J</math> or <math>2.38X10^-16 kJ</math><br><br />
===Difficult===<br />
An proton is moving through space at constant velocity. It is found to have kinetic energy <math>2.38X10^-15 J</math>. The proton has mass <math>1.67X10^-27 kg</math>. What is the proton's velocity?<br><br />
<br>Solution:<br><br />
#Because we are dealing with a subatomic particle, we should probably use the relativistic kinetic energy formula as the approximate kinetic energy formula may be very inaccurate if the particle is moving at a velocity near the speed of sound.<br> <br />
#<math>KE=mc²(\frac{1}{\sqrt{1-\frac{v²}{c²}}} -1)</math><br><br />
#<math>\frac{KE}{mc²}=(\frac{1}{\sqrt{1-\frac{v²}{c²}}} -1)</math><br><br />
#<math>\frac{KE}{mc²} + 1=\frac{1}{\sqrt{1-\frac{v²}{c²}}}</math><br><br />
#<math>\frac{1}{\frac{KE}{mc²} + 1}=\sqrt{1-\frac{v²}{c²}}</math><br><br />
#<math>1-\frac{v²}{c²}=(\frac{1}{\frac{KE}{mc²} + 1})²</math><br><br />
#<math>\frac{v²}{c²}=1 - (\frac{1}{\frac{KE}{mc²} + 1})²</math><br><br />
#<math>v²=c²*(1 - (\frac{1}{\frac{KE}{mc²} + 1})²)</math><br><br />
#<math>v=\sqrt{c²*(1 - (\frac{1}{\frac{KE}{mc²} + 1})²)}</math><br><br />
#<math>v=c*\sqrt{1 - (\frac{1}{\frac{KE}{mc²} + 1})²}</math><br><br />
#<math>v=3X10^8 m/s*\sqrt{1 - (\frac{1}{\frac{(2.38X10^-15 J)}{(1.67X10^-27 kg)(3X10^8 m/s)²} + 1})²}</math><br><br />
#v = <math>2.999X10^8 m/s</math>. This means that the proton is moving very close to the speed of light and hence our choice to use the relativistic kinetic energy equation was a good one.<br><br />
<br />
==Connectedness==<br />
Kinetic energy is very important in baseball. When a batter hits a ball, kinetic energy from the bat is transferred into the ball. leading to the ball flying out into the field. Assuming solid contact between the ball and the bat, the amount of kinetic energy transferred to the ball is directly proportional to the kinetic energy of the bat. As this is a non-relativistic application, there are only two variables to consider when calculating the kinetic energy of the bad: m and v. While a heaver bat would have greater kinetic energy than a bat of smaller mass traveling at the same velocity, heavier bats are more difficult to swing quickly and generally have lower velocity. Hence, batters must try to choose the a bat that they can swing very fast that still has reasonable mass to it to try to maximize kinetic energy (<math>\frac{1}{2}mv²</math>).<br />
https://www.youtube.com/watch?v=zDcf7eEaP0M<br />
<br />
Kinetic energy also plays a role in horse-back riding. For instance, a horse's jump happens in three parts.<br />
<br />
<br />
First, at the beginning of the jump, the horse tenses its muscles and kicks off the ground to lift into the air, converting chemical energy into horizontal and vertical kinetic energy. While in the air, the horse's horizontal kinetic energy doesn't change, but its vertical kinetic energy is converted into gravitational potential energy until it reaches the apex of the jump. As the horse falls, this gravitational potential is converted back into kinetic energy. Finally, when the horse lands, the horse's legs absorb the impact and slow down slightly by converting the vertical and some of the horizontal kinetic energy into heat.<br />
<br />
Kinetic energy is also important to biomedical engineering. Biomedical engineering, such as the design of prosthetics, must focus on how humans move and produce kinetic energy in order to make the best prosthetic for a person. <br />
<br />
<br />
The biomedical engineer must consider how much energy the person will need to move the prosthetics, how heavy the mass of the prosthetic must be to be efficient, and how much energy must continuously be applied to keep the prosthetic moving.<br />
<br />
==History==<br />
<br />
Kinetic energy can be traced all the way back to Aristotle (384 BCE to 322 BCE) who first proposed the concept of actuality and potentiality (actuality being kinetic energy and potentiality being [[Potential Energy]]). The connection between energy and mv² was first developed in the 1600-1700s by Gottfried Leibniz and Johann Bernoulli, who described it as the "living force." William Gravesande tested this by dropping weights from different heights into a block of clay, discovering a proportionality between penetration depth and impact velocity squared. However, William Thomson is credited for devising the term "kinetic energy" in the mid-1800s.<br />
<br />
== See also ==<br />
<br />
[[Potential Energy]]<br> <br />
[[Rest Mass Energy]]<br><br />
[[Translational, Rotational and Vibrational Energy]]<br />
<br />
===Further reading===<br />
<br />
Matter and Interactions By Ruth W. Chabay, Bruce A. Sherwood - Chapter 9 <br />
<br />
===External links===<br />
<br />
http://www.physicsclassroom.com/class/energy/Lesson-1/Kinetic-Energy<br> <br />
https://en.wikipedia.org/wiki/Kinetic_energy<br />
http://hyperphysics.phy-astr.gsu.edu/hbase/ke.html<br />
http://www.livescience.com/46278-kinetic-energy.html<br />
https://www.khanacademy.org/science/physics/work-and-energy/work-and-energy-tutorial/a/what-is-kinetic-energy<br />
<br />
==References==<br />
Matter and Interactions By Ruth W. Chabay, Bruce A. Sherwood - Chapter 9<br> <br />
http://www.physicsclassroom.com/class/energy/Lesson-1/Kinetic-Energy<br> <br />
https://en.wikipedia.org/wiki/Kinetic_energy<br> <br />
https://courses.lumenlearning.com/physics/chapter/28-6-relativistic-energy/<br> <br />
<br />
[[Category:Energy]]</div>Ilukowiecki3http://www.physicsbook.gatech.edu/index.php?title=Kinetic_Energy&diff=38597Kinetic Energy2020-03-24T17:32:36Z<p>Ilukowiecki3: </p>
<hr />
<div>'''Claimed by ilukowiecki3 Spring 2020'''<br />
== Introduction==<br />
Objects in motion have energy associated with them. This energy of motion is called kinetic energy. Kinetic energy, often abbreviated as KE, is usually given in the standard S.I. units of Joules (J). KE is also given in units of kilo Joules (kJ). <math>1 kJ = 1000 J</math>. <math>1 J = 1 kg*(m²/s²)</math>. Other types of energy include [[Rest Mass Energy]] and [[Potential Energy]]. Kinetic energy is also known as the work needed to accelerate a mass from rest to a final velocity. Once the object reaches this speed, the kinetic energy remains constant unless the speed is changed. <br />
<br />
Kinetic energy can take many forms, such as vibrational kinetic energy, rotational kinetic energy, and translational kinetic energy. Vibrational kinetic energy is the energy due to vibrational motion, while rotational kinetic energy is the energy due to rotational motion. Translational kinetic energy, the focus of this wiki page, is energy due to motion from place to place. More information on these three kinds of kinetic motions can be found at [[Translational, Rotational, and Vibrational Energy]]. Kinetic energy does not have a direction and is a scalar value. It is not a vector quantity but is rather a magnitude. Kinetic energy can be also be transferred into other types of energy, such as potential energy or chemical energy. In order to gain kinetic energy, a force must be applied, which causes work to be done on an object. This work causes the object to move, which is referred to as kinetic energy. The value of kinetic energy depends on the relationship between the object in question and the point of observation/the related frame of reference.<br />
<br />
==Rest Energy==<br />
Rest energy can be defined as the energy of an object at rest. To find the rest energy of an object/particle, multiply the mass by its velocity squared '''(E = mv²)'''. This is the correct form of Einstein’s most famous equation, which shows that energy is related to the mass of an object at rest. For example, if energy is stored in the object, its rest mass increases. This also implies that mass can be destroyed to release energy. <br />
Does this mean that only objects with mass have energy? NO! there is an equation that helps us understand this topic, Einstein's Relativistic Mass Formula.<br />
E² = (pc)² + (mc²)²<br />
When the mass = 0, E = pc<br />
<br />
==Mathematical Model==<br />
The relativistic equation for kinetic energy according to Einstein's Theory of Relativity is <math>KE=mc²(\frac{1}{\sqrt{1-\frac{v²}{c²}}} -1)</math>. However, for cases where an object's velocity is far less than the speed of light (<math>3X10^8 m/s</math>), one can use the simplified kinetic energy formula as an approximation: <math>KE=\frac{1}{2}mv²</math>. In most cases, the simplified kinetic energy formula gives a result with only minimal error. However, for near light-speed calculations, such as those involving subatomic particles such as electrons, protons, or photons, the relativistic equation must be used. Usually, the simplified kinetic energy formula is used to calculate the kinetic energy of an average object and will be used in almost all cases in introductory physics. <br />
<br />
==A Computational Model==<br />
By [[Conservation of Energy]], energy can be converted, but it cannot be created nor destroyed. Hence, in an isolated system, energy can continuously be converted back and forth between potential and kinetic energy without any loss. The following link is an excellent visualization of energy that can be demonstrated with v-python. The spring will oscillate up and down constantly converting between [[Elastic Potential Energy]] and Kinetic Energy. https://trinket.io/glowscript/87a35d5778<br />
<br />
<br />
<br />
[[File:springkinetic.gif|frame|center]]<br />
<br />
==Examples==<br />
===Simple===<br />
[[File:KEproblem.jpg|400px|center]]<br />
A ball is rolling along a frictionless surface at a constant <math>19 m/s</math>. The ball has mass <math>12 kg</math>. What is the kinetic energy of the ball in Joules?<br />
<br>Solution:<br><br />
#Because the ball's velocity is far less than the speed of light, we can use the simplified kinetic energy formula.<br> <br />
#<math>KE=\frac{1}{2}mv²</math><br> <br />
#<math>KE=\frac{1}{2}(12 kg)*(19 m/s)²</math><br> <br />
#Hence, KE = <math>2166 J</math> or <math>2.166 kJ</math><br><br />
<br />
===Middling===<br />
An electron is moving through space at a constant <math>2.9X10^8 m/s</math>. The electron has mass <math>9.1X10^-31 kg</math>. What is the kinetic energy of the ball in Joules?<br><br />
<br>Solution:<br><br />
#Because the electron's velocity is close to the speed of light, we must use the relativistic kinetic energy formula.<br> <br />
#<math>KE=mc²(\frac{1}{\sqrt{1-\frac{v²}{c²}}} -1)</math><br><br />
#<math>KE=(9.1X10^-31 kg)(3X10^8 m/s)²(\frac{1}{\sqrt{1-\frac{(2.9X10^8 m/s)²}{(3X10^8 m/s)²}}} -1)</math><br><br />
#Hence, KE = <math>2.38X10^-13 J</math> or <math>2.38X10^-16 kJ</math><br><br />
===Difficult===<br />
An proton is moving through space at constant velocity. It is found to have kinetic energy <math>2.38X10^-15 J</math>. The proton has mass <math>1.67X10^-27 kg</math>. What is the proton's velocity?<br><br />
<br>Solution:<br><br />
#Because we are dealing with a subatomic particle, we should probably use the relativistic kinetic energy formula as the approximate kinetic energy formula may be very inaccurate if the particle is moving at a velocity near the speed of sound.<br> <br />
#<math>KE=mc²(\frac{1}{\sqrt{1-\frac{v²}{c²}}} -1)</math><br><br />
#<math>\frac{KE}{mc²}=(\frac{1}{\sqrt{1-\frac{v²}{c²}}} -1)</math><br><br />
#<math>\frac{KE}{mc²} + 1=\frac{1}{\sqrt{1-\frac{v²}{c²}}}</math><br><br />
#<math>\frac{1}{\frac{KE}{mc²} + 1}=\sqrt{1-\frac{v²}{c²}}</math><br><br />
#<math>1-\frac{v²}{c²}=(\frac{1}{\frac{KE}{mc²} + 1})²</math><br><br />
#<math>\frac{v²}{c²}=1 - (\frac{1}{\frac{KE}{mc²} + 1})²</math><br><br />
#<math>v²=c²*(1 - (\frac{1}{\frac{KE}{mc²} + 1})²)</math><br><br />
#<math>v=\sqrt{c²*(1 - (\frac{1}{\frac{KE}{mc²} + 1})²)}</math><br><br />
#<math>v=c*\sqrt{1 - (\frac{1}{\frac{KE}{mc²} + 1})²}</math><br><br />
#<math>v=3X10^8 m/s*\sqrt{1 - (\frac{1}{\frac{(2.38X10^-15 J)}{(1.67X10^-27 kg)(3X10^8 m/s)²} + 1})²}</math><br><br />
#v = <math>2.999X10^8 m/s</math>. This means that the proton is moving very close to the speed of light and hence our choice to use the relativistic kinetic energy equation was a good one.<br><br />
<br />
==Connectedness==<br />
Kinetic energy is very important in baseball. When a batter hits a ball, kinetic energy from the bat is transferred into the ball. leading to the ball flying out into the field. Assuming solid contact between the ball and the bat, the amount of kinetic energy transferred to the ball is directly proportional to the kinetic energy of the bat. As this is a non-relativistic application, there are only two variables to consider when calculating the kinetic energy of the bad: m and v. While a heaver bat would have greater kinetic energy than a bat of smaller mass traveling at the same velocity, heavier bats are more difficult to swing quickly and generally have lower velocity. Hence, batters must try to choose the a bat that they can swing very fast that still has reasonable mass to it to try to maximize kinetic energy (<math>\frac{1}{2}mv²</math>).<br />
https://www.youtube.com/watch?v=zDcf7eEaP0M<br />
<br />
Kinetic energy also plays a role in horse-back riding. For instance, a horse's jump happens in three parts.<br />
<br />
<br />
First, at the beginning of the jump, the horse tenses its muscles and kicks off the ground to lift into the air, converting chemical energy into horizontal and vertical kinetic energy. While in the air, the horse's horizontal kinetic energy doesn't change, but its vertical kinetic energy is converted into gravitational potential energy until it reaches the apex of the jump. As the horse falls, this gravitational potential is converted back into kinetic energy. Finally, when the horse lands, the horse's legs absorb the impact and slow down slightly by converting the vertical and some of the horizontal kinetic energy into heat.<br />
<br />
Kinetic energy is also important to biomedical engineering. Biomedical engineering, such as the design of prosthetics, must focus on how humans move and produce kinetic energy in order to make the best prosthetic for a person. <br />
<br />
<br />
The biomedical engineer must consider how much energy the person will need to move the prosthetics, how heavy the mass of the prosthetic must be to be efficient, and how much energy must continuously be applied to keep the prosthetic moving.<br />
<br />
==History==<br />
<br />
Kinetic energy can be traced all the way back to Aristotle (384 BCE to 322 BCE) who first proposed the concept of actuality and potentiality (actuality being kinetic energy and potentiality being [[Potential Energy]]). The connection between energy and mv² was first developed in the 1600-1700s by Gottfried Leibniz and Johann Bernoulli, who described it as the "living force." William Gravesande tested this by dropping weights from different heights into a block of clay, discovering a proportionality between penetration depth and impact velocity squared. However, William Thomson is credited for devising the term "kinetic energy" in the mid-1800s.<br />
<br />
== See also ==<br />
<br />
[[Potential Energy]]<br> <br />
[[Rest Mass Energy]]<br><br />
[[Translational, Rotational and Vibrational Energy]]<br />
<br />
===Further reading===<br />
<br />
Matter and Interactions By Ruth W. Chabay, Bruce A. Sherwood - Chapter 9 <br />
<br />
===External links===<br />
<br />
http://www.physicsclassroom.com/class/energy/Lesson-1/Kinetic-Energy<br> <br />
https://en.wikipedia.org/wiki/Kinetic_energy<br />
http://hyperphysics.phy-astr.gsu.edu/hbase/ke.html<br />
http://www.livescience.com/46278-kinetic-energy.html<br />
https://www.khanacademy.org/science/physics/work-and-energy/work-and-energy-tutorial/a/what-is-kinetic-energy<br />
<br />
==References==<br />
Matter and Interactions By Ruth W. Chabay, Bruce A. Sherwood - Chapter 9<br> <br />
http://www.physicsclassroom.com/class/energy/Lesson-1/Kinetic-Energy<br> <br />
https://en.wikipedia.org/wiki/Kinetic_energy<br> <br />
https://courses.lumenlearning.com/physics/chapter/28-6-relativistic-energy/<br> <br />
<br />
[[Category:Energy]]</div>Ilukowiecki3http://www.physicsbook.gatech.edu/index.php?title=Kinetic_Energy&diff=38596Kinetic Energy2020-03-24T17:32:03Z<p>Ilukowiecki3: </p>
<hr />
<div>'''Claimed by ilukowiecki3 Spring 2020'''<br />
== Introduction==<br />
Objects in motion have energy associated with them. This energy of motion is called kinetic energy. Kinetic energy, often abbreviated as KE, is usually given in the standard S.I. units of Joules (J). KE is also given in units of kilo Joules (kJ). <math>1 kJ = 1000 J</math>. <math>1 J = 1 kg*(m²/s²)</math>. Other types of energy include [[Rest Mass Energy]] and [[Potential Energy]]. Kinetic energy is also known as the work needed to accelerate a mass from rest to a final velocity. Once the object reaches this speed, the kinetic energy remains constant unless the speed is changed. <br />
<br />
Kinetic energy can take many forms, such as vibrational kinetic energy, rotational kinetic energy, and translational kinetic energy. Vibrational kinetic energy is the energy due to vibrational motion, while rotational kinetic energy is the energy due to rotational motion. Translational kinetic energy, the focus of this wiki page, is energy due to motion from place to place. More information on these three kinds of kinetic motions can be found at [[Translational, Rotational, and Vibrational Energy]]. Kinetic energy does not have a direction and is a scalar value. It is not a vector quantity but is rather a magnitude. Kinetic energy can be also be transferred into other types of energy, such as potential energy or chemical energy. In order to gain kinetic energy, a force must be applied, which causes work to be done on an object. This work causes the object to move, which is referred to as kinetic energy. The value of kinetic energy depends on the relationship between the object in question and the point of observation/the related frame of reference.<br />
<br />
==Rest Energy==<br />
Rest energy can be defined as the energy of an object at rest. To find the rest energy of an object/particle, multiply the mass by its velocity squared '''(E = mv²)'''. This is the correct form of Einstein’s most famous equation, which shows that energy is related to the mass of an object at rest. For example, if energy is stored in the object, its rest mass increases. This also implies that mass can be destroyed to release energy. <br />
Does this mean that only objects with mass have energy? NO! there is an equation that helps us understand this topic, Einstein's Relativistic Mass Formula.<br />
E² = (pc)² + (mc²)²<br />
When the mas = 0, E = pc<br />
<br />
==Mathematical Model==<br />
The relativistic equation for kinetic energy according to Einstein's Theory of Relativity is <math>KE=mc²(\frac{1}{\sqrt{1-\frac{v²}{c²}}} -1)</math>. However, for cases where an object's velocity is far less than the speed of light (<math>3X10^8 m/s</math>), one can use the simplified kinetic energy formula as an approximation: <math>KE=\frac{1}{2}mv²</math>. In most cases, the simplified kinetic energy formula gives a result with only minimal error. However, for near light-speed calculations, such as those involving subatomic particles such as electrons, protons, or photons, the relativistic equation must be used. Usually, the simplified kinetic energy formula is used to calculate the kinetic energy of an average object and will be used in almost all cases in introductory physics. <br />
<br />
==A Computational Model==<br />
By [[Conservation of Energy]], energy can be converted, but it cannot be created nor destroyed. Hence, in an isolated system, energy can continuously be converted back and forth between potential and kinetic energy without any loss. The following link is an excellent visualization of energy that can be demonstrated with v-python. The spring will oscillate up and down constantly converting between [[Elastic Potential Energy]] and Kinetic Energy. https://trinket.io/glowscript/87a35d5778<br />
<br />
<br />
<br />
[[File:springkinetic.gif|frame|center]]<br />
<br />
==Examples==<br />
===Simple===<br />
[[File:KEproblem.jpg|400px|center]]<br />
A ball is rolling along a frictionless surface at a constant <math>19 m/s</math>. The ball has mass <math>12 kg</math>. What is the kinetic energy of the ball in Joules?<br />
<br>Solution:<br><br />
#Because the ball's velocity is far less than the speed of light, we can use the simplified kinetic energy formula.<br> <br />
#<math>KE=\frac{1}{2}mv²</math><br> <br />
#<math>KE=\frac{1}{2}(12 kg)*(19 m/s)²</math><br> <br />
#Hence, KE = <math>2166 J</math> or <math>2.166 kJ</math><br><br />
<br />
===Middling===<br />
An electron is moving through space at a constant <math>2.9X10^8 m/s</math>. The electron has mass <math>9.1X10^-31 kg</math>. What is the kinetic energy of the ball in Joules?<br><br />
<br>Solution:<br><br />
#Because the electron's velocity is close to the speed of light, we must use the relativistic kinetic energy formula.<br> <br />
#<math>KE=mc²(\frac{1}{\sqrt{1-\frac{v²}{c²}}} -1)</math><br><br />
#<math>KE=(9.1X10^-31 kg)(3X10^8 m/s)²(\frac{1}{\sqrt{1-\frac{(2.9X10^8 m/s)²}{(3X10^8 m/s)²}}} -1)</math><br><br />
#Hence, KE = <math>2.38X10^-13 J</math> or <math>2.38X10^-16 kJ</math><br><br />
===Difficult===<br />
An proton is moving through space at constant velocity. It is found to have kinetic energy <math>2.38X10^-15 J</math>. The proton has mass <math>1.67X10^-27 kg</math>. What is the proton's velocity?<br><br />
<br>Solution:<br><br />
#Because we are dealing with a subatomic particle, we should probably use the relativistic kinetic energy formula as the approximate kinetic energy formula may be very inaccurate if the particle is moving at a velocity near the speed of sound.<br> <br />
#<math>KE=mc²(\frac{1}{\sqrt{1-\frac{v²}{c²}}} -1)</math><br><br />
#<math>\frac{KE}{mc²}=(\frac{1}{\sqrt{1-\frac{v²}{c²}}} -1)</math><br><br />
#<math>\frac{KE}{mc²} + 1=\frac{1}{\sqrt{1-\frac{v²}{c²}}}</math><br><br />
#<math>\frac{1}{\frac{KE}{mc²} + 1}=\sqrt{1-\frac{v²}{c²}}</math><br><br />
#<math>1-\frac{v²}{c²}=(\frac{1}{\frac{KE}{mc²} + 1})²</math><br><br />
#<math>\frac{v²}{c²}=1 - (\frac{1}{\frac{KE}{mc²} + 1})²</math><br><br />
#<math>v²=c²*(1 - (\frac{1}{\frac{KE}{mc²} + 1})²)</math><br><br />
#<math>v=\sqrt{c²*(1 - (\frac{1}{\frac{KE}{mc²} + 1})²)}</math><br><br />
#<math>v=c*\sqrt{1 - (\frac{1}{\frac{KE}{mc²} + 1})²}</math><br><br />
#<math>v=3X10^8 m/s*\sqrt{1 - (\frac{1}{\frac{(2.38X10^-15 J)}{(1.67X10^-27 kg)(3X10^8 m/s)²} + 1})²}</math><br><br />
#v = <math>2.999X10^8 m/s</math>. This means that the proton is moving very close to the speed of light and hence our choice to use the relativistic kinetic energy equation was a good one.<br><br />
<br />
==Connectedness==<br />
Kinetic energy is very important in baseball. When a batter hits a ball, kinetic energy from the bat is transferred into the ball. leading to the ball flying out into the field. Assuming solid contact between the ball and the bat, the amount of kinetic energy transferred to the ball is directly proportional to the kinetic energy of the bat. As this is a non-relativistic application, there are only two variables to consider when calculating the kinetic energy of the bad: m and v. While a heaver bat would have greater kinetic energy than a bat of smaller mass traveling at the same velocity, heavier bats are more difficult to swing quickly and generally have lower velocity. Hence, batters must try to choose the a bat that they can swing very fast that still has reasonable mass to it to try to maximize kinetic energy (<math>\frac{1}{2}mv²</math>).<br />
https://www.youtube.com/watch?v=zDcf7eEaP0M<br />
<br />
Kinetic energy also plays a role in horse-back riding. For instance, a horse's jump happens in three parts.<br />
<br />
<br />
First, at the beginning of the jump, the horse tenses its muscles and kicks off the ground to lift into the air, converting chemical energy into horizontal and vertical kinetic energy. While in the air, the horse's horizontal kinetic energy doesn't change, but its vertical kinetic energy is converted into gravitational potential energy until it reaches the apex of the jump. As the horse falls, this gravitational potential is converted back into kinetic energy. Finally, when the horse lands, the horse's legs absorb the impact and slow down slightly by converting the vertical and some of the horizontal kinetic energy into heat.<br />
<br />
Kinetic energy is also important to biomedical engineering. Biomedical engineering, such as the design of prosthetics, must focus on how humans move and produce kinetic energy in order to make the best prosthetic for a person. <br />
<br />
<br />
The biomedical engineer must consider how much energy the person will need to move the prosthetics, how heavy the mass of the prosthetic must be to be efficient, and how much energy must continuously be applied to keep the prosthetic moving.<br />
<br />
==History==<br />
<br />
Kinetic energy can be traced all the way back to Aristotle (384 BCE to 322 BCE) who first proposed the concept of actuality and potentiality (actuality being kinetic energy and potentiality being [[Potential Energy]]). The connection between energy and mv² was first developed in the 1600-1700s by Gottfried Leibniz and Johann Bernoulli, who described it as the "living force." William Gravesande tested this by dropping weights from different heights into a block of clay, discovering a proportionality between penetration depth and impact velocity squared. However, William Thomson is credited for devising the term "kinetic energy" in the mid-1800s.<br />
<br />
== See also ==<br />
<br />
[[Potential Energy]]<br> <br />
[[Rest Mass Energy]]<br><br />
[[Translational, Rotational and Vibrational Energy]]<br />
<br />
===Further reading===<br />
<br />
Matter and Interactions By Ruth W. Chabay, Bruce A. Sherwood - Chapter 9 <br />
<br />
===External links===<br />
<br />
http://www.physicsclassroom.com/class/energy/Lesson-1/Kinetic-Energy<br> <br />
https://en.wikipedia.org/wiki/Kinetic_energy<br />
http://hyperphysics.phy-astr.gsu.edu/hbase/ke.html<br />
http://www.livescience.com/46278-kinetic-energy.html<br />
https://www.khanacademy.org/science/physics/work-and-energy/work-and-energy-tutorial/a/what-is-kinetic-energy<br />
<br />
==References==<br />
Matter and Interactions By Ruth W. Chabay, Bruce A. Sherwood - Chapter 9<br> <br />
http://www.physicsclassroom.com/class/energy/Lesson-1/Kinetic-Energy<br> <br />
https://en.wikipedia.org/wiki/Kinetic_energy<br> <br />
https://courses.lumenlearning.com/physics/chapter/28-6-relativistic-energy/<br> <br />
<br />
[[Category:Energy]]</div>Ilukowiecki3http://www.physicsbook.gatech.edu/index.php?title=Kinetic_Energy&diff=38595Kinetic Energy2020-03-24T17:28:49Z<p>Ilukowiecki3: /* Rest Energy */</p>
<hr />
<div>'''Claimed by ilukowiecki3 Spring 2020'''<br />
== Introduction==<br />
Objects in motion have energy associated with them. This energy of motion is called kinetic energy. Kinetic energy, often abbreviated as KE, is usually given in the standard S.I. units of Joules (J). KE is also given in units of kilo Joules (kJ). <math>1 kJ = 1000 J</math>. <math>1 J = 1 kg*(m²/s²)</math>. Other types of energy include [[Rest Mass Energy]] and [[Potential Energy]]. Kinetic energy is also known as the work needed to accelerate a mass from rest to a final velocity. Once the object reaches this speed, the kinetic energy remains constant unless the speed is changed. <br />
<br />
Kinetic energy can take many forms, such as vibrational kinetic energy, rotational kinetic energy, and translational kinetic energy. Vibrational kinetic energy is the energy due to vibrational motion, while rotational kinetic energy is the energy due to rotational motion. Translational kinetic energy, the focus of this wiki page, is energy due to motion from place to place. More information on these three kinds of kinetic motions can be found at [[Translational, Rotational, and Vibrational Energy]]. Kinetic energy does not have a direction and is a scalar value. It is not a vector quantity but is rather a magnitude. Kinetic energy can be also be transferred into other types of energy, such as potential energy or chemical energy. In order to gain kinetic energy, a force must be applied, which causes work to be done on an object. This work causes the object to move, which is referred to as kinetic energy. The value of kinetic energy depends on the relationship between the object in question and the point of observation/the related frame of reference.<br />
<br />
==Rest Energy==<br />
Rest energy can be defined as the energy of an object at rest. To find the rest energy of an object/particle, multiply the mass by its velocity squared '''(E = mv²)'''. This is the correct form of Einstein’s most famous equation, which shows that energy is related to the mass of an object at rest. For example, if energy is stored in the object, its rest mass increases. This also implies that mass can be destroyed to release energy. <br />
Does this mean that only objects with mass have energy? NO! there is an equation that helps us understand this topic.<br />
<br />
==Mathematical Model==<br />
The relativistic equation for kinetic energy according to Einstein's Theory of Relativity is <math>KE=mc²(\frac{1}{\sqrt{1-\frac{v²}{c²}}} -1)</math>. However, for cases where an object's velocity is far less than the speed of light (<math>3X10^8 m/s</math>), one can use the simplified kinetic energy formula as an approximation: <math>KE=\frac{1}{2}mv²</math>. In most cases, the simplified kinetic energy formula gives a result with only minimal error. However, for near light-speed calculations, such as those involving subatomic particles such as electrons, protons, or photons, the relativistic equation must be used. Usually, the simplified kinetic energy formula is used to calculate the kinetic energy of an average object and will be used in almost all cases in introductory physics. <br />
<br />
==A Computational Model==<br />
By [[Conservation of Energy]], energy can be converted, but it cannot be created nor destroyed. Hence, in an isolated system, energy can continuously be converted back and forth between potential and kinetic energy without any loss. The following link is an excellent visualization of energy that can be demonstrated with v-python. The spring will oscillate up and down constantly converting between [[Elastic Potential Energy]] and Kinetic Energy. https://trinket.io/glowscript/87a35d5778<br />
<br />
<br />
<br />
[[File:springkinetic.gif|frame|center]]<br />
<br />
==Examples==<br />
===Simple===<br />
[[File:KEproblem.jpg|400px|center]]<br />
A ball is rolling along a frictionless surface at a constant <math>19 m/s</math>. The ball has mass <math>12 kg</math>. What is the kinetic energy of the ball in Joules?<br />
<br>Solution:<br><br />
#Because the ball's velocity is far less than the speed of light, we can use the simplified kinetic energy formula.<br> <br />
#<math>KE=\frac{1}{2}mv²</math><br> <br />
#<math>KE=\frac{1}{2}(12 kg)*(19 m/s)²</math><br> <br />
#Hence, KE = <math>2166 J</math> or <math>2.166 kJ</math><br><br />
<br />
===Middling===<br />
An electron is moving through space at a constant <math>2.9X10^8 m/s</math>. The electron has mass <math>9.1X10^-31 kg</math>. What is the kinetic energy of the ball in Joules?<br><br />
<br>Solution:<br><br />
#Because the electron's velocity is close to the speed of light, we must use the relativistic kinetic energy formula.<br> <br />
#<math>KE=mc²(\frac{1}{\sqrt{1-\frac{v²}{c²}}} -1)</math><br><br />
#<math>KE=(9.1X10^-31 kg)(3X10^8 m/s)²(\frac{1}{\sqrt{1-\frac{(2.9X10^8 m/s)²}{(3X10^8 m/s)²}}} -1)</math><br><br />
#Hence, KE = <math>2.38X10^-13 J</math> or <math>2.38X10^-16 kJ</math><br><br />
===Difficult===<br />
An proton is moving through space at constant velocity. It is found to have kinetic energy <math>2.38X10^-15 J</math>. The proton has mass <math>1.67X10^-27 kg</math>. What is the proton's velocity?<br><br />
<br>Solution:<br><br />
#Because we are dealing with a subatomic particle, we should probably use the relativistic kinetic energy formula as the approximate kinetic energy formula may be very inaccurate if the particle is moving at a velocity near the speed of sound.<br> <br />
#<math>KE=mc²(\frac{1}{\sqrt{1-\frac{v²}{c²}}} -1)</math><br><br />
#<math>\frac{KE}{mc²}=(\frac{1}{\sqrt{1-\frac{v²}{c²}}} -1)</math><br><br />
#<math>\frac{KE}{mc²} + 1=\frac{1}{\sqrt{1-\frac{v²}{c²}}}</math><br><br />
#<math>\frac{1}{\frac{KE}{mc²} + 1}=\sqrt{1-\frac{v²}{c²}}</math><br><br />
#<math>1-\frac{v²}{c²}=(\frac{1}{\frac{KE}{mc²} + 1})²</math><br><br />
#<math>\frac{v²}{c²}=1 - (\frac{1}{\frac{KE}{mc²} + 1})²</math><br><br />
#<math>v²=c²*(1 - (\frac{1}{\frac{KE}{mc²} + 1})²)</math><br><br />
#<math>v=\sqrt{c²*(1 - (\frac{1}{\frac{KE}{mc²} + 1})²)}</math><br><br />
#<math>v=c*\sqrt{1 - (\frac{1}{\frac{KE}{mc²} + 1})²}</math><br><br />
#<math>v=3X10^8 m/s*\sqrt{1 - (\frac{1}{\frac{(2.38X10^-15 J)}{(1.67X10^-27 kg)(3X10^8 m/s)²} + 1})²}</math><br><br />
#v = <math>2.999X10^8 m/s</math>. This means that the proton is moving very close to the speed of light and hence our choice to use the relativistic kinetic energy equation was a good one.<br><br />
<br />
==Connectedness==<br />
Kinetic energy is very important in baseball. When a batter hits a ball, kinetic energy from the bat is transferred into the ball. leading to the ball flying out into the field. Assuming solid contact between the ball and the bat, the amount of kinetic energy transferred to the ball is directly proportional to the kinetic energy of the bat. As this is a non-relativistic application, there are only two variables to consider when calculating the kinetic energy of the bad: m and v. While a heaver bat would have greater kinetic energy than a bat of smaller mass traveling at the same velocity, heavier bats are more difficult to swing quickly and generally have lower velocity. Hence, batters must try to choose the a bat that they can swing very fast that still has reasonable mass to it to try to maximize kinetic energy (<math>\frac{1}{2}mv²</math>).<br />
https://www.youtube.com/watch?v=zDcf7eEaP0M<br />
<br />
Kinetic energy also plays a role in horse-back riding. For instance, a horse's jump happens in three parts.<br />
<br />
<br />
First, at the beginning of the jump, the horse tenses its muscles and kicks off the ground to lift into the air, converting chemical energy into horizontal and vertical kinetic energy. While in the air, the horse's horizontal kinetic energy doesn't change, but its vertical kinetic energy is converted into gravitational potential energy until it reaches the apex of the jump. As the horse falls, this gravitational potential is converted back into kinetic energy. Finally, when the horse lands, the horse's legs absorb the impact and slow down slightly by converting the vertical and some of the horizontal kinetic energy into heat.<br />
<br />
Kinetic energy is also important to biomedical engineering. Biomedical engineering, such as the design of prosthetics, must focus on how humans move and produce kinetic energy in order to make the best prosthetic for a person. <br />
<br />
<br />
The biomedical engineer must consider how much energy the person will need to move the prosthetics, how heavy the mass of the prosthetic must be to be efficient, and how much energy must continuously be applied to keep the prosthetic moving.<br />
<br />
==History==<br />
<br />
Kinetic energy can be traced all the way back to Aristotle (384 BCE to 322 BCE) who first proposed the concept of actuality and potentiality (actuality being kinetic energy and potentiality being [[Potential Energy]]). The connection between energy and mv² was first developed in the 1600-1700s by Gottfried Leibniz and Johann Bernoulli, who described it as the "living force." William Gravesande tested this by dropping weights from different heights into a block of clay, discovering a proportionality between penetration depth and impact velocity squared. However, William Thomson is credited for devising the term "kinetic energy" in the mid-1800s.<br />
<br />
== See also ==<br />
<br />
[[Potential Energy]]<br> <br />
[[Rest Mass Energy]]<br><br />
[[Translational, Rotational and Vibrational Energy]]<br />
<br />
===Further reading===<br />
<br />
Matter and Interactions By Ruth W. Chabay, Bruce A. Sherwood - Chapter 9 <br />
<br />
===External links===<br />
<br />
http://www.physicsclassroom.com/class/energy/Lesson-1/Kinetic-Energy<br> <br />
https://en.wikipedia.org/wiki/Kinetic_energy<br />
http://hyperphysics.phy-astr.gsu.edu/hbase/ke.html<br />
http://www.livescience.com/46278-kinetic-energy.html<br />
https://www.khanacademy.org/science/physics/work-and-energy/work-and-energy-tutorial/a/what-is-kinetic-energy<br />
<br />
==References==<br />
Matter and Interactions By Ruth W. Chabay, Bruce A. Sherwood - Chapter 9<br> <br />
http://www.physicsclassroom.com/class/energy/Lesson-1/Kinetic-Energy<br> <br />
https://en.wikipedia.org/wiki/Kinetic_energy<br> <br />
https://courses.lumenlearning.com/physics/chapter/28-6-relativistic-energy/<br> <br />
<br />
[[Category:Energy]]</div>Ilukowiecki3http://www.physicsbook.gatech.edu/index.php?title=Kinetic_Energy&diff=38594Kinetic Energy2020-03-24T17:21:30Z<p>Ilukowiecki3: /* References */</p>
<hr />
<div>'''Claimed by ilukowiecki3 Spring 2020'''<br />
== Introduction==<br />
Objects in motion have energy associated with them. This energy of motion is called kinetic energy. Kinetic energy, often abbreviated as KE, is usually given in the standard S.I. units of Joules (J). KE is also given in units of kilo Joules (kJ). <math>1 kJ = 1000 J</math>. <math>1 J = 1 kg*(m²/s²)</math>. Other types of energy include [[Rest Mass Energy]] and [[Potential Energy]]. Kinetic energy is also known as the work needed to accelerate a mass from rest to a final velocity. Once the object reaches this speed, the kinetic energy remains constant unless the speed is changed. <br />
<br />
Kinetic energy can take many forms, such as vibrational kinetic energy, rotational kinetic energy, and translational kinetic energy. Vibrational kinetic energy is the energy due to vibrational motion, while rotational kinetic energy is the energy due to rotational motion. Translational kinetic energy, the focus of this wiki page, is energy due to motion from place to place. More information on these three kinds of kinetic motions can be found at [[Translational, Rotational, and Vibrational Energy]]. Kinetic energy does not have a direction and is a scalar value. It is not a vector quantity but is rather a magnitude. Kinetic energy can be also be transferred into other types of energy, such as potential energy or chemical energy. In order to gain kinetic energy, a force must be applied, which causes work to be done on an object. This work causes the object to move, which is referred to as kinetic energy. The value of kinetic energy depends on the relationship between the object in question and the point of observation/the related frame of reference.<br />
<br />
==Rest Energy==<br />
Rest energy can be defined as the energy of an object at rest. To find the rest energy of an object/particle, multiply the mass by its velocity squared '''(E = mv²)'''. This is the correct form of Einstein’s most famous equation, which shows that energy is related to the mass of an object at rest. For example, if energy is stored in the object, its rest mass increases. This also implies that mass can be destroyed to release energy.<br />
<br />
==Mathematical Model==<br />
The relativistic equation for kinetic energy according to Einstein's Theory of Relativity is <math>KE=mc²(\frac{1}{\sqrt{1-\frac{v²}{c²}}} -1)</math>. However, for cases where an object's velocity is far less than the speed of light (<math>3X10^8 m/s</math>), one can use the simplified kinetic energy formula as an approximation: <math>KE=\frac{1}{2}mv²</math>. In most cases, the simplified kinetic energy formula gives a result with only minimal error. However, for near light-speed calculations, such as those involving subatomic particles such as electrons, protons, or photons, the relativistic equation must be used. Usually, the simplified kinetic energy formula is used to calculate the kinetic energy of an average object and will be used in almost all cases in introductory physics. <br />
<br />
==A Computational Model==<br />
By [[Conservation of Energy]], energy can be converted, but it cannot be created nor destroyed. Hence, in an isolated system, energy can continuously be converted back and forth between potential and kinetic energy without any loss. The following link is an excellent visualization of energy that can be demonstrated with v-python. The spring will oscillate up and down constantly converting between [[Elastic Potential Energy]] and Kinetic Energy. https://trinket.io/glowscript/87a35d5778<br />
<br />
<br />
<br />
[[File:springkinetic.gif|frame|center]]<br />
<br />
==Examples==<br />
===Simple===<br />
[[File:KEproblem.jpg|400px|center]]<br />
A ball is rolling along a frictionless surface at a constant <math>19 m/s</math>. The ball has mass <math>12 kg</math>. What is the kinetic energy of the ball in Joules?<br />
<br>Solution:<br><br />
#Because the ball's velocity is far less than the speed of light, we can use the simplified kinetic energy formula.<br> <br />
#<math>KE=\frac{1}{2}mv²</math><br> <br />
#<math>KE=\frac{1}{2}(12 kg)*(19 m/s)²</math><br> <br />
#Hence, KE = <math>2166 J</math> or <math>2.166 kJ</math><br><br />
<br />
===Middling===<br />
An electron is moving through space at a constant <math>2.9X10^8 m/s</math>. The electron has mass <math>9.1X10^-31 kg</math>. What is the kinetic energy of the ball in Joules?<br><br />
<br>Solution:<br><br />
#Because the electron's velocity is close to the speed of light, we must use the relativistic kinetic energy formula.<br> <br />
#<math>KE=mc²(\frac{1}{\sqrt{1-\frac{v²}{c²}}} -1)</math><br><br />
#<math>KE=(9.1X10^-31 kg)(3X10^8 m/s)²(\frac{1}{\sqrt{1-\frac{(2.9X10^8 m/s)²}{(3X10^8 m/s)²}}} -1)</math><br><br />
#Hence, KE = <math>2.38X10^-13 J</math> or <math>2.38X10^-16 kJ</math><br><br />
===Difficult===<br />
An proton is moving through space at constant velocity. It is found to have kinetic energy <math>2.38X10^-15 J</math>. The proton has mass <math>1.67X10^-27 kg</math>. What is the proton's velocity?<br><br />
<br>Solution:<br><br />
#Because we are dealing with a subatomic particle, we should probably use the relativistic kinetic energy formula as the approximate kinetic energy formula may be very inaccurate if the particle is moving at a velocity near the speed of sound.<br> <br />
#<math>KE=mc²(\frac{1}{\sqrt{1-\frac{v²}{c²}}} -1)</math><br><br />
#<math>\frac{KE}{mc²}=(\frac{1}{\sqrt{1-\frac{v²}{c²}}} -1)</math><br><br />
#<math>\frac{KE}{mc²} + 1=\frac{1}{\sqrt{1-\frac{v²}{c²}}}</math><br><br />
#<math>\frac{1}{\frac{KE}{mc²} + 1}=\sqrt{1-\frac{v²}{c²}}</math><br><br />
#<math>1-\frac{v²}{c²}=(\frac{1}{\frac{KE}{mc²} + 1})²</math><br><br />
#<math>\frac{v²}{c²}=1 - (\frac{1}{\frac{KE}{mc²} + 1})²</math><br><br />
#<math>v²=c²*(1 - (\frac{1}{\frac{KE}{mc²} + 1})²)</math><br><br />
#<math>v=\sqrt{c²*(1 - (\frac{1}{\frac{KE}{mc²} + 1})²)}</math><br><br />
#<math>v=c*\sqrt{1 - (\frac{1}{\frac{KE}{mc²} + 1})²}</math><br><br />
#<math>v=3X10^8 m/s*\sqrt{1 - (\frac{1}{\frac{(2.38X10^-15 J)}{(1.67X10^-27 kg)(3X10^8 m/s)²} + 1})²}</math><br><br />
#v = <math>2.999X10^8 m/s</math>. This means that the proton is moving very close to the speed of light and hence our choice to use the relativistic kinetic energy equation was a good one.<br><br />
<br />
==Connectedness==<br />
Kinetic energy is very important in baseball. When a batter hits a ball, kinetic energy from the bat is transferred into the ball. leading to the ball flying out into the field. Assuming solid contact between the ball and the bat, the amount of kinetic energy transferred to the ball is directly proportional to the kinetic energy of the bat. As this is a non-relativistic application, there are only two variables to consider when calculating the kinetic energy of the bad: m and v. While a heaver bat would have greater kinetic energy than a bat of smaller mass traveling at the same velocity, heavier bats are more difficult to swing quickly and generally have lower velocity. Hence, batters must try to choose the a bat that they can swing very fast that still has reasonable mass to it to try to maximize kinetic energy (<math>\frac{1}{2}mv²</math>).<br />
https://www.youtube.com/watch?v=zDcf7eEaP0M<br />
<br />
Kinetic energy also plays a role in horse-back riding. For instance, a horse's jump happens in three parts.<br />
<br />
<br />
First, at the beginning of the jump, the horse tenses its muscles and kicks off the ground to lift into the air, converting chemical energy into horizontal and vertical kinetic energy. While in the air, the horse's horizontal kinetic energy doesn't change, but its vertical kinetic energy is converted into gravitational potential energy until it reaches the apex of the jump. As the horse falls, this gravitational potential is converted back into kinetic energy. Finally, when the horse lands, the horse's legs absorb the impact and slow down slightly by converting the vertical and some of the horizontal kinetic energy into heat.<br />
<br />
Kinetic energy is also important to biomedical engineering. Biomedical engineering, such as the design of prosthetics, must focus on how humans move and produce kinetic energy in order to make the best prosthetic for a person. <br />
<br />
<br />
The biomedical engineer must consider how much energy the person will need to move the prosthetics, how heavy the mass of the prosthetic must be to be efficient, and how much energy must continuously be applied to keep the prosthetic moving.<br />
<br />
==History==<br />
<br />
Kinetic energy can be traced all the way back to Aristotle (384 BCE to 322 BCE) who first proposed the concept of actuality and potentiality (actuality being kinetic energy and potentiality being [[Potential Energy]]). The connection between energy and mv² was first developed in the 1600-1700s by Gottfried Leibniz and Johann Bernoulli, who described it as the "living force." William Gravesande tested this by dropping weights from different heights into a block of clay, discovering a proportionality between penetration depth and impact velocity squared. However, William Thomson is credited for devising the term "kinetic energy" in the mid-1800s.<br />
<br />
== See also ==<br />
<br />
[[Potential Energy]]<br> <br />
[[Rest Mass Energy]]<br><br />
[[Translational, Rotational and Vibrational Energy]]<br />
<br />
===Further reading===<br />
<br />
Matter and Interactions By Ruth W. Chabay, Bruce A. Sherwood - Chapter 9 <br />
<br />
===External links===<br />
<br />
http://www.physicsclassroom.com/class/energy/Lesson-1/Kinetic-Energy<br> <br />
https://en.wikipedia.org/wiki/Kinetic_energy<br />
http://hyperphysics.phy-astr.gsu.edu/hbase/ke.html<br />
http://www.livescience.com/46278-kinetic-energy.html<br />
https://www.khanacademy.org/science/physics/work-and-energy/work-and-energy-tutorial/a/what-is-kinetic-energy<br />
<br />
==References==<br />
Matter and Interactions By Ruth W. Chabay, Bruce A. Sherwood - Chapter 9<br> <br />
http://www.physicsclassroom.com/class/energy/Lesson-1/Kinetic-Energy<br> <br />
https://en.wikipedia.org/wiki/Kinetic_energy<br> <br />
https://courses.lumenlearning.com/physics/chapter/28-6-relativistic-energy/<br> <br />
<br />
[[Category:Energy]]</div>Ilukowiecki3http://www.physicsbook.gatech.edu/index.php?title=Kinetic_Energy&diff=38593Kinetic Energy2020-03-24T17:19:40Z<p>Ilukowiecki3: </p>
<hr />
<div>'''Claimed by ilukowiecki3 Spring 2020'''<br />
== Introduction==<br />
Objects in motion have energy associated with them. This energy of motion is called kinetic energy. Kinetic energy, often abbreviated as KE, is usually given in the standard S.I. units of Joules (J). KE is also given in units of kilo Joules (kJ). <math>1 kJ = 1000 J</math>. <math>1 J = 1 kg*(m²/s²)</math>. Other types of energy include [[Rest Mass Energy]] and [[Potential Energy]]. Kinetic energy is also known as the work needed to accelerate a mass from rest to a final velocity. Once the object reaches this speed, the kinetic energy remains constant unless the speed is changed. <br />
<br />
Kinetic energy can take many forms, such as vibrational kinetic energy, rotational kinetic energy, and translational kinetic energy. Vibrational kinetic energy is the energy due to vibrational motion, while rotational kinetic energy is the energy due to rotational motion. Translational kinetic energy, the focus of this wiki page, is energy due to motion from place to place. More information on these three kinds of kinetic motions can be found at [[Translational, Rotational, and Vibrational Energy]]. Kinetic energy does not have a direction and is a scalar value. It is not a vector quantity but is rather a magnitude. Kinetic energy can be also be transferred into other types of energy, such as potential energy or chemical energy. In order to gain kinetic energy, a force must be applied, which causes work to be done on an object. This work causes the object to move, which is referred to as kinetic energy. The value of kinetic energy depends on the relationship between the object in question and the point of observation/the related frame of reference.<br />
<br />
==Rest Energy==<br />
Rest energy can be defined as the energy of an object at rest. To find the rest energy of an object/particle, multiply the mass by its velocity squared '''(E = mv²)'''. This is the correct form of Einstein’s most famous equation, which shows that energy is related to the mass of an object at rest. For example, if energy is stored in the object, its rest mass increases. This also implies that mass can be destroyed to release energy.<br />
<br />
==Mathematical Model==<br />
The relativistic equation for kinetic energy according to Einstein's Theory of Relativity is <math>KE=mc²(\frac{1}{\sqrt{1-\frac{v²}{c²}}} -1)</math>. However, for cases where an object's velocity is far less than the speed of light (<math>3X10^8 m/s</math>), one can use the simplified kinetic energy formula as an approximation: <math>KE=\frac{1}{2}mv²</math>. In most cases, the simplified kinetic energy formula gives a result with only minimal error. However, for near light-speed calculations, such as those involving subatomic particles such as electrons, protons, or photons, the relativistic equation must be used. Usually, the simplified kinetic energy formula is used to calculate the kinetic energy of an average object and will be used in almost all cases in introductory physics. <br />
<br />
==A Computational Model==<br />
By [[Conservation of Energy]], energy can be converted, but it cannot be created nor destroyed. Hence, in an isolated system, energy can continuously be converted back and forth between potential and kinetic energy without any loss. The following link is an excellent visualization of energy that can be demonstrated with v-python. The spring will oscillate up and down constantly converting between [[Elastic Potential Energy]] and Kinetic Energy. https://trinket.io/glowscript/87a35d5778<br />
<br />
<br />
<br />
[[File:springkinetic.gif|frame|center]]<br />
<br />
==Examples==<br />
===Simple===<br />
[[File:KEproblem.jpg|400px|center]]<br />
A ball is rolling along a frictionless surface at a constant <math>19 m/s</math>. The ball has mass <math>12 kg</math>. What is the kinetic energy of the ball in Joules?<br />
<br>Solution:<br><br />
#Because the ball's velocity is far less than the speed of light, we can use the simplified kinetic energy formula.<br> <br />
#<math>KE=\frac{1}{2}mv²</math><br> <br />
#<math>KE=\frac{1}{2}(12 kg)*(19 m/s)²</math><br> <br />
#Hence, KE = <math>2166 J</math> or <math>2.166 kJ</math><br><br />
<br />
===Middling===<br />
An electron is moving through space at a constant <math>2.9X10^8 m/s</math>. The electron has mass <math>9.1X10^-31 kg</math>. What is the kinetic energy of the ball in Joules?<br><br />
<br>Solution:<br><br />
#Because the electron's velocity is close to the speed of light, we must use the relativistic kinetic energy formula.<br> <br />
#<math>KE=mc²(\frac{1}{\sqrt{1-\frac{v²}{c²}}} -1)</math><br><br />
#<math>KE=(9.1X10^-31 kg)(3X10^8 m/s)²(\frac{1}{\sqrt{1-\frac{(2.9X10^8 m/s)²}{(3X10^8 m/s)²}}} -1)</math><br><br />
#Hence, KE = <math>2.38X10^-13 J</math> or <math>2.38X10^-16 kJ</math><br><br />
===Difficult===<br />
An proton is moving through space at constant velocity. It is found to have kinetic energy <math>2.38X10^-15 J</math>. The proton has mass <math>1.67X10^-27 kg</math>. What is the proton's velocity?<br><br />
<br>Solution:<br><br />
#Because we are dealing with a subatomic particle, we should probably use the relativistic kinetic energy formula as the approximate kinetic energy formula may be very inaccurate if the particle is moving at a velocity near the speed of sound.<br> <br />
#<math>KE=mc²(\frac{1}{\sqrt{1-\frac{v²}{c²}}} -1)</math><br><br />
#<math>\frac{KE}{mc²}=(\frac{1}{\sqrt{1-\frac{v²}{c²}}} -1)</math><br><br />
#<math>\frac{KE}{mc²} + 1=\frac{1}{\sqrt{1-\frac{v²}{c²}}}</math><br><br />
#<math>\frac{1}{\frac{KE}{mc²} + 1}=\sqrt{1-\frac{v²}{c²}}</math><br><br />
#<math>1-\frac{v²}{c²}=(\frac{1}{\frac{KE}{mc²} + 1})²</math><br><br />
#<math>\frac{v²}{c²}=1 - (\frac{1}{\frac{KE}{mc²} + 1})²</math><br><br />
#<math>v²=c²*(1 - (\frac{1}{\frac{KE}{mc²} + 1})²)</math><br><br />
#<math>v=\sqrt{c²*(1 - (\frac{1}{\frac{KE}{mc²} + 1})²)}</math><br><br />
#<math>v=c*\sqrt{1 - (\frac{1}{\frac{KE}{mc²} + 1})²}</math><br><br />
#<math>v=3X10^8 m/s*\sqrt{1 - (\frac{1}{\frac{(2.38X10^-15 J)}{(1.67X10^-27 kg)(3X10^8 m/s)²} + 1})²}</math><br><br />
#v = <math>2.999X10^8 m/s</math>. This means that the proton is moving very close to the speed of light and hence our choice to use the relativistic kinetic energy equation was a good one.<br><br />
<br />
==Connectedness==<br />
Kinetic energy is very important in baseball. When a batter hits a ball, kinetic energy from the bat is transferred into the ball. leading to the ball flying out into the field. Assuming solid contact between the ball and the bat, the amount of kinetic energy transferred to the ball is directly proportional to the kinetic energy of the bat. As this is a non-relativistic application, there are only two variables to consider when calculating the kinetic energy of the bad: m and v. While a heaver bat would have greater kinetic energy than a bat of smaller mass traveling at the same velocity, heavier bats are more difficult to swing quickly and generally have lower velocity. Hence, batters must try to choose the a bat that they can swing very fast that still has reasonable mass to it to try to maximize kinetic energy (<math>\frac{1}{2}mv²</math>).<br />
https://www.youtube.com/watch?v=zDcf7eEaP0M<br />
<br />
Kinetic energy also plays a role in horse-back riding. For instance, a horse's jump happens in three parts.<br />
<br />
<br />
First, at the beginning of the jump, the horse tenses its muscles and kicks off the ground to lift into the air, converting chemical energy into horizontal and vertical kinetic energy. While in the air, the horse's horizontal kinetic energy doesn't change, but its vertical kinetic energy is converted into gravitational potential energy until it reaches the apex of the jump. As the horse falls, this gravitational potential is converted back into kinetic energy. Finally, when the horse lands, the horse's legs absorb the impact and slow down slightly by converting the vertical and some of the horizontal kinetic energy into heat.<br />
<br />
Kinetic energy is also important to biomedical engineering. Biomedical engineering, such as the design of prosthetics, must focus on how humans move and produce kinetic energy in order to make the best prosthetic for a person. <br />
<br />
<br />
The biomedical engineer must consider how much energy the person will need to move the prosthetics, how heavy the mass of the prosthetic must be to be efficient, and how much energy must continuously be applied to keep the prosthetic moving.<br />
<br />
==History==<br />
<br />
Kinetic energy can be traced all the way back to Aristotle (384 BCE to 322 BCE) who first proposed the concept of actuality and potentiality (actuality being kinetic energy and potentiality being [[Potential Energy]]). The connection between energy and mv² was first developed in the 1600-1700s by Gottfried Leibniz and Johann Bernoulli, who described it as the "living force." William Gravesande tested this by dropping weights from different heights into a block of clay, discovering a proportionality between penetration depth and impact velocity squared. However, William Thomson is credited for devising the term "kinetic energy" in the mid-1800s.<br />
<br />
== See also ==<br />
<br />
[[Potential Energy]]<br> <br />
[[Rest Mass Energy]]<br><br />
[[Translational, Rotational and Vibrational Energy]]<br />
<br />
===Further reading===<br />
<br />
Matter and Interactions By Ruth W. Chabay, Bruce A. Sherwood - Chapter 9 <br />
<br />
===External links===<br />
<br />
http://www.physicsclassroom.com/class/energy/Lesson-1/Kinetic-Energy<br> <br />
https://en.wikipedia.org/wiki/Kinetic_energy<br />
http://hyperphysics.phy-astr.gsu.edu/hbase/ke.html<br />
http://www.livescience.com/46278-kinetic-energy.html<br />
https://www.khanacademy.org/science/physics/work-and-energy/work-and-energy-tutorial/a/what-is-kinetic-energy<br />
<br />
==References==<br />
Matter and Interactions By Ruth W. Chabay, Bruce A. Sherwood - Chapter 9<br> <br />
http://www.physicsclassroom.com/class/energy/Lesson-1/Kinetic-Energy<br> <br />
https://en.wikipedia.org/wiki/Kinetic_energy<br> <br />
<br />
[[Category:Energy]]</div>Ilukowiecki3http://www.physicsbook.gatech.edu/index.php?title=Kinetic_Energy&diff=38592Kinetic Energy2020-03-24T17:18:52Z<p>Ilukowiecki3: </p>
<hr />
<div>'''Claimed by ilukowiecki3 Spring 2020'''<br />
== Introduction==<br />
Objects in motion have energy associated with them. This energy of motion is called kinetic energy. Kinetic energy, often abbreviated as KE, is usually given in the standard S.I. units of Joules (J). KE is also given in units of kilo Joules (kJ). <math>1 kJ = 1000 J</math>. <math>1 J = 1 kg*(m²/s²)</math>. Other types of energy include [[Rest Mass Energy]] and [[Potential Energy]]. Kinetic energy is also known as the work needed to accelerate a mass from rest to a final velocity. Once the object reaches this speed, the kinetic energy remains constant unless the speed is changed. <br />
<br />
Kinetic energy can take many forms, such as vibrational kinetic energy, rotational kinetic energy, and translational kinetic energy. Vibrational kinetic energy is the energy due to vibrational motion, while rotational kinetic energy is the energy due to rotational motion. Translational kinetic energy, the focus of this wiki page, is energy due to motion from place to place. More information on these three kinds of kinetic motions can be found at [[Translational, Rotational, and Vibrational Energy]]. Kinetic energy does not have a direction and is a scalar value. It is not a vector quantity but is rather a magnitude. Kinetic energy can be also be transferred into other types of energy, such as potential energy or chemical energy. In order to gain kinetic energy, a force must be applied, which causes work to be done on an object. This work causes the object to move, which is referred to as kinetic energy. The value of kinetic energy depends on the relationship between the object in question and the point of observation/the related frame of reference.<br />
<br />
==Rest Energy==<br />
Rest energy can be defined as the energy of an object at rest. To find the rest energy of an object/particle, multiply the mass by its velocity squared (E=mv²). This is the correct form of Einstein’s most famous equation, which shows that energy is related to the mass of an object at rest. For example, if energy is stored in the object, its rest mass increases. This also implies that mass can be destroyed to release energy.<br />
<br />
==Mathematical Model==<br />
The relativistic equation for kinetic energy according to Einstein's Theory of Relativity is <math>KE=mc²(\frac{1}{\sqrt{1-\frac{v²}{c²}}} -1)</math>. However, for cases where an object's velocity is far less than the speed of light (<math>3X10^8 m/s</math>), one can use the simplified kinetic energy formula as an approximation: <math>KE=\frac{1}{2}mv²</math>. In most cases, the simplified kinetic energy formula gives a result with only minimal error. However, for near light-speed calculations, such as those involving subatomic particles such as electrons, protons, or photons, the relativistic equation must be used. Usually, the simplified kinetic energy formula is used to calculate the kinetic energy of an average object and will be used in almost all cases in introductory physics. <br />
<br />
==A Computational Model==<br />
By [[Conservation of Energy]], energy can be converted, but it cannot be created nor destroyed. Hence, in an isolated system, energy can continuously be converted back and forth between potential and kinetic energy without any loss. The following link is an excellent visualization of energy that can be demonstrated with v-python. The spring will oscillate up and down constantly converting between [[Elastic Potential Energy]] and Kinetic Energy. https://trinket.io/glowscript/87a35d5778<br />
<br />
<br />
<br />
[[File:springkinetic.gif|frame|center]]<br />
<br />
==Examples==<br />
===Simple===<br />
[[File:KEproblem.jpg|400px|center]]<br />
A ball is rolling along a frictionless surface at a constant <math>19 m/s</math>. The ball has mass <math>12 kg</math>. What is the kinetic energy of the ball in Joules?<br />
<br>Solution:<br><br />
#Because the ball's velocity is far less than the speed of light, we can use the simplified kinetic energy formula.<br> <br />
#<math>KE=\frac{1}{2}mv²</math><br> <br />
#<math>KE=\frac{1}{2}(12 kg)*(19 m/s)²</math><br> <br />
#Hence, KE = <math>2166 J</math> or <math>2.166 kJ</math><br><br />
<br />
===Middling===<br />
An electron is moving through space at a constant <math>2.9X10^8 m/s</math>. The electron has mass <math>9.1X10^-31 kg</math>. What is the kinetic energy of the ball in Joules?<br><br />
<br>Solution:<br><br />
#Because the electron's velocity is close to the speed of light, we must use the relativistic kinetic energy formula.<br> <br />
#<math>KE=mc²(\frac{1}{\sqrt{1-\frac{v²}{c²}}} -1)</math><br><br />
#<math>KE=(9.1X10^-31 kg)(3X10^8 m/s)²(\frac{1}{\sqrt{1-\frac{(2.9X10^8 m/s)²}{(3X10^8 m/s)²}}} -1)</math><br><br />
#Hence, KE = <math>2.38X10^-13 J</math> or <math>2.38X10^-16 kJ</math><br><br />
===Difficult===<br />
An proton is moving through space at constant velocity. It is found to have kinetic energy <math>2.38X10^-15 J</math>. The proton has mass <math>1.67X10^-27 kg</math>. What is the proton's velocity?<br><br />
<br>Solution:<br><br />
#Because we are dealing with a subatomic particle, we should probably use the relativistic kinetic energy formula as the approximate kinetic energy formula may be very inaccurate if the particle is moving at a velocity near the speed of sound.<br> <br />
#<math>KE=mc²(\frac{1}{\sqrt{1-\frac{v²}{c²}}} -1)</math><br><br />
#<math>\frac{KE}{mc²}=(\frac{1}{\sqrt{1-\frac{v²}{c²}}} -1)</math><br><br />
#<math>\frac{KE}{mc²} + 1=\frac{1}{\sqrt{1-\frac{v²}{c²}}}</math><br><br />
#<math>\frac{1}{\frac{KE}{mc²} + 1}=\sqrt{1-\frac{v²}{c²}}</math><br><br />
#<math>1-\frac{v²}{c²}=(\frac{1}{\frac{KE}{mc²} + 1})²</math><br><br />
#<math>\frac{v²}{c²}=1 - (\frac{1}{\frac{KE}{mc²} + 1})²</math><br><br />
#<math>v²=c²*(1 - (\frac{1}{\frac{KE}{mc²} + 1})²)</math><br><br />
#<math>v=\sqrt{c²*(1 - (\frac{1}{\frac{KE}{mc²} + 1})²)}</math><br><br />
#<math>v=c*\sqrt{1 - (\frac{1}{\frac{KE}{mc²} + 1})²}</math><br><br />
#<math>v=3X10^8 m/s*\sqrt{1 - (\frac{1}{\frac{(2.38X10^-15 J)}{(1.67X10^-27 kg)(3X10^8 m/s)²} + 1})²}</math><br><br />
#v = <math>2.999X10^8 m/s</math>. This means that the proton is moving very close to the speed of light and hence our choice to use the relativistic kinetic energy equation was a good one.<br><br />
<br />
==Connectedness==<br />
Kinetic energy is very important in baseball. When a batter hits a ball, kinetic energy from the bat is transferred into the ball. leading to the ball flying out into the field. Assuming solid contact between the ball and the bat, the amount of kinetic energy transferred to the ball is directly proportional to the kinetic energy of the bat. As this is a non-relativistic application, there are only two variables to consider when calculating the kinetic energy of the bad: m and v. While a heaver bat would have greater kinetic energy than a bat of smaller mass traveling at the same velocity, heavier bats are more difficult to swing quickly and generally have lower velocity. Hence, batters must try to choose the a bat that they can swing very fast that still has reasonable mass to it to try to maximize kinetic energy (<math>\frac{1}{2}mv²</math>).<br />
https://www.youtube.com/watch?v=zDcf7eEaP0M<br />
<br />
Kinetic energy also plays a role in horse-back riding. For instance, a horse's jump happens in three parts.<br />
<br />
<br />
First, at the beginning of the jump, the horse tenses its muscles and kicks off the ground to lift into the air, converting chemical energy into horizontal and vertical kinetic energy. While in the air, the horse's horizontal kinetic energy doesn't change, but its vertical kinetic energy is converted into gravitational potential energy until it reaches the apex of the jump. As the horse falls, this gravitational potential is converted back into kinetic energy. Finally, when the horse lands, the horse's legs absorb the impact and slow down slightly by converting the vertical and some of the horizontal kinetic energy into heat.<br />
<br />
Kinetic energy is also important to biomedical engineering. Biomedical engineering, such as the design of prosthetics, must focus on how humans move and produce kinetic energy in order to make the best prosthetic for a person. <br />
<br />
<br />
The biomedical engineer must consider how much energy the person will need to move the prosthetics, how heavy the mass of the prosthetic must be to be efficient, and how much energy must continuously be applied to keep the prosthetic moving.<br />
<br />
==History==<br />
<br />
Kinetic energy can be traced all the way back to Aristotle (384 BCE to 322 BCE) who first proposed the concept of actuality and potentiality (actuality being kinetic energy and potentiality being [[Potential Energy]]). The connection between energy and mv² was first developed in the 1600-1700s by Gottfried Leibniz and Johann Bernoulli, who described it as the "living force." William Gravesande tested this by dropping weights from different heights into a block of clay, discovering a proportionality between penetration depth and impact velocity squared. However, William Thomson is credited for devising the term "kinetic energy" in the mid-1800s.<br />
<br />
== See also ==<br />
<br />
[[Potential Energy]]<br> <br />
[[Rest Mass Energy]]<br><br />
[[Translational, Rotational and Vibrational Energy]]<br />
<br />
===Further reading===<br />
<br />
Matter and Interactions By Ruth W. Chabay, Bruce A. Sherwood - Chapter 9 <br />
<br />
===External links===<br />
<br />
http://www.physicsclassroom.com/class/energy/Lesson-1/Kinetic-Energy<br> <br />
https://en.wikipedia.org/wiki/Kinetic_energy<br />
http://hyperphysics.phy-astr.gsu.edu/hbase/ke.html<br />
http://www.livescience.com/46278-kinetic-energy.html<br />
https://www.khanacademy.org/science/physics/work-and-energy/work-and-energy-tutorial/a/what-is-kinetic-energy<br />
<br />
==References==<br />
Matter and Interactions By Ruth W. Chabay, Bruce A. Sherwood - Chapter 9<br> <br />
http://www.physicsclassroom.com/class/energy/Lesson-1/Kinetic-Energy<br> <br />
https://en.wikipedia.org/wiki/Kinetic_energy<br> <br />
<br />
[[Category:Energy]]</div>Ilukowiecki3http://www.physicsbook.gatech.edu/index.php?title=Kinetic_Energy&diff=38591Kinetic Energy2020-03-24T17:13:37Z<p>Ilukowiecki3: </p>
<hr />
<div>'''Claimed by ilukowiecki3 Spring 2020'''<br />
== Introduction==<br />
Objects in motion have energy associated with them. This energy of motion is called kinetic energy. Kinetic energy, often abbreviated as KE, is usually given in the standard S.I. units of Joules (J). KE is also given in units of kilo Joules (kJ). <math>1 kJ = 1000 J</math>. <math>1 J = 1 kg*(m²/s²)</math>. Other types of energy include [[Rest Mass Energy]] and [[Potential Energy]]. Kinetic energy is also known as the work needed to accelerate a mass from rest to a final velocity. Once the object reaches this speed, the kinetic energy remains constant unless the speed is changed. <br />
<br />
Kinetic energy can take many forms, such as vibrational kinetic energy, rotational kinetic energy, and translational kinetic energy. Vibrational kinetic energy is the energy due to vibrational motion, while rotational kinetic energy is the energy due to rotational motion. Translational kinetic energy, the focus of this wiki page, is energy due to motion from place to place. More information on these three kinds of kinetic motions can be found at [[Translational, Rotational and Vibrational Energy]]. Kinetic energy does not have a direction and is a scalar value. It is not a vector quantity, but is rather a magnitude. Kinetic energy can be also be transferred into other types of energy, such as potential energy or chemical energy. In order to gain kinetic energy, a force must be applied, which causes work to be done on an object. This work causes the object to move, which is referred to as kinetic energy. The value of kinetic energy depends on the relationship between the obect in question and the point of observation/the related frame of reference.<br />
<br />
==Rest Energy==<br />
Rest energy can be defined as the energy of a mass at rest. To find the rest energy of an object/particle, multiply the mass by its velocity squared (E=mv²).<br />
<br />
==Mathematical Model==<br />
The relativistic equation for kinetic energy according to Einstein's Theory of Relativity is <math>KE=mc²(\frac{1}{\sqrt{1-\frac{v²}{c²}}} -1)</math>. However, for cases where an object's velocity is far less than the speed of light (<math>3X10^8 m/s</math>), one can use the simplified kinetic energy formula as an approximation: <math>KE=\frac{1}{2}mv²</math>. In most cases the simplified kinetic energy formula gives a result with only minimal error. However, for near light speed calculations, such as those involving subatomic particles such as electrons, protons, or photons, the relativistic equation must be used. Usually the simplified kinetic energy formula is used to calculate the kinetic energy of an average object, and will be used in almost all cases in introductory physics. <br />
<br />
==A Computational Model==<br />
By [[Conservation of Energy]], energy can be converted, but it cannot be created nor destroyed. Hence, in an isolated system, energy can continuously be converted back and forth between potential and kinetic energy without any loss. The following link is an excellent visualization of energy that can be demonstrated with vpython. The spring will oscillate up and down constantly converting between [[Elastic Potential Energy]] and Kinetic Energy. https://trinket.io/glowscript/87a35d5778<br />
<br />
<br />
<br />
[[File:springkinetic.gif|frame|center]]<br />
<br />
==Examples==<br />
===Simple===<br />
[[File:KEproblem.jpg|400px|center]]<br />
A ball is rolling along a frictionless surface at a constant <math>19 m/s</math>. The ball has mass <math>12 kg</math>. What is the kinetic energy of the ball in Joules?<br />
<br>Solution:<br><br />
#Because the ball's velocity is far less than the speed of light, we can use the simplified kinetic energy formula.<br> <br />
#<math>KE=\frac{1}{2}mv²</math><br> <br />
#<math>KE=\frac{1}{2}(12 kg)*(19 m/s)²</math><br> <br />
#Hence, KE = <math>2166 J</math> or <math>2.166 kJ</math><br><br />
<br />
===Middling===<br />
An electron is moving through space at a constant <math>2.9X10^8 m/s</math>. The electron has mass <math>9.1X10^-31 kg</math>. What is the kinetic energy of the ball in Joules?<br><br />
<br>Solution:<br><br />
#Because the electron's velocity is close to the speed of light, we must use the relativistic kinetic energy formula.<br> <br />
#<math>KE=mc²(\frac{1}{\sqrt{1-\frac{v²}{c²}}} -1)</math><br><br />
#<math>KE=(9.1X10^-31 kg)(3X10^8 m/s)²(\frac{1}{\sqrt{1-\frac{(2.9X10^8 m/s)²}{(3X10^8 m/s)²}}} -1)</math><br><br />
#Hence, KE = <math>2.38X10^-13 J</math> or <math>2.38X10^-16 kJ</math><br><br />
===Difficult===<br />
An proton is moving through space at constant velocity. It is found to have kinetic energy <math>2.38X10^-15 J</math>. The proton has mass <math>1.67X10^-27 kg</math>. What is the proton's velocity?<br><br />
<br>Solution:<br><br />
#Because we are dealing with a subatomic particle, we should probably use the relativistic kinetic energy formula as the approximate kinetic energy formula may be very inaccurate if the particle is moving at a velocity near the speed of sound.<br> <br />
#<math>KE=mc²(\frac{1}{\sqrt{1-\frac{v²}{c²}}} -1)</math><br><br />
#<math>\frac{KE}{mc²}=(\frac{1}{\sqrt{1-\frac{v²}{c²}}} -1)</math><br><br />
#<math>\frac{KE}{mc²} + 1=\frac{1}{\sqrt{1-\frac{v²}{c²}}}</math><br><br />
#<math>\frac{1}{\frac{KE}{mc²} + 1}=\sqrt{1-\frac{v²}{c²}}</math><br><br />
#<math>1-\frac{v²}{c²}=(\frac{1}{\frac{KE}{mc²} + 1})²</math><br><br />
#<math>\frac{v²}{c²}=1 - (\frac{1}{\frac{KE}{mc²} + 1})²</math><br><br />
#<math>v²=c²*(1 - (\frac{1}{\frac{KE}{mc²} + 1})²)</math><br><br />
#<math>v=\sqrt{c²*(1 - (\frac{1}{\frac{KE}{mc²} + 1})²)}</math><br><br />
#<math>v=c*\sqrt{1 - (\frac{1}{\frac{KE}{mc²} + 1})²}</math><br><br />
#<math>v=3X10^8 m/s*\sqrt{1 - (\frac{1}{\frac{(2.38X10^-15 J)}{(1.67X10^-27 kg)(3X10^8 m/s)²} + 1})²}</math><br><br />
#v = <math>2.999X10^8 m/s</math>. This means that the proton is moving very close to the speed of light and hence our choice to use the relativistic kinetic energy equation was a good one.<br><br />
<br />
==Connectedness==<br />
Kinetic energy is very important in baseball. When a batter hits a ball, kinetic energy from the bat is transferred into the ball. leading to the ball flying out into the field. Assuming solid contact between the ball and the bat, the amount of kinetic energy transferred to the ball is directly proportional to the kinetic energy of the bat. As this is a non-relativistic application, there are only two variables to consider when calculating the kinetic energy of the bad: m and v. While a heaver bat would have greater kinetic energy than a bat of smaller mass traveling at the same velocity, heavier bats are more difficult to swing quickly and generally have lower velocity. Hence, batters must try to choose the a bat that they can swing very fast that still has reasonable mass to it to try to maximize kinetic energy (<math>\frac{1}{2}mv²</math>).<br />
https://www.youtube.com/watch?v=zDcf7eEaP0M<br />
<br />
Kinetic energy also plays a role in horse-back riding. For instance, a horse's jump happens in three parts.<br />
<br />
<br />
First, at the beginning of the jump, the horse tenses its muscles and kicks off the ground to lift into the air, converting chemical energy into horizontal and vertical kinetic energy. While in the air, the horse's horizontal kinetic energy doesn't change, but its vertical kinetic energy is converted into gravitational potential energy until it reaches the apex of the jump. As the horse falls, this gravitational potential is converted back into kinetic energy. Finally, when the horse lands, the horse's legs absorb the impact and slow down slightly by converting the vertical and some of the horizontal kinetic energy into heat.<br />
<br />
Kinetic energy is also important to biomedical engineering. Biomedical engineering, such as the design of prosthetics, must focus on how humans move and produce kinetic energy in order to make the best prosthetic for a person. <br />
<br />
<br />
The biomedical engineer must consider how much energy the person will need to move the prosthetics, how heavy the mass of the prosthetic must be to be efficient, and how much energy must continuously be applied to keep the prosthetic moving.<br />
<br />
==History==<br />
<br />
Kinetic energy can be traced all the way back to Aristotle (384 BCE to 322 BCE) who first proposed the concept of actuality and potentiality (actuality being kinetic energy and potentiality being [[Potential Energy]]). The connection between energy and mv² was first developed in the 1600-1700's by Gottfried Leibniz and Johann Bernoulli, whom described it as the "living force." William Gravesande tested this by dropping weights from different heights into a block of clay, discovering a proportionality between penetration depth and impact velocity squared. However, William Thomson is credited for devising the term "kinetic energy" in the mid 1800's.<br />
<br />
== See also ==<br />
<br />
[[Potential Energy]]<br> <br />
[[Rest Mass Energy]]<br><br />
[[Translational, Rotational and Vibrational Energy]]<br />
<br />
===Further reading===<br />
<br />
Matter and Interactions By Ruth W. Chabay, Bruce A. Sherwood - Chapter 9 <br />
<br />
===External links===<br />
<br />
http://www.physicsclassroom.com/class/energy/Lesson-1/Kinetic-Energy<br> <br />
https://en.wikipedia.org/wiki/Kinetic_energy<br />
http://hyperphysics.phy-astr.gsu.edu/hbase/ke.html<br />
http://www.livescience.com/46278-kinetic-energy.html<br />
https://www.khanacademy.org/science/physics/work-and-energy/work-and-energy-tutorial/a/what-is-kinetic-energy<br />
<br />
==References==<br />
Matter and Interactions By Ruth W. Chabay, Bruce A. Sherwood - Chapter 9<br> <br />
http://www.physicsclassroom.com/class/energy/Lesson-1/Kinetic-Energy<br> <br />
https://en.wikipedia.org/wiki/Kinetic_energy<br> <br />
<br />
[[Category:Energy]]</div>Ilukowiecki3http://www.physicsbook.gatech.edu/index.php?title=Kinetic_Energy&diff=38590Kinetic Energy2020-03-24T17:12:45Z<p>Ilukowiecki3: </p>
<hr />
<div>'''Claimed by ilukowiecki3 Spring 2020'''<br />
== Introduction==<br />
Objects in motion have energy associated with them. This energy of motion is called kinetic energy. Kinetic energy, often abbreviated as KE, is usually given in the standard S.I. units of Joules (J). KE is also given in units of kilo Joules (kJ). <math>1 kJ = 1000 J</math>. <math>1 J = 1 kg*(m²/s²)</math>. Other types of energy include [[Rest Mass Energy]] and [[Potential Energy]]. Kinetic energy is also known as the work needed to accelerate a mass from rest to a final velocity. Once the object reaches this speed, the kinetic energy remains constant unless the speed is changed. <br />
<br />
Kinetic energy can take many forms, such as vibrational kinetic energy, rotational kinetic energy, and translational kinetic energy. Vibrational kinetic energy is the energy due to vibrational motion, while rotational kinetic energy is the energy due to rotational motion. Translational kinetic energy, the focus of this wiki page, is energy due to motion from place to place. More information on these three kinds of kinetic motions can be found at [[Translational, Rotational and Vibrational Energy]]. Kinetic energy does not have a direction and is a scalar value. It is not a vector quantity, but is rather a magnitude. Kinetic energy can be also be transferred into other types of energy, such as potential energy or chemical energy. In order to gain kinetic energy, a force must be applied, which causes work to be done on an object. This work causes the object to move, which is referred to as kinetic energy. The value of kinetic energy depends on the relationship between the obect in question and the point of observation/the related frame of reference.<br />
<br />
==Rest Energy==<br />
Rest energy can be defined as the energy of a mass at rest. To find the rest energy of an object/particle, multiply the mass by its velocity squared (<math>E=mv²<math>).<br />
<br />
==Mathematical Model==<br />
The relativistic equation for kinetic energy according to Einstein's Theory of Relativity is <math>KE=mc²(\frac{1}{\sqrt{1-\frac{v²}{c²}}} -1)</math>. However, for cases where an object's velocity is far less than the speed of light (<math>3X10^8 m/s</math>), one can use the simplified kinetic energy formula as an approximation: <math>KE=\frac{1}{2}mv²</math>. In most cases the simplified kinetic energy formula gives a result with only minimal error. However, for near light speed calculations, such as those involving subatomic particles such as electrons, protons, or photons, the relativistic equation must be used. Usually the simplified kinetic energy formula is used to calculate the kinetic energy of an average object, and will be used in almost all cases in introductory physics. <br />
<br />
==A Computational Model==<br />
By [[Conservation of Energy]], energy can be converted, but it cannot be created nor destroyed. Hence, in an isolated system, energy can continuously be converted back and forth between potential and kinetic energy without any loss. The following link is an excellent visualization of energy that can be demonstrated with vpython. The spring will oscillate up and down constantly converting between [[Elastic Potential Energy]] and Kinetic Energy. https://trinket.io/glowscript/87a35d5778<br />
<br />
<br />
<br />
[[File:springkinetic.gif|frame|center]]<br />
<br />
==Examples==<br />
===Simple===<br />
[[File:KEproblem.jpg|400px|center]]<br />
A ball is rolling along a frictionless surface at a constant <math>19 m/s</math>. The ball has mass <math>12 kg</math>. What is the kinetic energy of the ball in Joules?<br />
<br>Solution:<br><br />
#Because the ball's velocity is far less than the speed of light, we can use the simplified kinetic energy formula.<br> <br />
#<math>KE=\frac{1}{2}mv²</math><br> <br />
#<math>KE=\frac{1}{2}(12 kg)*(19 m/s)²</math><br> <br />
#Hence, KE = <math>2166 J</math> or <math>2.166 kJ</math><br><br />
<br />
===Middling===<br />
An electron is moving through space at a constant <math>2.9X10^8 m/s</math>. The electron has mass <math>9.1X10^-31 kg</math>. What is the kinetic energy of the ball in Joules?<br><br />
<br>Solution:<br><br />
#Because the electron's velocity is close to the speed of light, we must use the relativistic kinetic energy formula.<br> <br />
#<math>KE=mc²(\frac{1}{\sqrt{1-\frac{v²}{c²}}} -1)</math><br><br />
#<math>KE=(9.1X10^-31 kg)(3X10^8 m/s)²(\frac{1}{\sqrt{1-\frac{(2.9X10^8 m/s)²}{(3X10^8 m/s)²}}} -1)</math><br><br />
#Hence, KE = <math>2.38X10^-13 J</math> or <math>2.38X10^-16 kJ</math><br><br />
===Difficult===<br />
An proton is moving through space at constant velocity. It is found to have kinetic energy <math>2.38X10^-15 J</math>. The proton has mass <math>1.67X10^-27 kg</math>. What is the proton's velocity?<br><br />
<br>Solution:<br><br />
#Because we are dealing with a subatomic particle, we should probably use the relativistic kinetic energy formula as the approximate kinetic energy formula may be very inaccurate if the particle is moving at a velocity near the speed of sound.<br> <br />
#<math>KE=mc²(\frac{1}{\sqrt{1-\frac{v²}{c²}}} -1)</math><br><br />
#<math>\frac{KE}{mc²}=(\frac{1}{\sqrt{1-\frac{v²}{c²}}} -1)</math><br><br />
#<math>\frac{KE}{mc²} + 1=\frac{1}{\sqrt{1-\frac{v²}{c²}}}</math><br><br />
#<math>\frac{1}{\frac{KE}{mc²} + 1}=\sqrt{1-\frac{v²}{c²}}</math><br><br />
#<math>1-\frac{v²}{c²}=(\frac{1}{\frac{KE}{mc²} + 1})²</math><br><br />
#<math>\frac{v²}{c²}=1 - (\frac{1}{\frac{KE}{mc²} + 1})²</math><br><br />
#<math>v²=c²*(1 - (\frac{1}{\frac{KE}{mc²} + 1})²)</math><br><br />
#<math>v=\sqrt{c²*(1 - (\frac{1}{\frac{KE}{mc²} + 1})²)}</math><br><br />
#<math>v=c*\sqrt{1 - (\frac{1}{\frac{KE}{mc²} + 1})²}</math><br><br />
#<math>v=3X10^8 m/s*\sqrt{1 - (\frac{1}{\frac{(2.38X10^-15 J)}{(1.67X10^-27 kg)(3X10^8 m/s)²} + 1})²}</math><br><br />
#v = <math>2.999X10^8 m/s</math>. This means that the proton is moving very close to the speed of light and hence our choice to use the relativistic kinetic energy equation was a good one.<br><br />
<br />
==Connectedness==<br />
Kinetic energy is very important in baseball. When a batter hits a ball, kinetic energy from the bat is transferred into the ball. leading to the ball flying out into the field. Assuming solid contact between the ball and the bat, the amount of kinetic energy transferred to the ball is directly proportional to the kinetic energy of the bat. As this is a non-relativistic application, there are only two variables to consider when calculating the kinetic energy of the bad: m and v. While a heaver bat would have greater kinetic energy than a bat of smaller mass traveling at the same velocity, heavier bats are more difficult to swing quickly and generally have lower velocity. Hence, batters must try to choose the a bat that they can swing very fast that still has reasonable mass to it to try to maximize kinetic energy (<math>\frac{1}{2}mv²</math>).<br />
https://www.youtube.com/watch?v=zDcf7eEaP0M<br />
<br />
Kinetic energy also plays a role in horse-back riding. For instance, a horse's jump happens in three parts.<br />
<br />
<br />
First, at the beginning of the jump, the horse tenses its muscles and kicks off the ground to lift into the air, converting chemical energy into horizontal and vertical kinetic energy. While in the air, the horse's horizontal kinetic energy doesn't change, but its vertical kinetic energy is converted into gravitational potential energy until it reaches the apex of the jump. As the horse falls, this gravitational potential is converted back into kinetic energy. Finally, when the horse lands, the horse's legs absorb the impact and slow down slightly by converting the vertical and some of the horizontal kinetic energy into heat.<br />
<br />
Kinetic energy is also important to biomedical engineering. Biomedical engineering, such as the design of prosthetics, must focus on how humans move and produce kinetic energy in order to make the best prosthetic for a person. <br />
<br />
<br />
The biomedical engineer must consider how much energy the person will need to move the prosthetics, how heavy the mass of the prosthetic must be to be efficient, and how much energy must continuously be applied to keep the prosthetic moving.<br />
<br />
==History==<br />
<br />
Kinetic energy can be traced all the way back to Aristotle (384 BCE to 322 BCE) who first proposed the concept of actuality and potentiality (actuality being kinetic energy and potentiality being [[Potential Energy]]). The connection between energy and mv² was first developed in the 1600-1700's by Gottfried Leibniz and Johann Bernoulli, whom described it as the "living force." William Gravesande tested this by dropping weights from different heights into a block of clay, discovering a proportionality between penetration depth and impact velocity squared. However, William Thomson is credited for devising the term "kinetic energy" in the mid 1800's.<br />
<br />
== See also ==<br />
<br />
[[Potential Energy]]<br> <br />
[[Rest Mass Energy]]<br><br />
[[Translational, Rotational and Vibrational Energy]]<br />
<br />
===Further reading===<br />
<br />
Matter and Interactions By Ruth W. Chabay, Bruce A. Sherwood - Chapter 9 <br />
<br />
===External links===<br />
<br />
http://www.physicsclassroom.com/class/energy/Lesson-1/Kinetic-Energy<br> <br />
https://en.wikipedia.org/wiki/Kinetic_energy<br />
http://hyperphysics.phy-astr.gsu.edu/hbase/ke.html<br />
http://www.livescience.com/46278-kinetic-energy.html<br />
https://www.khanacademy.org/science/physics/work-and-energy/work-and-energy-tutorial/a/what-is-kinetic-energy<br />
<br />
==References==<br />
Matter and Interactions By Ruth W. Chabay, Bruce A. Sherwood - Chapter 9<br> <br />
http://www.physicsclassroom.com/class/energy/Lesson-1/Kinetic-Energy<br> <br />
https://en.wikipedia.org/wiki/Kinetic_energy<br> <br />
<br />
[[Category:Energy]]</div>Ilukowiecki3http://www.physicsbook.gatech.edu/index.php?title=Kinetic_Energy&diff=38589Kinetic Energy2020-03-24T17:08:45Z<p>Ilukowiecki3: </p>
<hr />
<div>'''Claimed by ilukowiecki3 Spring 2020'''<br />
== Introduction==<br />
Objects in motion have energy associated with them. This energy of motion is called kinetic energy. Kinetic energy, often abbreviated as KE, is usually given in the standard S.I. units of Joules (J). KE is also given in units of kilo Joules (kJ). <math>1 kJ = 1000 J</math>. <math>1 J = 1 kg*(m²/s²)</math>. Other types of energy include [[Rest Mass Energy]] and [[Potential Energy]]. Kinetic energy is also known as the work needed to accelerate a mass from rest to a final velocity. Once the object reaches this speed, the kinetic energy remains constant unless the speed is changed. <br />
<br />
Kinetic energy can take many forms, such as vibrational kinetic energy, rotational kinetic energy, and translational kinetic energy. Vibrational kinetic energy is the energy due to vibrational motion, while rotational kinetic energy is the energy due to rotational motion. Translational kinetic energy, the focus of this wiki page, is energy due to motion from place to place. More information on these three kinds of kinetic motions can be found at [[Translational, Rotational and Vibrational Energy]]. Kinetic energy does not have a direction and is a scalar value. It is not a vector quantity, but is rather a magnitude. Kinetic energy can be also be transferred into other types of energy, such as potential energy or chemical energy. In order to gain kinetic energy, a force must be applied, which causes work to be done on an object. This work causes the object to move, which is referred to as kinetic energy. The value of kinetic energy depends on the relationship between the obect in question and the point of observation/the related frame of reference.<br />
<br />
==Rest Energy==<br />
<br />
==Mathematical Model==<br />
The relativistic equation for kinetic energy according to Einstein's Theory of Relativity is <math>KE=mc²(\frac{1}{\sqrt{1-\frac{v²}{c²}}} -1)</math>. However, for cases where an object's velocity is far less than the speed of light (<math>3X10^8 m/s</math>), one can use the simplified kinetic energy formula as an approximation: <math>KE=\frac{1}{2}mv²</math>. In most cases the simplified kinetic energy formula gives a result with only minimal error. However, for near light speed calculations, such as those involving subatomic particles such as electrons, protons, or photons, the relativistic equation must be used. Usually the simplified kinetic energy formula is used to calculate the kinetic energy of an average object, and will be used in almost all cases in introductory physics. <br />
<br />
==A Computational Model==<br />
By [[Conservation of Energy]], energy can be converted, but it cannot be created nor destroyed. Hence, in an isolated system, energy can continuously be converted back and forth between potential and kinetic energy without any loss. The following link is an excellent visualization of energy that can be demonstrated with vpython. The spring will oscillate up and down constantly converting between [[Elastic Potential Energy]] and Kinetic Energy. https://trinket.io/glowscript/87a35d5778<br />
<br />
<br />
<br />
[[File:springkinetic.gif|frame|center]]<br />
<br />
==Examples==<br />
===Simple===<br />
[[File:KEproblem.jpg|400px|center]]<br />
A ball is rolling along a frictionless surface at a constant <math>19 m/s</math>. The ball has mass <math>12 kg</math>. What is the kinetic energy of the ball in Joules?<br />
<br>Solution:<br><br />
#Because the ball's velocity is far less than the speed of light, we can use the simplified kinetic energy formula.<br> <br />
#<math>KE=\frac{1}{2}mv²</math><br> <br />
#<math>KE=\frac{1}{2}(12 kg)*(19 m/s)²</math><br> <br />
#Hence, KE = <math>2166 J</math> or <math>2.166 kJ</math><br><br />
<br />
===Middling===<br />
An electron is moving through space at a constant <math>2.9X10^8 m/s</math>. The electron has mass <math>9.1X10^-31 kg</math>. What is the kinetic energy of the ball in Joules?<br><br />
<br>Solution:<br><br />
#Because the electron's velocity is close to the speed of light, we must use the relativistic kinetic energy formula.<br> <br />
#<math>KE=mc²(\frac{1}{\sqrt{1-\frac{v²}{c²}}} -1)</math><br><br />
#<math>KE=(9.1X10^-31 kg)(3X10^8 m/s)²(\frac{1}{\sqrt{1-\frac{(2.9X10^8 m/s)²}{(3X10^8 m/s)²}}} -1)</math><br><br />
#Hence, KE = <math>2.38X10^-13 J</math> or <math>2.38X10^-16 kJ</math><br><br />
===Difficult===<br />
An proton is moving through space at constant velocity. It is found to have kinetic energy <math>2.38X10^-15 J</math>. The proton has mass <math>1.67X10^-27 kg</math>. What is the proton's velocity?<br><br />
<br>Solution:<br><br />
#Because we are dealing with a subatomic particle, we should probably use the relativistic kinetic energy formula as the approximate kinetic energy formula may be very inaccurate if the particle is moving at a velocity near the speed of sound.<br> <br />
#<math>KE=mc²(\frac{1}{\sqrt{1-\frac{v²}{c²}}} -1)</math><br><br />
#<math>\frac{KE}{mc²}=(\frac{1}{\sqrt{1-\frac{v²}{c²}}} -1)</math><br><br />
#<math>\frac{KE}{mc²} + 1=\frac{1}{\sqrt{1-\frac{v²}{c²}}}</math><br><br />
#<math>\frac{1}{\frac{KE}{mc²} + 1}=\sqrt{1-\frac{v²}{c²}}</math><br><br />
#<math>1-\frac{v²}{c²}=(\frac{1}{\frac{KE}{mc²} + 1})²</math><br><br />
#<math>\frac{v²}{c²}=1 - (\frac{1}{\frac{KE}{mc²} + 1})²</math><br><br />
#<math>v²=c²*(1 - (\frac{1}{\frac{KE}{mc²} + 1})²)</math><br><br />
#<math>v=\sqrt{c²*(1 - (\frac{1}{\frac{KE}{mc²} + 1})²)}</math><br><br />
#<math>v=c*\sqrt{1 - (\frac{1}{\frac{KE}{mc²} + 1})²}</math><br><br />
#<math>v=3X10^8 m/s*\sqrt{1 - (\frac{1}{\frac{(2.38X10^-15 J)}{(1.67X10^-27 kg)(3X10^8 m/s)²} + 1})²}</math><br><br />
#v = <math>2.999X10^8 m/s</math>. This means that the proton is moving very close to the speed of light and hence our choice to use the relativistic kinetic energy equation was a good one.<br><br />
<br />
==Connectedness==<br />
Kinetic energy is very important in baseball. When a batter hits a ball, kinetic energy from the bat is transferred into the ball. leading to the ball flying out into the field. Assuming solid contact between the ball and the bat, the amount of kinetic energy transferred to the ball is directly proportional to the kinetic energy of the bat. As this is a non-relativistic application, there are only two variables to consider when calculating the kinetic energy of the bad: m and v. While a heaver bat would have greater kinetic energy than a bat of smaller mass traveling at the same velocity, heavier bats are more difficult to swing quickly and generally have lower velocity. Hence, batters must try to choose the a bat that they can swing very fast that still has reasonable mass to it to try to maximize kinetic energy (<math>\frac{1}{2}mv²</math>).<br />
https://www.youtube.com/watch?v=zDcf7eEaP0M<br />
<br />
Kinetic energy also plays a role in horse-back riding. For instance, a horse's jump happens in three parts.<br />
<br />
<br />
First, at the beginning of the jump, the horse tenses its muscles and kicks off the ground to lift into the air, converting chemical energy into horizontal and vertical kinetic energy. While in the air, the horse's horizontal kinetic energy doesn't change, but its vertical kinetic energy is converted into gravitational potential energy until it reaches the apex of the jump. As the horse falls, this gravitational potential is converted back into kinetic energy. Finally, when the horse lands, the horse's legs absorb the impact and slow down slightly by converting the vertical and some of the horizontal kinetic energy into heat.<br />
<br />
Kinetic energy is also important to biomedical engineering. Biomedical engineering, such as the design of prosthetics, must focus on how humans move and produce kinetic energy in order to make the best prosthetic for a person. <br />
<br />
<br />
The biomedical engineer must consider how much energy the person will need to move the prosthetics, how heavy the mass of the prosthetic must be to be efficient, and how much energy must continuously be applied to keep the prosthetic moving.<br />
<br />
==History==<br />
<br />
Kinetic energy can be traced all the way back to Aristotle (384 BCE to 322 BCE) who first proposed the concept of actuality and potentiality (actuality being kinetic energy and potentiality being [[Potential Energy]]). The connection between energy and mv² was first developed in the 1600-1700's by Gottfried Leibniz and Johann Bernoulli, whom described it as the "living force." William Gravesande tested this by dropping weights from different heights into a block of clay, discovering a proportionality between penetration depth and impact velocity squared. However, William Thomson is credited for devising the term "kinetic energy" in the mid 1800's.<br />
<br />
== See also ==<br />
<br />
[[Potential Energy]]<br> <br />
[[Rest Mass Energy]]<br><br />
[[Translational, Rotational and Vibrational Energy]]<br />
<br />
===Further reading===<br />
<br />
Matter and Interactions By Ruth W. Chabay, Bruce A. Sherwood - Chapter 9 <br />
<br />
===External links===<br />
<br />
http://www.physicsclassroom.com/class/energy/Lesson-1/Kinetic-Energy<br> <br />
https://en.wikipedia.org/wiki/Kinetic_energy<br />
http://hyperphysics.phy-astr.gsu.edu/hbase/ke.html<br />
http://www.livescience.com/46278-kinetic-energy.html<br />
https://www.khanacademy.org/science/physics/work-and-energy/work-and-energy-tutorial/a/what-is-kinetic-energy<br />
<br />
==References==<br />
Matter and Interactions By Ruth W. Chabay, Bruce A. Sherwood - Chapter 9<br> <br />
http://www.physicsclassroom.com/class/energy/Lesson-1/Kinetic-Energy<br> <br />
https://en.wikipedia.org/wiki/Kinetic_energy<br> <br />
<br />
[[Category:Energy]]</div>Ilukowiecki3http://www.physicsbook.gatech.edu/index.php?title=Kinetic_Energy&diff=38588Kinetic Energy2020-03-24T17:05:01Z<p>Ilukowiecki3: </p>
<hr />
<div>'''Claimed by ilukowiecki3 Spring 2020'''<br />
== Introduction==<br />
Objects in motion have energy associated with them. This energy of motion is called kinetic energy. Kinetic energy, often abbreviated as KE, is usually given in the standard S.I. units of Joules (J). KE is also given in units of kilo Joules (kJ). <math>1 kJ = 1000 J</math>. <math>1 J = 1 kg*(m²/s²)</math>. Other types of energy include [[Rest Mass Energy]] and [[Potential Energy]]. Kinetic energy is also known as the work needed to accelerate a mass from rest to a final velocity. Once the object reaches this speed, the kinetic energy remains constant unless the speed is changed. <br />
<br />
Kinetic energy can take many forms, such as vibrational kinetic energy, rotational kinetic energy, and translational kinetic energy. Vibrational kinetic energy is the energy due to vibrational motion, while rotational kinetic energy is the energy due to rotational motion. Translational kinetic energy, the focus of this wiki page, is energy due to motion from place to place. More information on these three kinds of kinetic motions can be found at [[Translational, Rotational and Vibrational Energy]]. Kinetic energy does not have a direction and is a scalar value. It is not a vector quantity, but is rather a magnitude. Kinetic energy can be also be transferred into other types of energy, such as potential energy or chemical energy. In order to gain kinetic energy, a force must be applied, which causes work to be done on an object. This work causes the object to move, which is referred to as kinetic energy. The value of kinetic energy depends on the relationship between the obect in question and the point of observation/the related frame of reference.<br />
<br />
==Mathematical Model==<br />
The relativistic equation for kinetic energy according to Einstein's Theory of Relativity is <math>KE=mc²(\frac{1}{\sqrt{1-\frac{v²}{c²}}} -1)</math>. However, for cases where an object's velocity is far less than the speed of light (<math>3X10^8 m/s</math>), one can use the simplified kinetic energy formula as an approximation: <math>KE=\frac{1}{2}mv²</math>. In most cases the simplified kinetic energy formula gives a result with only minimal error. However, for near light speed calculations, such as those involving subatomic particles such as electrons, protons, or photons, the relativistic equation must be used. Usually the simplified kinetic energy formula is used to calculate the kinetic energy of an average object, and will be used in almost all cases in introductory physics. <br />
<br />
==A Computational Model==<br />
By [[Conservation of Energy]], energy can be converted, but it cannot be created nor destroyed. Hence, in an isolated system, energy can continuously be converted back and forth between potential and kinetic energy without any loss. The following link is an excellent visualization of energy that can be demonstrated with vpython. The spring will oscillate up and down constantly converting between [[Elastic Potential Energy]] and Kinetic Energy. https://trinket.io/glowscript/87a35d5778<br />
<br />
<br />
<br />
[[File:springkinetic.gif|frame|center]]<br />
<br />
==Examples==<br />
===Simple===<br />
[[File:KEproblem.jpg|400px|center]]<br />
A ball is rolling along a frictionless surface at a constant <math>19 m/s</math>. The ball has mass <math>12 kg</math>. What is the kinetic energy of the ball in Joules?<br />
<br>Solution:<br><br />
#Because the ball's velocity is far less than the speed of light, we can use the simplified kinetic energy formula.<br> <br />
#<math>KE=\frac{1}{2}mv²</math><br> <br />
#<math>KE=\frac{1}{2}(12 kg)*(19 m/s)²</math><br> <br />
#Hence, KE = <math>2166 J</math> or <math>2.166 kJ</math><br><br />
<br />
===Middling===<br />
An electron is moving through space at a constant <math>2.9X10^8 m/s</math>. The electron has mass <math>9.1X10^-31 kg</math>. What is the kinetic energy of the ball in Joules?<br><br />
<br>Solution:<br><br />
#Because the electron's velocity is close to the speed of light, we must use the relativistic kinetic energy formula.<br> <br />
#<math>KE=mc²(\frac{1}{\sqrt{1-\frac{v²}{c²}}} -1)</math><br><br />
#<math>KE=(9.1X10^-31 kg)(3X10^8 m/s)²(\frac{1}{\sqrt{1-\frac{(2.9X10^8 m/s)²}{(3X10^8 m/s)²}}} -1)</math><br><br />
#Hence, KE = <math>2.38X10^-13 J</math> or <math>2.38X10^-16 kJ</math><br><br />
===Difficult===<br />
An proton is moving through space at constant velocity. It is found to have kinetic energy <math>2.38X10^-15 J</math>. The proton has mass <math>1.67X10^-27 kg</math>. What is the proton's velocity?<br><br />
<br>Solution:<br><br />
#Because we are dealing with a subatomic particle, we should probably use the relativistic kinetic energy formula as the approximate kinetic energy formula may be very inaccurate if the particle is moving at a velocity near the speed of sound.<br> <br />
#<math>KE=mc²(\frac{1}{\sqrt{1-\frac{v²}{c²}}} -1)</math><br><br />
#<math>\frac{KE}{mc²}=(\frac{1}{\sqrt{1-\frac{v²}{c²}}} -1)</math><br><br />
#<math>\frac{KE}{mc²} + 1=\frac{1}{\sqrt{1-\frac{v²}{c²}}}</math><br><br />
#<math>\frac{1}{\frac{KE}{mc²} + 1}=\sqrt{1-\frac{v²}{c²}}</math><br><br />
#<math>1-\frac{v²}{c²}=(\frac{1}{\frac{KE}{mc²} + 1})²</math><br><br />
#<math>\frac{v²}{c²}=1 - (\frac{1}{\frac{KE}{mc²} + 1})²</math><br><br />
#<math>v²=c²*(1 - (\frac{1}{\frac{KE}{mc²} + 1})²)</math><br><br />
#<math>v=\sqrt{c²*(1 - (\frac{1}{\frac{KE}{mc²} + 1})²)}</math><br><br />
#<math>v=c*\sqrt{1 - (\frac{1}{\frac{KE}{mc²} + 1})²}</math><br><br />
#<math>v=3X10^8 m/s*\sqrt{1 - (\frac{1}{\frac{(2.38X10^-15 J)}{(1.67X10^-27 kg)(3X10^8 m/s)²} + 1})²}</math><br><br />
#v = <math>2.999X10^8 m/s</math>. This means that the proton is moving very close to the speed of light and hence our choice to use the relativistic kinetic energy equation was a good one.<br><br />
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==Connectedness==<br />
Kinetic energy is very important in baseball. When a batter hits a ball, kinetic energy from the bat is transferred into the ball. leading to the ball flying out into the field. Assuming solid contact between the ball and the bat, the amount of kinetic energy transferred to the ball is directly proportional to the kinetic energy of the bat. As this is a non-relativistic application, there are only two variables to consider when calculating the kinetic energy of the bad: m and v. While a heaver bat would have greater kinetic energy than a bat of smaller mass traveling at the same velocity, heavier bats are more difficult to swing quickly and generally have lower velocity. Hence, batters must try to choose the a bat that they can swing very fast that still has reasonable mass to it to try to maximize kinetic energy (<math>\frac{1}{2}mv²</math>).<br />
https://www.youtube.com/watch?v=zDcf7eEaP0M<br />
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Kinetic energy also plays a role in horse-back riding. For instance, a horse's jump happens in three parts.<br />
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First, at the beginning of the jump, the horse tenses its muscles and kicks off the ground to lift into the air, converting chemical energy into horizontal and vertical kinetic energy. While in the air, the horse's horizontal kinetic energy doesn't change, but its vertical kinetic energy is converted into gravitational potential energy until it reaches the apex of the jump. As the horse falls, this gravitational potential is converted back into kinetic energy. Finally, when the horse lands, the horse's legs absorb the impact and slow down slightly by converting the vertical and some of the horizontal kinetic energy into heat.<br />
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Kinetic energy is also important to biomedical engineering. Biomedical engineering, such as the design of prosthetics, must focus on how humans move and produce kinetic energy in order to make the best prosthetic for a person. <br />
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The biomedical engineer must consider how much energy the person will need to move the prosthetics, how heavy the mass of the prosthetic must be to be efficient, and how much energy must continuously be applied to keep the prosthetic moving.<br />
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==History==<br />
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Kinetic energy can be traced all the way back to Aristotle (384 BCE to 322 BCE) who first proposed the concept of actuality and potentiality (actuality being kinetic energy and potentiality being [[Potential Energy]]). The connection between energy and mv² was first developed in the 1600-1700's by Gottfried Leibniz and Johann Bernoulli, whom described it as the "living force." William Gravesande tested this by dropping weights from different heights into a block of clay, discovering a proportionality between penetration depth and impact velocity squared. However, William Thomson is credited for devising the term "kinetic energy" in the mid 1800's.<br />
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== See also ==<br />
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[[Potential Energy]]<br> <br />
[[Rest Mass Energy]]<br><br />
[[Translational, Rotational and Vibrational Energy]]<br />
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===Further reading===<br />
<br />
Matter and Interactions By Ruth W. Chabay, Bruce A. Sherwood - Chapter 9 <br />
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===External links===<br />
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http://www.physicsclassroom.com/class/energy/Lesson-1/Kinetic-Energy<br> <br />
https://en.wikipedia.org/wiki/Kinetic_energy<br />
http://hyperphysics.phy-astr.gsu.edu/hbase/ke.html<br />
http://www.livescience.com/46278-kinetic-energy.html<br />
https://www.khanacademy.org/science/physics/work-and-energy/work-and-energy-tutorial/a/what-is-kinetic-energy<br />
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==References==<br />
Matter and Interactions By Ruth W. Chabay, Bruce A. Sherwood - Chapter 9<br> <br />
http://www.physicsclassroom.com/class/energy/Lesson-1/Kinetic-Energy<br> <br />
https://en.wikipedia.org/wiki/Kinetic_energy<br> <br />
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[[Category:Energy]]</div>Ilukowiecki3