Physics Book - User contributions [en]

Iterative Prediction

2019-08-23T22:26:23Z

Mfschatz:

'''MFS 23 aug 19: This page should explicitly indicate the connection to Newton's 2nd Law---The Momentum Principle---the page makes it sound like this is some kind of computational algorithm rather than the more fundamental idea of how does N 2nd law actually work to allow us to predict the motion future----the middling example is not good.....it uses the kinematic equations for constant net force motion as the "analytic" solution'''This page describes iterative prediction, a technique used to predict the motion of particles over a period of time often used in simulations.

==The Main Idea==

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. 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.

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:

<ol>
<li>The [[Net Force]] acting on each particle should be calculated,</li>
<li>The [[Linear Momentum]] of each particle should be updated from the last time step using the newly calculated net force, and</li>
<li>The position of each particle should be updated from the last time step using its newly calculated [[Velocity]].</li>
</ol>

[[File:Iterative_Model.png|center]]

<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>

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.

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.

===A Mathematical Model===

Below are the mathematical formulas required for each of the three steps outlined above.

<ol>

<li>The net force acting on each particle should be calculated.</li>

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.

<li>The momentum of each particle should be updated from the last time step using the newly calculated net force.</li>

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.

<li>The position of each particle should be updated from the last time step using its newly calculated velocity.</li>

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.

</ol>

===A Computational Model===

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 with 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.

[https://www.glowscript.org/#/user/YorickAndeweg/folder/PhysicsBookFolder/program/IterativePrediction1 high resolution ball trajectory simulation]

To demonstrate differing resolutions, here is a lower resolution version of the same simulation, which uses larger time steps:

[https://www.glowscript.org/#/user/YorickAndeweg/folder/PhysicsBookFolder/program/IterativePrediction2 low resolution ball trajectory simulation]

Note that the first program is more realistic; the more frequent position updates result in a smoother curve that more closely approximate 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.

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]].

[https://www.glowscript.org/#/user/YorickAndeweg/folder/PhysicsBookFolder/program/SpringMass spring mass simulation]

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.

==Examples==

In addition to the example below, know how to create vPython simulations using iterative prediction with both constant and varying forces.

===1. (Middling)===

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?

Solve this question

A.) analytically (using kinematic equations)

B.) using iterative prediction with 12 equal time steps

C.) using iterative prediction with 4 equal time steps

D.) using iterative prediction with a single time step

Pay attention to how the resolution of the iterative prediction affects the accuracy of the answer. The analytical answer is completely accurate.

A.) To solve this analytically, the following kinematic equation should be used:

<math>\Delta x = \frac{1}{2} a t^2 + v_0 t</math>

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

<math>\Delta x = \frac{1}{2} (3) (4)^2 = 24</math>m.

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.

[[File:12steps.PNG]]

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.

C.)

[[File:4steps.PNG]]

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.

D.)

[[File:1step.PNG]]

using 1 time step of 4 seconds results in a simulated displacement of 48m, which is twice the actual amount!

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.

==Connectedness==

===Application: video game industry===

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.

==History==

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.

== See also ==

*[[Linear Momentum]]
*[[Impulse and Momentum]]
*[[Analytical Prediction]]
*[[Iterative Prediction of Spring-Mass System]]
*[[Determinism]]

===External links===

*[http://p3server.pa.msu.edu/coursewiki/doku.php?id=183_notes:modeling_with_vpython Iterative Prediction with VPython]

==References==

*Georgia Institute of Technology. Physics Department. PHYS 2211. Fall 2016. '''Wednesday, Week 2 Lecture Slides. Fenton, Flavio H'''
*Georgia Institute of Technology. Physics Department. PHYS 2211. Fall 2016. '''Monday, Week 3 Lecture Slides. Fenton, Flavio H'''
*Georgia Institute of Technology. Physics Department. PHYS 2211. Fall 2016. Lab 07 Fancart Energy & Spring-Mass Instructions, Greco, Edwin
*http://p3server.pa.msu.edu/coursewiki/doku.php?id=183_notes:iterativepredict
*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

Iterative Prediction

2019-08-23T22:18:38Z

Mfschatz:

'''MFS 23 aug 19: This page should explicitly indicate the connection to Newton's 2nd Law---The Momentum Principle---the page makes it sound like this is some kind of computational algorithm rather than the more fundamental idea of how does N 2nd law actually work to allow us to predict the motion future'''This page describes iterative prediction, a technique used to predict the motion of particles over a period of time often used in simulations.

==The Main Idea==

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. 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.

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:

<ol>
<li>The [[Net Force]] acting on each particle should be calculated,</li>
<li>The [[Linear Momentum]] of each particle should be updated from the last time step using the newly calculated net force, and</li>
<li>The position of each particle should be updated from the last time step using its newly calculated [[Velocity]].</li>
</ol>

[[File:Iterative_Model.png|center]]

<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>

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.

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.

===A Mathematical Model===

Below are the mathematical formulas required for each of the three steps outlined above.

<ol>

<li>The net force acting on each particle should be calculated.</li>

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.

<li>The momentum of each particle should be updated from the last time step using the newly calculated net force.</li>

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.

<li>The position of each particle should be updated from the last time step using its newly calculated velocity.</li>

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.

</ol>

===A Computational Model===

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 with 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.

[https://www.glowscript.org/#/user/YorickAndeweg/folder/PhysicsBookFolder/program/IterativePrediction1 high resolution ball trajectory simulation]

To demonstrate differing resolutions, here is a lower resolution version of the same simulation, which uses larger time steps:

[https://www.glowscript.org/#/user/YorickAndeweg/folder/PhysicsBookFolder/program/IterativePrediction2 low resolution ball trajectory simulation]

Note that the first program is more realistic; the more frequent position updates result in a smoother curve that more closely approximate 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.

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]].

[https://www.glowscript.org/#/user/YorickAndeweg/folder/PhysicsBookFolder/program/SpringMass spring mass simulation]

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.

==Examples==

In addition to the example below, know how to create vPython simulations using iterative prediction with both constant and varying forces.

===1. (Middling)===

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?

Solve this question

A.) analytically (using kinematic equations)

B.) using iterative prediction with 12 equal time steps

C.) using iterative prediction with 4 equal time steps

D.) using iterative prediction with a single time step

Pay attention to how the resolution of the iterative prediction affects the accuracy of the answer. The analytical answer is completely accurate.

A.) To solve this analytically, the following kinematic equation should be used:

<math>\Delta x = \frac{1}{2} a t^2 + v_0 t</math>

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

<math>\Delta x = \frac{1}{2} (3) (4)^2 = 24</math>m.

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.

[[File:12steps.PNG]]

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.

C.)

[[File:4steps.PNG]]

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.

D.)

[[File:1step.PNG]]

using 1 time step of 4 seconds results in a simulated displacement of 48m, which is twice the actual amount!

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.

==Connectedness==

===Application: video game industry===

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.

==History==

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.

== See also ==

*[[Linear Momentum]]
*[[Impulse and Momentum]]
*[[Analytical Prediction]]
*[[Iterative Prediction of Spring-Mass System]]
*[[Determinism]]

===External links===

*[http://p3server.pa.msu.edu/coursewiki/doku.php?id=183_notes:modeling_with_vpython Iterative Prediction with VPython]

==References==

*Georgia Institute of Technology. Physics Department. PHYS 2211. Fall 2016. '''Wednesday, Week 2 Lecture Slides. Fenton, Flavio H'''
*Georgia Institute of Technology. Physics Department. PHYS 2211. Fall 2016. '''Monday, Week 3 Lecture Slides. Fenton, Flavio H'''
*Georgia Institute of Technology. Physics Department. PHYS 2211. Fall 2016. Lab 07 Fancart Energy & Spring-Mass Instructions, Greco, Edwin
*http://p3server.pa.msu.edu/coursewiki/doku.php?id=183_notes:iterativepredict
*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

Velocity

2019-08-23T22:12:41Z

Mfschatz:

'''MFS 23 AUG 2019---NEEDS EDITING TO MOTIVATE THE CONCEPT OF VELOCITY FOR NOVICES: HERE'S ONE WAY TO DO THIS...INSERT DISCUSSION THAT INDICATES HOW ONE TAKES OBSERVATIONS OF THE POSITION OF AN OBJECT AT DIFFERENT TIMES AND USES THAT INFORMATION TO OBTAIN A VECTOR QUANTITY WHOSE MAGNITUDE TELLS ONE HOW FAST THE OBJECT IS MOVING AND WHOSE DIRECTION INDICATES THE DIRECTION THAT THE OBJECT IS MOVING.'''This page defines and describes velocity. Be sure to distinguish between velocity and [[Speed|speed]]. See [[Speed vs Velocity]] for similarities and differences.

==Main Idea==

Velocity is a [[Vectors|vector]] quantity that describes both how fast an object is moving and its direction of motion. It is represented by the symbol <math>\vec{v}</math> or v, as opposed to <math>v</math>, which denotes [[Speed|speed]]. Velocity is defined as the rate of change of position with respect to time. In calculus terms, it is the time derivative of the position vector. The magnitude of a velocity vector is speed, and the direction of the vector is the direction of travel. Velocity is an instantaneous value, so it may change over time. Several properties in physics, such as [[Linear Momentum|momentum]] and [[Magnetic Force|magnetic force]], are functions of velocity. Velocity is given in unit distance per unit time. The [[SI Units|SI unit]] for velocity is the meter per second (m/s).

===A Mathematical Model===

Instantaneous velocity <math>\vec{v}</math> is defined as:

<math>\vec{v} = \frac{d\vec{r}}{dt}</math>

where <math>\vec{r}</math> is a position vector and <math>t</math> is time.

====Average Velocity====

Since velocity is an instantaneous quantity, it can change over time. Over any given time interval, there is an average velocity value denoted <math>\vec{v}_{avg}</math>. The average velocity over an interval of time <math>\Delta t</math> is given by

<math>\vec{v}_{avg} = \frac{\Delta \vec{r}}{\Delta t}</math>

where <math>\Delta \vec{r}</math> is the displacement (change in position) over that time interval (<math>\Delta \vec{r} = \vec{r}_f - \vec{r}_i</math>).

Average velocity is often confused with average speed. A section on the [[Speed vs Velocity]] page goes into depth about the difference.

As the duration of the time interval <math>\Delta t</math> becomes very small, the value of the average velocity over that time interval approaches the value of the instantaneous velocity at any point in time within that interval.

If velocity is constant over some time interval, the average velocity over that time interval equals that constant velocity.

Here is another equation giving average velocity, this time in terms of initial velocity <math>\vec{v}_i</math> and final velocity <math>\vec{v}_f</math>:

<math>\vec{v}_{avg} = \frac{\vec{v}_i + \vec{v}_f}{2}</math>

Unlike the first equation, this equation is only true if [[Acceleration|acceleration]] is constant.

====Derivative Relationships====

Velocity is the time derivative of position:

<math>\vec{v}(t) = \frac{d\vec{r}(t)}{dt}</math>.

[[Acceleration]], in turn, is the time derivative of velocity:

<math>\vec{a}(t) = \frac{d\vec{v}(t)}{dt}</math>

====Integral Relationships====

Position is the time integral of velocity:

<math>\vec{r}(t) = \int \vec{v}(t) \ dt</math>.

Integrating the velocity function as a definite integral from one time coordinate to another yields the displacement of the position over that time interval:

<math>\Delta \vec{r}_{t_1,t_2} = \int_{t_1}^{t_2} \vec{v}(t) \ dt</math>.

Velocity is, in turn, the time integral of acceleration:

<math>\vec{v}(t) = \int \vec{a}(t) \ dt</math>.

====Kinematic Equations====

The kinematic equations can be derived from the derivative and integral relationships between acceleration, velocity, and displacement. For the equations and more information, view the [[Kinematics]] page.

====In Physics====

According to [[Newton's First Law of Motion]], the velocity of an object must remain constant (whether that velocity is zero or nonzero) unless a force acts on it. [[Newton's Second Law: the Momentum Principle]] describes how velocity changes over time as a result of forces.

===A Computational Model===

In VPython simulations of physical systems, the [[Iterative Prediction]] algorithm is often used to move objects with specified velocities and evolve those velocities over time. The program below is an example of such a simulation. In this program, the ball's velocity is represented by a purple arrow.

[https://www.glowscript.org/#/user/YorickAndeweg/folder/PhysicsBookFolder/program/Velocity Click here for velocity simulation]

Click "view this program" in the top left corner to view the source code.

==Examples==

===Simple===

At time t=0s, a particle positioned at (4,-1)m began to move. It moved in a straight line at a constant speed until time t=3s, at which point it had arrived at the position (-2,8)m. What was the velocity of the particle at time t=2s?

Solution:

Since the particle moved at a constant speed in a constant direction, the velocity of the particle must have been constant. That is, the velocity of the particle at time t=2 must be the same as its velocity at any other time between t=0 and t=3. This constant velocity must also be the average velocity over the time interval from t=0 to t=3, so let us use the average velocity formula:

<math>\vec{v}_{avg} = \frac{\Delta \vec{r}}{\Delta t}</math>

<math>\vec{v}_{avg} = \frac{<-2, 8> - <4,-1>}{3}</math>

<math>\vec{v}_{avg} = \frac{<-6, 9>}{3}</math>

<math>\vec{v}_{avg} = <-2, 3></math>m/s

therefore

<math>\vec{v}(t=2) = <-2, 3></math>m/s

===Middling===

A particle's position as a function of time is as follows: <math>\vec{r}(t) = <6\sin(2 \pi t), -6t^3 + 10, e^{8t}></math>

What is the particle's velocity as a function of time?

Solution:

<math>\vec{v} = \frac{d \vec{r}}{dt}</math>

<math>\vec{v} = <12 \pi \cos(2 \pi t), -18t^2, 8e^{8t}></math>

===Difficult===

A particle moves along the x axis. At time t=0, its position is x=2. Its velocity <math>v</math> varies over time, obeying the following function:

<math>
v(t) =
\begin{cases}
4-2t & \text{if $t\leq 3$} \\
-2 & \text{if $t>3$}
\end{cases}
</math>m/s

What is its position as a function of time after time t=0?

Solution:

<math>x(t) = x(0) + \int_0^t v(t')dt'</math> (<math>t'</math> is a "dummy variable" since <math>t</math> is already our limit of integration)

for <math>t\leq 3</math>,

<math>x(t) = 2 + \int_0^t 4-2t' dt'</math>

<math>x(t) = 2 + [4t'-t'^2]_0^t</math>

<math>x(t) = 2 + 4t - t^2</math>

for <math>t>3</math>,

<math>x(t) = 2 + \int_0^3 4-2t' dt' + \int_3^t -2 dt'</math>

<math>x(t) = 5 + \int_3^t -2 dt'</math>

<math>x(t) = 5 + [-2t']_3^t</math>

<math>x(t) = 5 -2t + 6</math>

<math>x(t) = 11 - 2t</math>

final answer:

<math>
x(t) =
\begin{cases}
-t^2 + 4t + 2 & \text{if $t\leq 3$} \\
11-2t & \text{if $t>3$}
\end{cases}
</math>m

==Connectedness==

Since the goal of the field of mechanics is to predict the motion of systems, velocity is critical to this branch of physics. In addition to being a value of interest in and of itself, properties such as [[Linear Momentum]], [[Total Angular Momentum|Angular Momentum]], and [[Magnetic Force]] depend on the velocities of objects.

The use of velocity in calculations has a wide range of industrial applications including calculating travel time, determining safe operating speeds of vehicles and other machines, vehicle collision testing, determining the inertial forces experienced by roller coaster riders, and the design of equipment such as radar guns. The velocities of moving parts are important in virtually every type of machinery.

==See Also==

*[[Vectors]]
*[[Speed]]
*[[Speed vs Velocity]]
*[[Acceleration]]
*[[Kinematics]]
*[[Relative Velocity]]
*[[Derivation of Average Velocity]]

===External links===

[http://www.physicsclassroom.com/class/1DKin/Lesson-1/Speed-and-Velocity The Physics Classroom: Speed and Velocity]

[http://hyperphysics.phy-astr.gsu.edu/hbase/vel2.html HyperPhysics: Average Velocity]

[https://www.youtube.com/watch?v=Bxp0AWhs57g YouTube video explaining average vs instantaneous velocity]

==References==

1. Chabay, Ruth W., and Bruce A. Sherwood. Matter and Interactions. Hoboken, NJ: Wiley, 2011. Print.

2. "Velocity." Def. 2. Dictionary.com. N.p., n.d. Web. 29 Nov. 2015.

[[Category:Properties of Matter]]

Velocity

2019-08-23T22:11:42Z

Mfschatz:

This page defines and describes velocity. Be sure to distinguish between velocity and [[Speed|speed]]. See [[Speed vs Velocity]] for similarities and differences.

==Main Idea==
MFS 23 AUG 2019---NEEDS EDITING TO MOTIVATE THE CONCEPT OF VELOCITY FOR NOVICES: HERE'S ONE WAY TO DO THIS...INSERT DISCUSSION THAT INDICATES HOW ONE TAKES OBSERVATIONS OF THE POSITION OF AN OBJECT AT DIFFERENT TIMES AND USES THAT INFORMATION TO OBTAIN A VECTOR QUANTITY WHOSE MAGNITUDE TELLS ONE HOW FAST THE OBJECT IS MOVING AND WHOSE DIRECTION INDICATES THE DIRECTION THAT THE OBJECT IS MOVING.
Velocity is a [[Vectors|vector]] quantity that describes both how fast an object is moving and its direction of motion. It is represented by the symbol <math>\vec{v}</math> or v, as opposed to <math>v</math>, which denotes [[Speed|speed]]. Velocity is defined as the rate of change of position with respect to time. In calculus terms, it is the time derivative of the position vector. The magnitude of a velocity vector is speed, and the direction of the vector is the direction of travel. Velocity is an instantaneous value, so it may change over time. Several properties in physics, such as [[Linear Momentum|momentum]] and [[Magnetic Force|magnetic force]], are functions of velocity. Velocity is given in unit distance per unit time. The [[SI Units|SI unit]] for velocity is the meter per second (m/s).

===A Mathematical Model===

Instantaneous velocity <math>\vec{v}</math> is defined as:

<math>\vec{v} = \frac{d\vec{r}}{dt}</math>

where <math>\vec{r}</math> is a position vector and <math>t</math> is time.

====Average Velocity====

Since velocity is an instantaneous quantity, it can change over time. Over any given time interval, there is an average velocity value denoted <math>\vec{v}_{avg}</math>. The average velocity over an interval of time <math>\Delta t</math> is given by

<math>\vec{v}_{avg} = \frac{\Delta \vec{r}}{\Delta t}</math>

where <math>\Delta \vec{r}</math> is the displacement (change in position) over that time interval (<math>\Delta \vec{r} = \vec{r}_f - \vec{r}_i</math>).

Average velocity is often confused with average speed. A section on the [[Speed vs Velocity]] page goes into depth about the difference.

As the duration of the time interval <math>\Delta t</math> becomes very small, the value of the average velocity over that time interval approaches the value of the instantaneous velocity at any point in time within that interval.

If velocity is constant over some time interval, the average velocity over that time interval equals that constant velocity.

Here is another equation giving average velocity, this time in terms of initial velocity <math>\vec{v}_i</math> and final velocity <math>\vec{v}_f</math>:

<math>\vec{v}_{avg} = \frac{\vec{v}_i + \vec{v}_f}{2}</math>

Unlike the first equation, this equation is only true if [[Acceleration|acceleration]] is constant.

====Derivative Relationships====

Velocity is the time derivative of position:

<math>\vec{v}(t) = \frac{d\vec{r}(t)}{dt}</math>.

[[Acceleration]], in turn, is the time derivative of velocity:

<math>\vec{a}(t) = \frac{d\vec{v}(t)}{dt}</math>

====Integral Relationships====

Position is the time integral of velocity:

<math>\vec{r}(t) = \int \vec{v}(t) \ dt</math>.

Integrating the velocity function as a definite integral from one time coordinate to another yields the displacement of the position over that time interval:

<math>\Delta \vec{r}_{t_1,t_2} = \int_{t_1}^{t_2} \vec{v}(t) \ dt</math>.

Velocity is, in turn, the time integral of acceleration:

<math>\vec{v}(t) = \int \vec{a}(t) \ dt</math>.

====Kinematic Equations====

The kinematic equations can be derived from the derivative and integral relationships between acceleration, velocity, and displacement. For the equations and more information, view the [[Kinematics]] page.

====In Physics====

According to [[Newton's First Law of Motion]], the velocity of an object must remain constant (whether that velocity is zero or nonzero) unless a force acts on it. [[Newton's Second Law: the Momentum Principle]] describes how velocity changes over time as a result of forces.

===A Computational Model===

In VPython simulations of physical systems, the [[Iterative Prediction]] algorithm is often used to move objects with specified velocities and evolve those velocities over time. The program below is an example of such a simulation. In this program, the ball's velocity is represented by a purple arrow.

[https://www.glowscript.org/#/user/YorickAndeweg/folder/PhysicsBookFolder/program/Velocity Click here for velocity simulation]

Click "view this program" in the top left corner to view the source code.

==Examples==

===Simple===

At time t=0s, a particle positioned at (4,-1)m began to move. It moved in a straight line at a constant speed until time t=3s, at which point it had arrived at the position (-2,8)m. What was the velocity of the particle at time t=2s?

Solution:

Since the particle moved at a constant speed in a constant direction, the velocity of the particle must have been constant. That is, the velocity of the particle at time t=2 must be the same as its velocity at any other time between t=0 and t=3. This constant velocity must also be the average velocity over the time interval from t=0 to t=3, so let us use the average velocity formula:

<math>\vec{v}_{avg} = \frac{\Delta \vec{r}}{\Delta t}</math>

<math>\vec{v}_{avg} = \frac{<-2, 8> - <4,-1>}{3}</math>

<math>\vec{v}_{avg} = \frac{<-6, 9>}{3}</math>

<math>\vec{v}_{avg} = <-2, 3></math>m/s

therefore

<math>\vec{v}(t=2) = <-2, 3></math>m/s

===Middling===

A particle's position as a function of time is as follows: <math>\vec{r}(t) = <6\sin(2 \pi t), -6t^3 + 10, e^{8t}></math>

What is the particle's velocity as a function of time?

Solution:

<math>\vec{v} = \frac{d \vec{r}}{dt}</math>

<math>\vec{v} = <12 \pi \cos(2 \pi t), -18t^2, 8e^{8t}></math>

===Difficult===

A particle moves along the x axis. At time t=0, its position is x=2. Its velocity <math>v</math> varies over time, obeying the following function:

<math>
v(t) =
\begin{cases}
4-2t & \text{if $t\leq 3$} \\
-2 & \text{if $t>3$}
\end{cases}
</math>m/s

What is its position as a function of time after time t=0?

Solution:

<math>x(t) = x(0) + \int_0^t v(t')dt'</math> (<math>t'</math> is a "dummy variable" since <math>t</math> is already our limit of integration)

for <math>t\leq 3</math>,

<math>x(t) = 2 + \int_0^t 4-2t' dt'</math>

<math>x(t) = 2 + [4t'-t'^2]_0^t</math>

<math>x(t) = 2 + 4t - t^2</math>

for <math>t>3</math>,

<math>x(t) = 2 + \int_0^3 4-2t' dt' + \int_3^t -2 dt'</math>

<math>x(t) = 5 + \int_3^t -2 dt'</math>

<math>x(t) = 5 + [-2t']_3^t</math>

<math>x(t) = 5 -2t + 6</math>

<math>x(t) = 11 - 2t</math>

final answer:

<math>
x(t) =
\begin{cases}
-t^2 + 4t + 2 & \text{if $t\leq 3$} \\
11-2t & \text{if $t>3$}
\end{cases}
</math>m

==Connectedness==

Since the goal of the field of mechanics is to predict the motion of systems, velocity is critical to this branch of physics. In addition to being a value of interest in and of itself, properties such as [[Linear Momentum]], [[Total Angular Momentum|Angular Momentum]], and [[Magnetic Force]] depend on the velocities of objects.

The use of velocity in calculations has a wide range of industrial applications including calculating travel time, determining safe operating speeds of vehicles and other machines, vehicle collision testing, determining the inertial forces experienced by roller coaster riders, and the design of equipment such as radar guns. The velocities of moving parts are important in virtually every type of machinery.

==See Also==

*[[Vectors]]
*[[Speed]]
*[[Speed vs Velocity]]
*[[Acceleration]]
*[[Kinematics]]
*[[Relative Velocity]]
*[[Derivation of Average Velocity]]

===External links===

[http://www.physicsclassroom.com/class/1DKin/Lesson-1/Speed-and-Velocity The Physics Classroom: Speed and Velocity]

[http://hyperphysics.phy-astr.gsu.edu/hbase/vel2.html HyperPhysics: Average Velocity]

[https://www.youtube.com/watch?v=Bxp0AWhs57g YouTube video explaining average vs instantaneous velocity]

==References==

1. Chabay, Ruth W., and Bruce A. Sherwood. Matter and Interactions. Hoboken, NJ: Wiley, 2011. Print.

2. "Velocity." Def. 2. Dictionary.com. N.p., n.d. Web. 29 Nov. 2015.

[[Category:Properties of Matter]]