Physics Book - User contributions [en]

Iterative Prediction

2016-04-16T06:43:46Z

Jbarua:

Editted by Joshua Barua (jbarua3)

By tconnors3

==An Overview==
Iterative Prediction is a process used to approximate the effects of a force, be it constant or varying, on the motion of an object. This process does this approximation splitting the time interval up into many small intervals, finding the force on the object at each of the intervals, and then calculation the change of momentum and change of position for the object across each of the time intervals. Understanding Iterative Prediction also requires an understanding of the Momentum Principle, because you are constantly using the equation dp = F * deltat to update your momentum followed by your position.

Iterative Prediction is a necessary technique to employ when doing physics because, even though our fundamental principles are based on calculus and derivatives and you should be able to solve these problems using integrals, it is often impossible to solve the integrals required to solve complex problems. Iterative Prediction also becomes much more convenient when you implement it using a program, such as vPython, which allows you to make the approximation use as many steps as your processing power can handle, which can make your prediction much more accurate.

===A Mathematical Model===

By starting from the general equation for the Momentum Principle, a formula can derived to predict the momentum of a given system at a specified point in the future. This is often referred to as the ''momentum update form'' of the Momentum Principle:

<div style="text-align: center;"><math>{{(1)}\frac{d\vec{p}}{dt}}_{system} = \vec{F}_{net}</math></div>

<div style="text-align: center;"><math>{{(2)}\vec{p}_{f} - \vec{p}_{i} = \vec{F}_{net}{Δt}}</math></div>

<div style="text-align: center;"><math>{{(3)}\vec{p}_{f} = \vec{p}_{i} + \vec{F}_{net}{Δt}}</math></div>

Equation <math>{(3)}</math> gives the momentum update form of the momentum principle. As it shows, the momentum of a system can be predicted if the time period of the interaction and the external net force on the system are known. Momentum can be iteratively predicted like such for uniform or non-uniform time periods. Additionally, with update of momentum, velocity and position can be updated similarly to better reflect the motion over time of the system:

<div style="text-align: center;">'''Velocity Update Formula''':

<math>{\vec{v}_{f} = \vec{v}_{i} + \frac{\vec{F}_{net}}{m}}{Δt}</math>

'''Position Update Formula''':

<math>{\vec{r}_{f} = \vec{r}_{i} + \vec{v}_{avg}{Δt}}</math></div>

===A Visual Model===

As the momentum update formula suggests, the final momentum (shown here as <math>\vec{p}_{future}</math>) of a system after a given change in time should be the vector sum of the initial momentum (denoted <math>\vec{p}_{now}</math>) and the net force on the system multiplied by the scalar time change (<math>\vec{F}_{net}{Δt}</math>).
[[Image:MOTION.png|center|650x300px|]]

As the diagram suggests, each individual step can be analyzed through application of the momentum update form of the Momentum Principle. Once the final momentum is calculated, by utilizing the position and velocity update formula, the final velocity and position of the system can be determined at the end of the time interval of interest.

You can observe this Iterative Prediction in action by viewing this Vpython Script[http://www.glowscript.org/#/user/JBarua/folder/Public/program/ProjectilePublic] (http://www.glowscript.org/#/user/JBarua/folder/Public/program/ProjectilePublic). It shows the flight path of a ball given an initial positive x and y velocity until it hits the ground, calculated using Iterative Prediction. In order to get a visual cue on how changing the time step of an Iterative Prediction calculation, try changing the deltat value from .15 to .05. You will see that making deltat smaller will place the approximation points closer together, which will give the ball's flight path a more consistent curve.

==Example Calculations==

There are a variety of different problems that can be solved by utilizing the momentum update form of the Momentum Principle. They can vary in difficulty and require any number of iterations. It is often prudent to calculate these iterations in a program loop to save time and avoid miscalculations.

===Example #1 - Momentum Update===

'''A boy standing on level ground throws a 2 kg ball into the air at an initial velocity of <math>{<8,6,0> m/s}</math>. If the only force acting on the ball is gravity, what is the final momentum of the ball after 0.2 seconds?'''

This problem can be solved simply by solving for final momentum in the momentum update equation:

<div style="text-align: center;"><math>{\vec{p}_{f} = {m}\vec{v} + \vec{F}_{grav}{Δt}}</math></div>

<div style="text-align: center;"><math>{\vec{p}_{f} = <16,12,0> + <0,-3.92,0>}</math></div>

<div style="text-align: center;"><math>{\vec{p}_{f} = <16,8.08,0>}</math></div>

===Example #2 - Trajectory Maxima===

'''A 1 kg ball is kicked from location <math>{<9,0,0> m}</math> giving it an initial velocity of <math>{<-10,13,0> m/s}</math>. What is the maximum height that the ball will reach along its trajectory?'''

This problem is a bit more complicated than the previous one. Before calculating the maximum height of the ball, we must know the time it takes for the ball to reach its maximum height by analyzing the motion of the ball in the +y direction through the momentum update equation:

<div style="text-align: center;"><math>{t}_{f} = {v}_{i,y}{m}/{F}_{net,y}</math></div>

<div style="text-align: center;"><math>{t}_{f} = \frac{(13)(1)}{(1)(9.8)} = 1.326 s</math></div>

Now that we know how long it takes for the ball to reach its maximum height, we can solve for the final height of the ball by utilizing the position update equation:

<div style="text-align: center;"><math>{r}_{f,y} = {r}_{i,y} + {v}_{avg,y}{Δt}</math></div>

<div style="text-align: center;"><math>{r}_{f,y} = 0 + \frac{13+0}{2}(1.326) = 8.64 m</math></div>

===Example #3 - Varying Force===
'''Model the motion of a spring with a ball attached to the end, taken into consideration of two forces, gravity (constant) and spring force (non-constant)'''

This is a good example of a problem that can most easily be solved using a computer program and knowledge of Iterative Prediction. In order to iteratively update the position of the ball and spring, three steps need to be taken inside of our while loop.

Step 1: Calculate net force
Calculating the net force requires you first to calculate both the spring force vector and gravitational force vector based on the current position of the ball, then add them to get a net force vector.

[[File:Spring1.PNG]]

Step 2: Apply Momentum Principle
Now that we know our net force vector, we need to use the Momentum Principle to find the change in momentum when that force is applied for our time step.

[[File:Spring2.PNG]]

Step 3: Update Position
Finally, now that we know the change in momentum(and in turn the change in velocity), we can use our position update formula to update the position of both the ball and the axis of the spring. Don't forget to update your time!

[[File:Spring4.PNG]]

==Connectedness==

Iterative Prediction serves as a basis for modeling complex motion throughout time. While some of the basic examples do not appear to be too trivial, iterative prediction can be used to solve more difficult problems. Having now almost completed a semester of mechanics, one topic that I became interested in almost right away was this idea of being able to solve problems in which we aren't necessarily concerned with the physics of the system at the current moment in time. Iterative Prediction of motion is just one of the many different techniques we have used to study the "physics of the future."

Iterative Prediction can have many applications to the field of Chemical Engineering, namely in the principles governing momentum transfer in chemical processes. A simple example is in studying fluid flow in a process. By making use of iterative prediction and the momentum principle, a chemical engineer can model the change in momentum of the fluid flow by analyzing the net force on the fluid (likely due to pressure) and the initial momentum of the fluid flow. In industry, this can be important so as to predict accumulation in a process and thus allow for modeling of an efficient process to maximize output.

==History==

The physics behind iterative prediction is nothing more than simple application of the momentum principle and projectile motion, which has existed in classical mechanics for quite some time. However, the practical application of iterative prediction for analyzing systems has evolved due to advances in computational methods and technology. With the computational power of a computer to iteratively calculate changes in momentum through time, one can analyze a system's motion extremely quickly without tedious and difficult mathematical calculations.

== See also ==

There are many public resources that delve further into iterative prediction examples of more complex motion.

===External links===

[http://en.wikipedia.org/wiki/Portal:Physics Physics Portal]

[http://hyperphysics.phy-astr.gsu.edu/hbase/hph.html HyperPhysics Concept Map]

[http://ocw.mit.edu/resources/res-8-002-a-wikitextbook-for-introductory-mechanics-fall-2009/index.htm MIT Open Courseware Introductory Mechanics]

[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 2015. Wednesday, Week 2 Lecture Slides. Fenton, Flavio H

Georgia Institute of Technology. Physics Department. PHYS 2211. Fall 2015. Monday, Week 3 Lecture Slides. Fenton, Flavio H

Georgia Institute of Technology. Physics Department. PHYS 2211. Fall 2015. Lab 07 Fancart Energy & Spring-Mass Instructions, Greco, Edwin

Iterative Prediction

2016-04-16T06:42:06Z

Jbarua: /* References */

Claimed by Joshua Barua (jbarua3)

By tconnors3

==An Overview==
Iterative Prediction is a process used to approximate the effects of a force, be it constant or varying, on the motion of an object. This process does this approximation splitting the time interval up into many small intervals, finding the force on the object at each of the intervals, and then calculation the change of momentum and change of position for the object across each of the time intervals. Understanding Iterative Prediction also requires an understanding of the Momentum Principle, because you are constantly using the equation dp = F * deltat to update your momentum followed by your position.

Iterative Prediction is a necessary technique to employ when doing physics because, even though our fundamental principles are based on calculus and derivatives and you should be able to solve these problems using integrals, it is often impossible to solve the integrals required to solve complex problems. Iterative Prediction also becomes much more convenient when you implement it using a program, such as vPython, which allows you to make the approximation use as many steps as your processing power can handle, which can make your prediction much more accurate.

===A Mathematical Model===

By starting from the general equation for the Momentum Principle, a formula can derived to predict the momentum of a given system at a specified point in the future. This is often referred to as the ''momentum update form'' of the Momentum Principle:

<div style="text-align: center;"><math>{{(1)}\frac{d\vec{p}}{dt}}_{system} = \vec{F}_{net}</math></div>

<div style="text-align: center;"><math>{{(2)}\vec{p}_{f} - \vec{p}_{i} = \vec{F}_{net}{Δt}}</math></div>

<div style="text-align: center;"><math>{{(3)}\vec{p}_{f} = \vec{p}_{i} + \vec{F}_{net}{Δt}}</math></div>

Equation <math>{(3)}</math> gives the momentum update form of the momentum principle. As it shows, the momentum of a system can be predicted if the time period of the interaction and the external net force on the system are known. Momentum can be iteratively predicted like such for uniform or non-uniform time periods. Additionally, with update of momentum, velocity and position can be updated similarly to better reflect the motion over time of the system:

<div style="text-align: center;">'''Velocity Update Formula''':

<math>{\vec{v}_{f} = \vec{v}_{i} + \frac{\vec{F}_{net}}{m}}{Δt}</math>

'''Position Update Formula''':

<math>{\vec{r}_{f} = \vec{r}_{i} + \vec{v}_{avg}{Δt}}</math></div>

===A Visual Model===

As the momentum update formula suggests, the final momentum (shown here as <math>\vec{p}_{future}</math>) of a system after a given change in time should be the vector sum of the initial momentum (denoted <math>\vec{p}_{now}</math>) and the net force on the system multiplied by the scalar time change (<math>\vec{F}_{net}{Δt}</math>).
[[Image:MOTION.png|center|650x300px|]]

As the diagram suggests, each individual step can be analyzed through application of the momentum update form of the Momentum Principle. Once the final momentum is calculated, by utilizing the position and velocity update formula, the final velocity and position of the system can be determined at the end of the time interval of interest.

You can observe this Iterative Prediction in action by viewing this Vpython Script[http://www.glowscript.org/#/user/JBarua/folder/Public/program/ProjectilePublic] (http://www.glowscript.org/#/user/JBarua/folder/Public/program/ProjectilePublic). It shows the flight path of a ball given an initial positive x and y velocity until it hits the ground, calculated using Iterative Prediction. In order to get a visual cue on how changing the time step of an Iterative Prediction calculation, try changing the deltat value from .15 to .05. You will see that making deltat smaller will place the approximation points closer together, which will give the ball's flight path a more consistent curve.

==Example Calculations==

There are a variety of different problems that can be solved by utilizing the momentum update form of the Momentum Principle. They can vary in difficulty and require any number of iterations. It is often prudent to calculate these iterations in a program loop to save time and avoid miscalculations.

===Example #1 - Momentum Update===

'''A boy standing on level ground throws a 2 kg ball into the air at an initial velocity of <math>{<8,6,0> m/s}</math>. If the only force acting on the ball is gravity, what is the final momentum of the ball after 0.2 seconds?'''

This problem can be solved simply by solving for final momentum in the momentum update equation:

<div style="text-align: center;"><math>{\vec{p}_{f} = {m}\vec{v} + \vec{F}_{grav}{Δt}}</math></div>

<div style="text-align: center;"><math>{\vec{p}_{f} = <16,12,0> + <0,-3.92,0>}</math></div>

<div style="text-align: center;"><math>{\vec{p}_{f} = <16,8.08,0>}</math></div>

===Example #2 - Trajectory Maxima===

'''A 1 kg ball is kicked from location <math>{<9,0,0> m}</math> giving it an initial velocity of <math>{<-10,13,0> m/s}</math>. What is the maximum height that the ball will reach along its trajectory?'''

This problem is a bit more complicated than the previous one. Before calculating the maximum height of the ball, we must know the time it takes for the ball to reach its maximum height by analyzing the motion of the ball in the +y direction through the momentum update equation:

<div style="text-align: center;"><math>{t}_{f} = {v}_{i,y}{m}/{F}_{net,y}</math></div>

<div style="text-align: center;"><math>{t}_{f} = \frac{(13)(1)}{(1)(9.8)} = 1.326 s</math></div>

Now that we know how long it takes for the ball to reach its maximum height, we can solve for the final height of the ball by utilizing the position update equation:

<div style="text-align: center;"><math>{r}_{f,y} = {r}_{i,y} + {v}_{avg,y}{Δt}</math></div>

<div style="text-align: center;"><math>{r}_{f,y} = 0 + \frac{13+0}{2}(1.326) = 8.64 m</math></div>

===Example #3 - Varying Force===
'''Model the motion of a spring with a ball attached to the end, taken into consideration of two forces, gravity (constant) and spring force (non-constant)'''

This is a good example of a problem that can most easily be solved using a computer program and knowledge of Iterative Prediction. In order to iteratively update the position of the ball and spring, three steps need to be taken inside of our while loop.

Step 1: Calculate net force
Calculating the net force requires you first to calculate both the spring force vector and gravitational force vector based on the current position of the ball, then add them to get a net force vector.

[[File:Spring1.PNG]]

Step 2: Apply Momentum Principle
Now that we know our net force vector, we need to use the Momentum Principle to find the change in momentum when that force is applied for our time step.

[[File:Spring2.PNG]]

Step 3: Update Position
Finally, now that we know the change in momentum(and in turn the change in velocity), we can use our position update formula to update the position of both the ball and the axis of the spring. Don't forget to update your time!

[[File:Spring4.PNG]]

==Connectedness==

Iterative Prediction serves as a basis for modeling complex motion throughout time. While some of the basic examples do not appear to be too trivial, iterative prediction can be used to solve more difficult problems. Having now almost completed a semester of mechanics, one topic that I became interested in almost right away was this idea of being able to solve problems in which we aren't necessarily concerned with the physics of the system at the current moment in time. Iterative Prediction of motion is just one of the many different techniques we have used to study the "physics of the future."

Iterative Prediction can have many applications to the field of Chemical Engineering, namely in the principles governing momentum transfer in chemical processes. A simple example is in studying fluid flow in a process. By making use of iterative prediction and the momentum principle, a chemical engineer can model the change in momentum of the fluid flow by analyzing the net force on the fluid (likely due to pressure) and the initial momentum of the fluid flow. In industry, this can be important so as to predict accumulation in a process and thus allow for modeling of an efficient process to maximize output.

==History==

The physics behind iterative prediction is nothing more than simple application of the momentum principle and projectile motion, which has existed in classical mechanics for quite some time. However, the practical application of iterative prediction for analyzing systems has evolved due to advances in computational methods and technology. With the computational power of a computer to iteratively calculate changes in momentum through time, one can analyze a system's motion extremely quickly without tedious and difficult mathematical calculations.

== See also ==

There are many public resources that delve further into iterative prediction examples of more complex motion.

===External links===

[http://en.wikipedia.org/wiki/Portal:Physics Physics Portal]

[http://hyperphysics.phy-astr.gsu.edu/hbase/hph.html HyperPhysics Concept Map]

[http://ocw.mit.edu/resources/res-8-002-a-wikitextbook-for-introductory-mechanics-fall-2009/index.htm MIT Open Courseware Introductory Mechanics]

[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 2015. Wednesday, Week 2 Lecture Slides. Fenton, Flavio H

Georgia Institute of Technology. Physics Department. PHYS 2211. Fall 2015. Monday, Week 3 Lecture Slides. Fenton, Flavio H

Georgia Institute of Technology. Physics Department. PHYS 2211. Fall 2015. Lab 07 Fancart Energy & Spring-Mass Instructions, Greco, Edwin

Iterative Prediction

2016-04-16T06:39:32Z

Jbarua: /* Example #3 - Varying Force */

File:Spring4.PNG

2016-04-16T06:39:14Z

Jbarua:

Iterative Prediction

2016-04-16T06:37:34Z

Jbarua: /* Example #3 - Varying Force */

Claimed by Joshua Barua (jbarua3)

By tconnors3

==An Overview==
Iterative Prediction is a process used to approximate the effects of a force, be it constant or varying, on the motion of an object. This process does this approximation splitting the time interval up into many small intervals, finding the force on the object at each of the intervals, and then calculation the change of momentum and change of position for the object across each of the time intervals. Understanding Iterative Prediction also requires an understanding of the Momentum Principle, because you are constantly using the equation dp = F * deltat to update your momentum followed by your position.

Iterative Prediction is a necessary technique to employ when doing physics because, even though our fundamental principles are based on calculus and derivatives and you should be able to solve these problems using integrals, it is often impossible to solve the integrals required to solve complex problems. Iterative Prediction also becomes much more convenient when you implement it using a program, such as vPython, which allows you to make the approximation use as many steps as your processing power can handle, which can make your prediction much more accurate.

===A Mathematical Model===

By starting from the general equation for the Momentum Principle, a formula can derived to predict the momentum of a given system at a specified point in the future. This is often referred to as the ''momentum update form'' of the Momentum Principle:

<div style="text-align: center;"><math>{{(1)}\frac{d\vec{p}}{dt}}_{system} = \vec{F}_{net}</math></div>

<div style="text-align: center;"><math>{{(2)}\vec{p}_{f} - \vec{p}_{i} = \vec{F}_{net}{Δt}}</math></div>

<div style="text-align: center;"><math>{{(3)}\vec{p}_{f} = \vec{p}_{i} + \vec{F}_{net}{Δt}}</math></div>

Equation <math>{(3)}</math> gives the momentum update form of the momentum principle. As it shows, the momentum of a system can be predicted if the time period of the interaction and the external net force on the system are known. Momentum can be iteratively predicted like such for uniform or non-uniform time periods. Additionally, with update of momentum, velocity and position can be updated similarly to better reflect the motion over time of the system:

<div style="text-align: center;">'''Velocity Update Formula''':

<math>{\vec{v}_{f} = \vec{v}_{i} + \frac{\vec{F}_{net}}{m}}{Δt}</math>

'''Position Update Formula''':

<math>{\vec{r}_{f} = \vec{r}_{i} + \vec{v}_{avg}{Δt}}</math></div>

===A Visual Model===

As the momentum update formula suggests, the final momentum (shown here as <math>\vec{p}_{future}</math>) of a system after a given change in time should be the vector sum of the initial momentum (denoted <math>\vec{p}_{now}</math>) and the net force on the system multiplied by the scalar time change (<math>\vec{F}_{net}{Δt}</math>).
[[Image:MOTION.png|center|650x300px|]]

As the diagram suggests, each individual step can be analyzed through application of the momentum update form of the Momentum Principle. Once the final momentum is calculated, by utilizing the position and velocity update formula, the final velocity and position of the system can be determined at the end of the time interval of interest.

You can observe this Iterative Prediction in action by viewing this Vpython Script[http://www.glowscript.org/#/user/JBarua/folder/Public/program/ProjectilePublic] (http://www.glowscript.org/#/user/JBarua/folder/Public/program/ProjectilePublic). It shows the flight path of a ball given an initial positive x and y velocity until it hits the ground, calculated using Iterative Prediction. In order to get a visual cue on how changing the time step of an Iterative Prediction calculation, try changing the deltat value from .15 to .05. You will see that making deltat smaller will place the approximation points closer together, which will give the ball's flight path a more consistent curve.

==Example Calculations==

There are a variety of different problems that can be solved by utilizing the momentum update form of the Momentum Principle. They can vary in difficulty and require any number of iterations. It is often prudent to calculate these iterations in a program loop to save time and avoid miscalculations.

===Example #1 - Momentum Update===

'''A boy standing on level ground throws a 2 kg ball into the air at an initial velocity of <math>{<8,6,0> m/s}</math>. If the only force acting on the ball is gravity, what is the final momentum of the ball after 0.2 seconds?'''

This problem can be solved simply by solving for final momentum in the momentum update equation:

<div style="text-align: center;"><math>{\vec{p}_{f} = {m}\vec{v} + \vec{F}_{grav}{Δt}}</math></div>

<div style="text-align: center;"><math>{\vec{p}_{f} = <16,12,0> + <0,-3.92,0>}</math></div>

<div style="text-align: center;"><math>{\vec{p}_{f} = <16,8.08,0>}</math></div>

===Example #2 - Trajectory Maxima===

'''A 1 kg ball is kicked from location <math>{<9,0,0> m}</math> giving it an initial velocity of <math>{<-10,13,0> m/s}</math>. What is the maximum height that the ball will reach along its trajectory?'''

This problem is a bit more complicated than the previous one. Before calculating the maximum height of the ball, we must know the time it takes for the ball to reach its maximum height by analyzing the motion of the ball in the +y direction through the momentum update equation:

<div style="text-align: center;"><math>{t}_{f} = {v}_{i,y}{m}/{F}_{net,y}</math></div>

<div style="text-align: center;"><math>{t}_{f} = \frac{(13)(1)}{(1)(9.8)} = 1.326 s</math></div>

Now that we know how long it takes for the ball to reach its maximum height, we can solve for the final height of the ball by utilizing the position update equation:

<div style="text-align: center;"><math>{r}_{f,y} = {r}_{i,y} + {v}_{avg,y}{Δt}</math></div>

<div style="text-align: center;"><math>{r}_{f,y} = 0 + \frac{13+0}{2}(1.326) = 8.64 m</math></div>

===Example #3 - Varying Force===
'''Model the motion of a spring with a ball attached to the end, taken into consideration of two forces, gravity (constant) and spring force (non-constant)'''

This is a good example of a problem that can most easily be solved using a computer program and knowledge of Iterative Prediction. In order to iteratively update the position of the ball and spring, three steps need to be taken inside of our while loop.

Step 1: Calculate net force
Calculating the net force requires you first to calculate both the spring force vector and gravitational force vector based on the current position of the ball, then add them to get a net force vector.

[[File:Spring1.PNG]]

Step 2: Apply Momentum Principle
Now that we know our net force vector, we need to use the Momentum Principle to find the change in momentum when that force is applied for our time step.

[[File:Spring2.PNG]]

Step 3: Update Position
Finally, now that we know the change in momentum(and in turn the change in velocity), we can use our position update formula to update the position of both the ball and the axis of the spring. Don't forget to update your time!

[[File:Spring3.PNG]]

==Connectedness==

Iterative Prediction serves as a basis for modeling complex motion throughout time. While some of the basic examples do not appear to be too trivial, iterative prediction can be used to solve more difficult problems. Having now almost completed a semester of mechanics, one topic that I became interested in almost right away was this idea of being able to solve problems in which we aren't necessarily concerned with the physics of the system at the current moment in time. Iterative Prediction of motion is just one of the many different techniques we have used to study the "physics of the future."

Iterative Prediction can have many applications to the field of Chemical Engineering, namely in the principles governing momentum transfer in chemical processes. A simple example is in studying fluid flow in a process. By making use of iterative prediction and the momentum principle, a chemical engineer can model the change in momentum of the fluid flow by analyzing the net force on the fluid (likely due to pressure) and the initial momentum of the fluid flow. In industry, this can be important so as to predict accumulation in a process and thus allow for modeling of an efficient process to maximize output.

==History==

The physics behind iterative prediction is nothing more than simple application of the momentum principle and projectile motion, which has existed in classical mechanics for quite some time. However, the practical application of iterative prediction for analyzing systems has evolved due to advances in computational methods and technology. With the computational power of a computer to iteratively calculate changes in momentum through time, one can analyze a system's motion extremely quickly without tedious and difficult mathematical calculations.

== See also ==

There are many public resources that delve further into iterative prediction examples of more complex motion.

===External links===

[http://en.wikipedia.org/wiki/Portal:Physics Physics Portal]

[http://hyperphysics.phy-astr.gsu.edu/hbase/hph.html HyperPhysics Concept Map]

[http://ocw.mit.edu/resources/res-8-002-a-wikitextbook-for-introductory-mechanics-fall-2009/index.htm MIT Open Courseware Introductory Mechanics]

[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 2015. Wednesday, Week 2 Lecture Slides. Fenton, Flavio H

Georgia Institute of Technology. Physics Department. PHYS 2211. Fall 2015. Monday, Week 3 Lecture Slides. Fenton, Flavio H

File:Spring3.PNG

2016-04-16T06:33:54Z

Jbarua:

File:Spring2.PNG

2016-04-16T06:33:39Z

Jbarua:

Iterative Prediction

2016-04-16T06:33:14Z

Jbarua: /* Example #3 - Varying Force */

Claimed by Joshua Barua (jbarua3)

By tconnors3

==An Overview==
Iterative Prediction is a process used to approximate the effects of a force, be it constant or varying, on the motion of an object. This process does this approximation splitting the time interval up into many small intervals, finding the force on the object at each of the intervals, and then calculation the change of momentum and change of position for the object across each of the time intervals. Understanding Iterative Prediction also requires an understanding of the Momentum Principle, because you are constantly using the equation dp = F * deltat to update your momentum followed by your position.

Iterative Prediction is a necessary technique to employ when doing physics because, even though our fundamental principles are based on calculus and derivatives and you should be able to solve these problems using integrals, it is often impossible to solve the integrals required to solve complex problems. Iterative Prediction also becomes much more convenient when you implement it using a program, such as vPython, which allows you to make the approximation use as many steps as your processing power can handle, which can make your prediction much more accurate.

===A Mathematical Model===

By starting from the general equation for the Momentum Principle, a formula can derived to predict the momentum of a given system at a specified point in the future. This is often referred to as the ''momentum update form'' of the Momentum Principle:

<div style="text-align: center;"><math>{{(1)}\frac{d\vec{p}}{dt}}_{system} = \vec{F}_{net}</math></div>

<div style="text-align: center;"><math>{{(2)}\vec{p}_{f} - \vec{p}_{i} = \vec{F}_{net}{Δt}}</math></div>

<div style="text-align: center;"><math>{{(3)}\vec{p}_{f} = \vec{p}_{i} + \vec{F}_{net}{Δt}}</math></div>

Equation <math>{(3)}</math> gives the momentum update form of the momentum principle. As it shows, the momentum of a system can be predicted if the time period of the interaction and the external net force on the system are known. Momentum can be iteratively predicted like such for uniform or non-uniform time periods. Additionally, with update of momentum, velocity and position can be updated similarly to better reflect the motion over time of the system:

<div style="text-align: center;">'''Velocity Update Formula''':

<math>{\vec{v}_{f} = \vec{v}_{i} + \frac{\vec{F}_{net}}{m}}{Δt}</math>

'''Position Update Formula''':

<math>{\vec{r}_{f} = \vec{r}_{i} + \vec{v}_{avg}{Δt}}</math></div>

===A Visual Model===

As the momentum update formula suggests, the final momentum (shown here as <math>\vec{p}_{future}</math>) of a system after a given change in time should be the vector sum of the initial momentum (denoted <math>\vec{p}_{now}</math>) and the net force on the system multiplied by the scalar time change (<math>\vec{F}_{net}{Δt}</math>).
[[Image:MOTION.png|center|650x300px|]]

As the diagram suggests, each individual step can be analyzed through application of the momentum update form of the Momentum Principle. Once the final momentum is calculated, by utilizing the position and velocity update formula, the final velocity and position of the system can be determined at the end of the time interval of interest.

You can observe this Iterative Prediction in action by viewing this Vpython Script[http://www.glowscript.org/#/user/JBarua/folder/Public/program/ProjectilePublic] (http://www.glowscript.org/#/user/JBarua/folder/Public/program/ProjectilePublic). It shows the flight path of a ball given an initial positive x and y velocity until it hits the ground, calculated using Iterative Prediction. In order to get a visual cue on how changing the time step of an Iterative Prediction calculation, try changing the deltat value from .15 to .05. You will see that making deltat smaller will place the approximation points closer together, which will give the ball's flight path a more consistent curve.

==Example Calculations==

There are a variety of different problems that can be solved by utilizing the momentum update form of the Momentum Principle. They can vary in difficulty and require any number of iterations. It is often prudent to calculate these iterations in a program loop to save time and avoid miscalculations.

===Example #1 - Momentum Update===

'''A boy standing on level ground throws a 2 kg ball into the air at an initial velocity of <math>{<8,6,0> m/s}</math>. If the only force acting on the ball is gravity, what is the final momentum of the ball after 0.2 seconds?'''

This problem can be solved simply by solving for final momentum in the momentum update equation:

<div style="text-align: center;"><math>{\vec{p}_{f} = {m}\vec{v} + \vec{F}_{grav}{Δt}}</math></div>

<div style="text-align: center;"><math>{\vec{p}_{f} = <16,12,0> + <0,-3.92,0>}</math></div>

<div style="text-align: center;"><math>{\vec{p}_{f} = <16,8.08,0>}</math></div>

===Example #2 - Trajectory Maxima===

'''A 1 kg ball is kicked from location <math>{<9,0,0> m}</math> giving it an initial velocity of <math>{<-10,13,0> m/s}</math>. What is the maximum height that the ball will reach along its trajectory?'''

This problem is a bit more complicated than the previous one. Before calculating the maximum height of the ball, we must know the time it takes for the ball to reach its maximum height by analyzing the motion of the ball in the +y direction through the momentum update equation:

<div style="text-align: center;"><math>{t}_{f} = {v}_{i,y}{m}/{F}_{net,y}</math></div>

<div style="text-align: center;"><math>{t}_{f} = \frac{(13)(1)}{(1)(9.8)} = 1.326 s</math></div>

Now that we know how long it takes for the ball to reach its maximum height, we can solve for the final height of the ball by utilizing the position update equation:

<div style="text-align: center;"><math>{r}_{f,y} = {r}_{i,y} + {v}_{avg,y}{Δt}</math></div>

<div style="text-align: center;"><math>{r}_{f,y} = 0 + \frac{13+0}{2}(1.326) = 8.64 m</math></div>

===Example #3 - Varying Force===
'''Model the motion of a spring with a ball attached to the end, taken into consideration of two forces, gravity (constant) and spring force (non-constant)'''

This is a good example of a problem that can most easily be solved using a computer program and knowledge of Iterative Prediction. In order to iteratively update the position of the ball and spring, three steps need to be taken inside of our while loop.

Step 1: Calculate net force
Calculating the net force requires you first to calculate both the spring force vector and gravitational force vector based on the current position of the ball, then add them to get a net force vector.

[[File:Spring1.PNG]]

Step 2: Apply Momentum Principle
Now that we know our net force vector, we need to use the Momentum Principle to find the change in momentum when that force is applied for our time step.
[[File.Spring2.PNG]]

==Connectedness==

Iterative Prediction serves as a basis for modeling complex motion throughout time. While some of the basic examples do not appear to be too trivial, iterative prediction can be used to solve more difficult problems. Having now almost completed a semester of mechanics, one topic that I became interested in almost right away was this idea of being able to solve problems in which we aren't necessarily concerned with the physics of the system at the current moment in time. Iterative Prediction of motion is just one of the many different techniques we have used to study the "physics of the future."

Iterative Prediction can have many applications to the field of Chemical Engineering, namely in the principles governing momentum transfer in chemical processes. A simple example is in studying fluid flow in a process. By making use of iterative prediction and the momentum principle, a chemical engineer can model the change in momentum of the fluid flow by analyzing the net force on the fluid (likely due to pressure) and the initial momentum of the fluid flow. In industry, this can be important so as to predict accumulation in a process and thus allow for modeling of an efficient process to maximize output.

==History==

The physics behind iterative prediction is nothing more than simple application of the momentum principle and projectile motion, which has existed in classical mechanics for quite some time. However, the practical application of iterative prediction for analyzing systems has evolved due to advances in computational methods and technology. With the computational power of a computer to iteratively calculate changes in momentum through time, one can analyze a system's motion extremely quickly without tedious and difficult mathematical calculations.

== See also ==

There are many public resources that delve further into iterative prediction examples of more complex motion.

===External links===

[http://en.wikipedia.org/wiki/Portal:Physics Physics Portal]

[http://hyperphysics.phy-astr.gsu.edu/hbase/hph.html HyperPhysics Concept Map]

[http://ocw.mit.edu/resources/res-8-002-a-wikitextbook-for-introductory-mechanics-fall-2009/index.htm MIT Open Courseware Introductory Mechanics]

[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 2015. Wednesday, Week 2 Lecture Slides. Fenton, Flavio H

Georgia Institute of Technology. Physics Department. PHYS 2211. Fall 2015. Monday, Week 3 Lecture Slides. Fenton, Flavio H

Iterative Prediction

2016-04-16T06:29:35Z

Jbarua: /* Example #3 - Varying Force */

Claimed by Joshua Barua (jbarua3)

By tconnors3

==An Overview==
Iterative Prediction is a process used to approximate the effects of a force, be it constant or varying, on the motion of an object. This process does this approximation splitting the time interval up into many small intervals, finding the force on the object at each of the intervals, and then calculation the change of momentum and change of position for the object across each of the time intervals. Understanding Iterative Prediction also requires an understanding of the Momentum Principle, because you are constantly using the equation dp = F * deltat to update your momentum followed by your position.

Iterative Prediction is a necessary technique to employ when doing physics because, even though our fundamental principles are based on calculus and derivatives and you should be able to solve these problems using integrals, it is often impossible to solve the integrals required to solve complex problems. Iterative Prediction also becomes much more convenient when you implement it using a program, such as vPython, which allows you to make the approximation use as many steps as your processing power can handle, which can make your prediction much more accurate.

===A Mathematical Model===

By starting from the general equation for the Momentum Principle, a formula can derived to predict the momentum of a given system at a specified point in the future. This is often referred to as the ''momentum update form'' of the Momentum Principle:

<div style="text-align: center;"><math>{{(1)}\frac{d\vec{p}}{dt}}_{system} = \vec{F}_{net}</math></div>

<div style="text-align: center;"><math>{{(2)}\vec{p}_{f} - \vec{p}_{i} = \vec{F}_{net}{Δt}}</math></div>

<div style="text-align: center;"><math>{{(3)}\vec{p}_{f} = \vec{p}_{i} + \vec{F}_{net}{Δt}}</math></div>

Equation <math>{(3)}</math> gives the momentum update form of the momentum principle. As it shows, the momentum of a system can be predicted if the time period of the interaction and the external net force on the system are known. Momentum can be iteratively predicted like such for uniform or non-uniform time periods. Additionally, with update of momentum, velocity and position can be updated similarly to better reflect the motion over time of the system:

<div style="text-align: center;">'''Velocity Update Formula''':

<math>{\vec{v}_{f} = \vec{v}_{i} + \frac{\vec{F}_{net}}{m}}{Δt}</math>

'''Position Update Formula''':

<math>{\vec{r}_{f} = \vec{r}_{i} + \vec{v}_{avg}{Δt}}</math></div>

===A Visual Model===

As the momentum update formula suggests, the final momentum (shown here as <math>\vec{p}_{future}</math>) of a system after a given change in time should be the vector sum of the initial momentum (denoted <math>\vec{p}_{now}</math>) and the net force on the system multiplied by the scalar time change (<math>\vec{F}_{net}{Δt}</math>).
[[Image:MOTION.png|center|650x300px|]]

As the diagram suggests, each individual step can be analyzed through application of the momentum update form of the Momentum Principle. Once the final momentum is calculated, by utilizing the position and velocity update formula, the final velocity and position of the system can be determined at the end of the time interval of interest.

You can observe this Iterative Prediction in action by viewing this Vpython Script[http://www.glowscript.org/#/user/JBarua/folder/Public/program/ProjectilePublic] (http://www.glowscript.org/#/user/JBarua/folder/Public/program/ProjectilePublic). It shows the flight path of a ball given an initial positive x and y velocity until it hits the ground, calculated using Iterative Prediction. In order to get a visual cue on how changing the time step of an Iterative Prediction calculation, try changing the deltat value from .15 to .05. You will see that making deltat smaller will place the approximation points closer together, which will give the ball's flight path a more consistent curve.

==Example Calculations==

There are a variety of different problems that can be solved by utilizing the momentum update form of the Momentum Principle. They can vary in difficulty and require any number of iterations. It is often prudent to calculate these iterations in a program loop to save time and avoid miscalculations.

===Example #1 - Momentum Update===

'''A boy standing on level ground throws a 2 kg ball into the air at an initial velocity of <math>{<8,6,0> m/s}</math>. If the only force acting on the ball is gravity, what is the final momentum of the ball after 0.2 seconds?'''

This problem can be solved simply by solving for final momentum in the momentum update equation:

<div style="text-align: center;"><math>{\vec{p}_{f} = {m}\vec{v} + \vec{F}_{grav}{Δt}}</math></div>

<div style="text-align: center;"><math>{\vec{p}_{f} = <16,12,0> + <0,-3.92,0>}</math></div>

<div style="text-align: center;"><math>{\vec{p}_{f} = <16,8.08,0>}</math></div>

===Example #2 - Trajectory Maxima===

'''A 1 kg ball is kicked from location <math>{<9,0,0> m}</math> giving it an initial velocity of <math>{<-10,13,0> m/s}</math>. What is the maximum height that the ball will reach along its trajectory?'''

This problem is a bit more complicated than the previous one. Before calculating the maximum height of the ball, we must know the time it takes for the ball to reach its maximum height by analyzing the motion of the ball in the +y direction through the momentum update equation:

<div style="text-align: center;"><math>{t}_{f} = {v}_{i,y}{m}/{F}_{net,y}</math></div>

<div style="text-align: center;"><math>{t}_{f} = \frac{(13)(1)}{(1)(9.8)} = 1.326 s</math></div>

Now that we know how long it takes for the ball to reach its maximum height, we can solve for the final height of the ball by utilizing the position update equation:

<div style="text-align: center;"><math>{r}_{f,y} = {r}_{i,y} + {v}_{avg,y}{Δt}</math></div>

<div style="text-align: center;"><math>{r}_{f,y} = 0 + \frac{13+0}{2}(1.326) = 8.64 m</math></div>

===Example #3 - Varying Force===
'''Model the motion of a spring with a ball attached to the end, taken into consideration of two forces, gravity (constant) and spring force (non-constant)'''

This is a good example of a problem that can most easily be solved using a computer program and knowledge of Iterative Prediction. In order to iteratively update the position of the ball and spring, three steps need to be taken inside of our while loop.

Step 1: Calculate net force
Calculating the net force requires you first to calculate both the spring force vector and gravitational force vector based on the current position of the ball, then add them to get a net force vector.

[[File:Spring1.png]]

==Connectedness==

Iterative Prediction serves as a basis for modeling complex motion throughout time. While some of the basic examples do not appear to be too trivial, iterative prediction can be used to solve more difficult problems. Having now almost completed a semester of mechanics, one topic that I became interested in almost right away was this idea of being able to solve problems in which we aren't necessarily concerned with the physics of the system at the current moment in time. Iterative Prediction of motion is just one of the many different techniques we have used to study the "physics of the future."

Iterative Prediction can have many applications to the field of Chemical Engineering, namely in the principles governing momentum transfer in chemical processes. A simple example is in studying fluid flow in a process. By making use of iterative prediction and the momentum principle, a chemical engineer can model the change in momentum of the fluid flow by analyzing the net force on the fluid (likely due to pressure) and the initial momentum of the fluid flow. In industry, this can be important so as to predict accumulation in a process and thus allow for modeling of an efficient process to maximize output.

==History==

The physics behind iterative prediction is nothing more than simple application of the momentum principle and projectile motion, which has existed in classical mechanics for quite some time. However, the practical application of iterative prediction for analyzing systems has evolved due to advances in computational methods and technology. With the computational power of a computer to iteratively calculate changes in momentum through time, one can analyze a system's motion extremely quickly without tedious and difficult mathematical calculations.

== See also ==

There are many public resources that delve further into iterative prediction examples of more complex motion.

===External links===

[http://en.wikipedia.org/wiki/Portal:Physics Physics Portal]

[http://hyperphysics.phy-astr.gsu.edu/hbase/hph.html HyperPhysics Concept Map]

[http://ocw.mit.edu/resources/res-8-002-a-wikitextbook-for-introductory-mechanics-fall-2009/index.htm MIT Open Courseware Introductory Mechanics]

[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 2015. Wednesday, Week 2 Lecture Slides. Fenton, Flavio H

Georgia Institute of Technology. Physics Department. PHYS 2211. Fall 2015. Monday, Week 3 Lecture Slides. Fenton, Flavio H

File:Spring1.PNG

2016-04-16T06:27:17Z

Jbarua:

Iterative Prediction

2016-04-16T06:22:05Z

Jbarua: /* Example Calculations */

Claimed by Joshua Barua (jbarua3)

By tconnors3

==An Overview==
Iterative Prediction is a process used to approximate the effects of a force, be it constant or varying, on the motion of an object. This process does this approximation splitting the time interval up into many small intervals, finding the force on the object at each of the intervals, and then calculation the change of momentum and change of position for the object across each of the time intervals. Understanding Iterative Prediction also requires an understanding of the Momentum Principle, because you are constantly using the equation dp = F * deltat to update your momentum followed by your position.

Iterative Prediction is a necessary technique to employ when doing physics because, even though our fundamental principles are based on calculus and derivatives and you should be able to solve these problems using integrals, it is often impossible to solve the integrals required to solve complex problems. Iterative Prediction also becomes much more convenient when you implement it using a program, such as vPython, which allows you to make the approximation use as many steps as your processing power can handle, which can make your prediction much more accurate.

===A Mathematical Model===

By starting from the general equation for the Momentum Principle, a formula can derived to predict the momentum of a given system at a specified point in the future. This is often referred to as the ''momentum update form'' of the Momentum Principle:

<div style="text-align: center;"><math>{{(1)}\frac{d\vec{p}}{dt}}_{system} = \vec{F}_{net}</math></div>

<div style="text-align: center;"><math>{{(2)}\vec{p}_{f} - \vec{p}_{i} = \vec{F}_{net}{Δt}}</math></div>

<div style="text-align: center;"><math>{{(3)}\vec{p}_{f} = \vec{p}_{i} + \vec{F}_{net}{Δt}}</math></div>

Equation <math>{(3)}</math> gives the momentum update form of the momentum principle. As it shows, the momentum of a system can be predicted if the time period of the interaction and the external net force on the system are known. Momentum can be iteratively predicted like such for uniform or non-uniform time periods. Additionally, with update of momentum, velocity and position can be updated similarly to better reflect the motion over time of the system:

<div style="text-align: center;">'''Velocity Update Formula''':

<math>{\vec{v}_{f} = \vec{v}_{i} + \frac{\vec{F}_{net}}{m}}{Δt}</math>

'''Position Update Formula''':

<math>{\vec{r}_{f} = \vec{r}_{i} + \vec{v}_{avg}{Δt}}</math></div>

===A Visual Model===

As the momentum update formula suggests, the final momentum (shown here as <math>\vec{p}_{future}</math>) of a system after a given change in time should be the vector sum of the initial momentum (denoted <math>\vec{p}_{now}</math>) and the net force on the system multiplied by the scalar time change (<math>\vec{F}_{net}{Δt}</math>).
[[Image:MOTION.png|center|650x300px|]]

As the diagram suggests, each individual step can be analyzed through application of the momentum update form of the Momentum Principle. Once the final momentum is calculated, by utilizing the position and velocity update formula, the final velocity and position of the system can be determined at the end of the time interval of interest.

You can observe this Iterative Prediction in action by viewing this Vpython Script[http://www.glowscript.org/#/user/JBarua/folder/Public/program/ProjectilePublic] (http://www.glowscript.org/#/user/JBarua/folder/Public/program/ProjectilePublic). It shows the flight path of a ball given an initial positive x and y velocity until it hits the ground, calculated using Iterative Prediction. In order to get a visual cue on how changing the time step of an Iterative Prediction calculation, try changing the deltat value from .15 to .05. You will see that making deltat smaller will place the approximation points closer together, which will give the ball's flight path a more consistent curve.

==Example Calculations==

There are a variety of different problems that can be solved by utilizing the momentum update form of the Momentum Principle. They can vary in difficulty and require any number of iterations. It is often prudent to calculate these iterations in a program loop to save time and avoid miscalculations.

===Example #1 - Momentum Update===

'''A boy standing on level ground throws a 2 kg ball into the air at an initial velocity of <math>{<8,6,0> m/s}</math>. If the only force acting on the ball is gravity, what is the final momentum of the ball after 0.2 seconds?'''

This problem can be solved simply by solving for final momentum in the momentum update equation:

<div style="text-align: center;"><math>{\vec{p}_{f} = {m}\vec{v} + \vec{F}_{grav}{Δt}}</math></div>

<div style="text-align: center;"><math>{\vec{p}_{f} = <16,12,0> + <0,-3.92,0>}</math></div>

<div style="text-align: center;"><math>{\vec{p}_{f} = <16,8.08,0>}</math></div>

===Example #2 - Trajectory Maxima===

'''A 1 kg ball is kicked from location <math>{<9,0,0> m}</math> giving it an initial velocity of <math>{<-10,13,0> m/s}</math>. What is the maximum height that the ball will reach along its trajectory?'''

This problem is a bit more complicated than the previous one. Before calculating the maximum height of the ball, we must know the time it takes for the ball to reach its maximum height by analyzing the motion of the ball in the +y direction through the momentum update equation:

<div style="text-align: center;"><math>{t}_{f} = {v}_{i,y}{m}/{F}_{net,y}</math></div>

<div style="text-align: center;"><math>{t}_{f} = \frac{(13)(1)}{(1)(9.8)} = 1.326 s</math></div>

Now that we know how long it takes for the ball to reach its maximum height, we can solve for the final height of the ball by utilizing the position update equation:

<div style="text-align: center;"><math>{r}_{f,y} = {r}_{i,y} + {v}_{avg,y}{Δt}</math></div>

<div style="text-align: center;"><math>{r}_{f,y} = 0 + \frac{13+0}{2}(1.326) = 8.64 m</math></div>

===Example #3 - Varying Force===
'''Model the motion of a spring with a ball attached to the end, taken into consideration of two forces, gravity (constant) and spring force (non-constant)'''

This is a good example of a problem that can most easily be solved using a computer program and knowledge of Iterative Prediction. In order to iteratively update the position of the ball and spring, three steps need to be taken inside of our while loop.

Step 1: Calculate net force
Calculating the net force requires you first to calculate both the spring force vector and gravitational force vector based on the current position of the ball, then add them to get a net force vector.

==Connectedness==

Iterative Prediction serves as a basis for modeling complex motion throughout time. While some of the basic examples do not appear to be too trivial, iterative prediction can be used to solve more difficult problems. Having now almost completed a semester of mechanics, one topic that I became interested in almost right away was this idea of being able to solve problems in which we aren't necessarily concerned with the physics of the system at the current moment in time. Iterative Prediction of motion is just one of the many different techniques we have used to study the "physics of the future."

Iterative Prediction can have many applications to the field of Chemical Engineering, namely in the principles governing momentum transfer in chemical processes. A simple example is in studying fluid flow in a process. By making use of iterative prediction and the momentum principle, a chemical engineer can model the change in momentum of the fluid flow by analyzing the net force on the fluid (likely due to pressure) and the initial momentum of the fluid flow. In industry, this can be important so as to predict accumulation in a process and thus allow for modeling of an efficient process to maximize output.

==History==

The physics behind iterative prediction is nothing more than simple application of the momentum principle and projectile motion, which has existed in classical mechanics for quite some time. However, the practical application of iterative prediction for analyzing systems has evolved due to advances in computational methods and technology. With the computational power of a computer to iteratively calculate changes in momentum through time, one can analyze a system's motion extremely quickly without tedious and difficult mathematical calculations.

== See also ==

There are many public resources that delve further into iterative prediction examples of more complex motion.

===External links===

[http://en.wikipedia.org/wiki/Portal:Physics Physics Portal]

[http://hyperphysics.phy-astr.gsu.edu/hbase/hph.html HyperPhysics Concept Map]

[http://ocw.mit.edu/resources/res-8-002-a-wikitextbook-for-introductory-mechanics-fall-2009/index.htm MIT Open Courseware Introductory Mechanics]

[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 2015. Wednesday, Week 2 Lecture Slides. Fenton, Flavio H

Georgia Institute of Technology. Physics Department. PHYS 2211. Fall 2015. Monday, Week 3 Lecture Slides. Fenton, Flavio H

Iterative Prediction

2016-04-16T06:15:48Z

Jbarua: /* Example Calculations */

Claimed by Joshua Barua (jbarua3)

By tconnors3

==An Overview==
Iterative Prediction is a process used to approximate the effects of a force, be it constant or varying, on the motion of an object. This process does this approximation splitting the time interval up into many small intervals, finding the force on the object at each of the intervals, and then calculation the change of momentum and change of position for the object across each of the time intervals. Understanding Iterative Prediction also requires an understanding of the Momentum Principle, because you are constantly using the equation dp = F * deltat to update your momentum followed by your position.

Iterative Prediction is a necessary technique to employ when doing physics because, even though our fundamental principles are based on calculus and derivatives and you should be able to solve these problems using integrals, it is often impossible to solve the integrals required to solve complex problems. Iterative Prediction also becomes much more convenient when you implement it using a program, such as vPython, which allows you to make the approximation use as many steps as your processing power can handle, which can make your prediction much more accurate.

===A Mathematical Model===

By starting from the general equation for the Momentum Principle, a formula can derived to predict the momentum of a given system at a specified point in the future. This is often referred to as the ''momentum update form'' of the Momentum Principle:

<div style="text-align: center;"><math>{{(1)}\frac{d\vec{p}}{dt}}_{system} = \vec{F}_{net}</math></div>

<div style="text-align: center;"><math>{{(2)}\vec{p}_{f} - \vec{p}_{i} = \vec{F}_{net}{Δt}}</math></div>

<div style="text-align: center;"><math>{{(3)}\vec{p}_{f} = \vec{p}_{i} + \vec{F}_{net}{Δt}}</math></div>

Equation <math>{(3)}</math> gives the momentum update form of the momentum principle. As it shows, the momentum of a system can be predicted if the time period of the interaction and the external net force on the system are known. Momentum can be iteratively predicted like such for uniform or non-uniform time periods. Additionally, with update of momentum, velocity and position can be updated similarly to better reflect the motion over time of the system:

<div style="text-align: center;">'''Velocity Update Formula''':

<math>{\vec{v}_{f} = \vec{v}_{i} + \frac{\vec{F}_{net}}{m}}{Δt}</math>

'''Position Update Formula''':

<math>{\vec{r}_{f} = \vec{r}_{i} + \vec{v}_{avg}{Δt}}</math></div>

===A Visual Model===

As the momentum update formula suggests, the final momentum (shown here as <math>\vec{p}_{future}</math>) of a system after a given change in time should be the vector sum of the initial momentum (denoted <math>\vec{p}_{now}</math>) and the net force on the system multiplied by the scalar time change (<math>\vec{F}_{net}{Δt}</math>).
[[Image:MOTION.png|center|650x300px|]]

As the diagram suggests, each individual step can be analyzed through application of the momentum update form of the Momentum Principle. Once the final momentum is calculated, by utilizing the position and velocity update formula, the final velocity and position of the system can be determined at the end of the time interval of interest.

You can observe this Iterative Prediction in action by viewing this Vpython Script[http://www.glowscript.org/#/user/JBarua/folder/Public/program/ProjectilePublic] (http://www.glowscript.org/#/user/JBarua/folder/Public/program/ProjectilePublic). It shows the flight path of a ball given an initial positive x and y velocity until it hits the ground, calculated using Iterative Prediction. In order to get a visual cue on how changing the time step of an Iterative Prediction calculation, try changing the deltat value from .15 to .05. You will see that making deltat smaller will place the approximation points closer together, which will give the ball's flight path a more consistent curve.

==Example Calculations==

There are a variety of different problems that can be solved by utilizing the momentum update form of the Momentum Principle. They can vary in difficulty and require any number of iterations. It is often prudent to calculate these iterations in a program loop to save time and avoid miscalculations.

===Example #1 - Momentum Update===

'''A boy standing on level ground throws a 2 kg ball into the air at an initial velocity of <math>{<8,6,0> m/s}</math>. If the only force acting on the ball is gravity, what is the final momentum of the ball after 0.2 seconds?'''

This problem can be solved simply by solving for final momentum in the momentum update equation:

<div style="text-align: center;"><math>{\vec{p}_{f} = {m}\vec{v} + \vec{F}_{grav}{Δt}}</math></div>

<div style="text-align: center;"><math>{\vec{p}_{f} = <16,12,0> + <0,-3.92,0>}</math></div>

<div style="text-align: center;"><math>{\vec{p}_{f} = <16,8.08,0>}</math></div>

===Example #2 - Trajectory Maxima===

'''A 1 kg ball is kicked from location <math>{<9,0,0> m}</math> giving it an initial velocity of <math>{<-10,13,0> m/s}</math>. What is the maximum height that the ball will reach along its trajectory?'''

This problem is a bit more complicated than the previous one. Before calculating the maximum height of the ball, we must know the time it takes for the ball to reach its maximum height by analyzing the motion of the ball in the +y direction through the momentum update equation:

<div style="text-align: center;"><math>{t}_{f} = {v}_{i,y}{m}/{F}_{net,y}</math></div>

<div style="text-align: center;"><math>{t}_{f} = \frac{(13)(1)}{(1)(9.8)} = 1.326 s</math></div>

Now that we know how long it takes for the ball to reach its maximum height, we can solve for the final height of the ball by utilizing the position update equation:

<div style="text-align: center;"><math>{r}_{f,y} = {r}_{i,y} + {v}_{avg,y}{Δt}</math></div>

<div style="text-align: center;"><math>{r}_{f,y} = 0 + \frac{13+0}{2}(1.326) = 8.64 m</math></div>

===Example #3 - Varying Force===
'''Model the motion of a spring with a ball attached to the end, taken into consideration of two forces, gravity (constant) and spring force (non-constant)'''

==Connectedness==

Iterative Prediction serves as a basis for modeling complex motion throughout time. While some of the basic examples do not appear to be too trivial, iterative prediction can be used to solve more difficult problems. Having now almost completed a semester of mechanics, one topic that I became interested in almost right away was this idea of being able to solve problems in which we aren't necessarily concerned with the physics of the system at the current moment in time. Iterative Prediction of motion is just one of the many different techniques we have used to study the "physics of the future."

Iterative Prediction can have many applications to the field of Chemical Engineering, namely in the principles governing momentum transfer in chemical processes. A simple example is in studying fluid flow in a process. By making use of iterative prediction and the momentum principle, a chemical engineer can model the change in momentum of the fluid flow by analyzing the net force on the fluid (likely due to pressure) and the initial momentum of the fluid flow. In industry, this can be important so as to predict accumulation in a process and thus allow for modeling of an efficient process to maximize output.

==History==

The physics behind iterative prediction is nothing more than simple application of the momentum principle and projectile motion, which has existed in classical mechanics for quite some time. However, the practical application of iterative prediction for analyzing systems has evolved due to advances in computational methods and technology. With the computational power of a computer to iteratively calculate changes in momentum through time, one can analyze a system's motion extremely quickly without tedious and difficult mathematical calculations.

== See also ==

There are many public resources that delve further into iterative prediction examples of more complex motion.

===External links===

[http://en.wikipedia.org/wiki/Portal:Physics Physics Portal]

[http://hyperphysics.phy-astr.gsu.edu/hbase/hph.html HyperPhysics Concept Map]

[http://ocw.mit.edu/resources/res-8-002-a-wikitextbook-for-introductory-mechanics-fall-2009/index.htm MIT Open Courseware Introductory Mechanics]

[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 2015. Wednesday, Week 2 Lecture Slides. Fenton, Flavio H

Georgia Institute of Technology. Physics Department. PHYS 2211. Fall 2015. Monday, Week 3 Lecture Slides. Fenton, Flavio H

Iterative Prediction

2016-04-16T06:15:15Z

Jbarua: /* Example Calculations */

Claimed by Joshua Barua (jbarua3)

By tconnors3

==An Overview==
Iterative Prediction is a process used to approximate the effects of a force, be it constant or varying, on the motion of an object. This process does this approximation splitting the time interval up into many small intervals, finding the force on the object at each of the intervals, and then calculation the change of momentum and change of position for the object across each of the time intervals. Understanding Iterative Prediction also requires an understanding of the Momentum Principle, because you are constantly using the equation dp = F * deltat to update your momentum followed by your position.

Iterative Prediction is a necessary technique to employ when doing physics because, even though our fundamental principles are based on calculus and derivatives and you should be able to solve these problems using integrals, it is often impossible to solve the integrals required to solve complex problems. Iterative Prediction also becomes much more convenient when you implement it using a program, such as vPython, which allows you to make the approximation use as many steps as your processing power can handle, which can make your prediction much more accurate.

===A Mathematical Model===

By starting from the general equation for the Momentum Principle, a formula can derived to predict the momentum of a given system at a specified point in the future. This is often referred to as the ''momentum update form'' of the Momentum Principle:

<div style="text-align: center;"><math>{{(1)}\frac{d\vec{p}}{dt}}_{system} = \vec{F}_{net}</math></div>

<div style="text-align: center;"><math>{{(2)}\vec{p}_{f} - \vec{p}_{i} = \vec{F}_{net}{Δt}}</math></div>

<div style="text-align: center;"><math>{{(3)}\vec{p}_{f} = \vec{p}_{i} + \vec{F}_{net}{Δt}}</math></div>

Equation <math>{(3)}</math> gives the momentum update form of the momentum principle. As it shows, the momentum of a system can be predicted if the time period of the interaction and the external net force on the system are known. Momentum can be iteratively predicted like such for uniform or non-uniform time periods. Additionally, with update of momentum, velocity and position can be updated similarly to better reflect the motion over time of the system:

<div style="text-align: center;">'''Velocity Update Formula''':

<math>{\vec{v}_{f} = \vec{v}_{i} + \frac{\vec{F}_{net}}{m}}{Δt}</math>

'''Position Update Formula''':

<math>{\vec{r}_{f} = \vec{r}_{i} + \vec{v}_{avg}{Δt}}</math></div>

===A Visual Model===

As the momentum update formula suggests, the final momentum (shown here as <math>\vec{p}_{future}</math>) of a system after a given change in time should be the vector sum of the initial momentum (denoted <math>\vec{p}_{now}</math>) and the net force on the system multiplied by the scalar time change (<math>\vec{F}_{net}{Δt}</math>).
[[Image:MOTION.png|center|650x300px|]]

As the diagram suggests, each individual step can be analyzed through application of the momentum update form of the Momentum Principle. Once the final momentum is calculated, by utilizing the position and velocity update formula, the final velocity and position of the system can be determined at the end of the time interval of interest.

You can observe this Iterative Prediction in action by viewing this Vpython Script[http://www.glowscript.org/#/user/JBarua/folder/Public/program/ProjectilePublic] (http://www.glowscript.org/#/user/JBarua/folder/Public/program/ProjectilePublic). It shows the flight path of a ball given an initial positive x and y velocity until it hits the ground, calculated using Iterative Prediction. In order to get a visual cue on how changing the time step of an Iterative Prediction calculation, try changing the deltat value from .15 to .05. You will see that making deltat smaller will place the approximation points closer together, which will give the ball's flight path a more consistent curve.

==Example Calculations==

There are a variety of different problems that can be solved by utilizing the momentum update form of the Momentum Principle. They can vary in difficulty and require any number of iterations. It is often prudent to calculate these iterations in a program loop to save time and avoid miscalculations.

===Example #1 - Momentum Update===

'''A boy standing on level ground throws a 2 kg ball into the air at an initial velocity of <math>{<8,6,0> m/s}</math>. If the only force acting on the ball is gravity, what is the final momentum of the ball after 0.2 seconds?'''

This problem can be solved simply by solving for final momentum in the momentum update equation:

<div style="text-align: center;"><math>{\vec{p}_{f} = {m}\vec{v} + \vec{F}_{grav}{Δt}}</math></div>

<div style="text-align: center;"><math>{\vec{p}_{f} = <16,12,0> + <0,-3.92,0>}</math></div>

<div style="text-align: center;"><math>{\vec{p}_{f} = <16,8.08,0>}</math></div>

===Example #2 - Trajectory Maxima===

'''A 1 kg ball is kicked from location <math>{<9,0,0> m}</math> giving it an initial velocity of <math>{<-10,13,0> m/s}</math>. What is the maximum height that the ball will reach along its trajectory?'''

This problem is a bit more complicated than the previous one. Before calculating the maximum height of the ball, we must know the time it takes for the ball to reach its maximum height by analyzing the motion of the ball in the +y direction through the momentum update equation:

<div style="text-align: center;"><math>{t}_{f} = {v}_{i,y}{m}/{F}_{net,y}</math></div>

<div style="text-align: center;"><math>{t}_{f} = \frac{(13)(1)}{(1)(9.8)} = 1.326 s</math></div>

Now that we know how long it takes for the ball to reach its maximum height, we can solve for the final height of the ball by utilizing the position update equation:

<div style="text-align: center;"><math>{r}_{f,y} = {r}_{i,y} + {v}_{avg,y}{Δt}</math></div>

<div style="text-align: center;"><math>{r}_{f,y} = 0 + \frac{13+0}{2}(1.326) = 8.64 m</math></div>

===Example #3 - Varying Force
'''Model the motion of a spring with a ball attached to the end, taken into consideration of two forces, gravity (constant) and spring force (non-constant)'''

==Connectedness==

Iterative Prediction serves as a basis for modeling complex motion throughout time. While some of the basic examples do not appear to be too trivial, iterative prediction can be used to solve more difficult problems. Having now almost completed a semester of mechanics, one topic that I became interested in almost right away was this idea of being able to solve problems in which we aren't necessarily concerned with the physics of the system at the current moment in time. Iterative Prediction of motion is just one of the many different techniques we have used to study the "physics of the future."

Iterative Prediction can have many applications to the field of Chemical Engineering, namely in the principles governing momentum transfer in chemical processes. A simple example is in studying fluid flow in a process. By making use of iterative prediction and the momentum principle, a chemical engineer can model the change in momentum of the fluid flow by analyzing the net force on the fluid (likely due to pressure) and the initial momentum of the fluid flow. In industry, this can be important so as to predict accumulation in a process and thus allow for modeling of an efficient process to maximize output.

==History==

The physics behind iterative prediction is nothing more than simple application of the momentum principle and projectile motion, which has existed in classical mechanics for quite some time. However, the practical application of iterative prediction for analyzing systems has evolved due to advances in computational methods and technology. With the computational power of a computer to iteratively calculate changes in momentum through time, one can analyze a system's motion extremely quickly without tedious and difficult mathematical calculations.

== See also ==

There are many public resources that delve further into iterative prediction examples of more complex motion.

===External links===

[http://en.wikipedia.org/wiki/Portal:Physics Physics Portal]

[http://hyperphysics.phy-astr.gsu.edu/hbase/hph.html HyperPhysics Concept Map]

[http://ocw.mit.edu/resources/res-8-002-a-wikitextbook-for-introductory-mechanics-fall-2009/index.htm MIT Open Courseware Introductory Mechanics]

[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 2015. Wednesday, Week 2 Lecture Slides. Fenton, Flavio H

Georgia Institute of Technology. Physics Department. PHYS 2211. Fall 2015. Monday, Week 3 Lecture Slides. Fenton, Flavio H

Iterative Prediction

2016-04-16T06:04:12Z

Jbarua: /* A Visual Model */

Claimed by Joshua Barua (jbarua3)

By tconnors3

==An Overview==
Iterative Prediction is a process used to approximate the effects of a force, be it constant or varying, on the motion of an object. This process does this approximation splitting the time interval up into many small intervals, finding the force on the object at each of the intervals, and then calculation the change of momentum and change of position for the object across each of the time intervals. Understanding Iterative Prediction also requires an understanding of the Momentum Principle, because you are constantly using the equation dp = F * deltat to update your momentum followed by your position.

Iterative Prediction is a necessary technique to employ when doing physics because, even though our fundamental principles are based on calculus and derivatives and you should be able to solve these problems using integrals, it is often impossible to solve the integrals required to solve complex problems. Iterative Prediction also becomes much more convenient when you implement it using a program, such as vPython, which allows you to make the approximation use as many steps as your processing power can handle, which can make your prediction much more accurate.

===A Mathematical Model===

By starting from the general equation for the Momentum Principle, a formula can derived to predict the momentum of a given system at a specified point in the future. This is often referred to as the ''momentum update form'' of the Momentum Principle:

<div style="text-align: center;"><math>{{(1)}\frac{d\vec{p}}{dt}}_{system} = \vec{F}_{net}</math></div>

<div style="text-align: center;"><math>{{(2)}\vec{p}_{f} - \vec{p}_{i} = \vec{F}_{net}{Δt}}</math></div>

<div style="text-align: center;"><math>{{(3)}\vec{p}_{f} = \vec{p}_{i} + \vec{F}_{net}{Δt}}</math></div>

Equation <math>{(3)}</math> gives the momentum update form of the momentum principle. As it shows, the momentum of a system can be predicted if the time period of the interaction and the external net force on the system are known. Momentum can be iteratively predicted like such for uniform or non-uniform time periods. Additionally, with update of momentum, velocity and position can be updated similarly to better reflect the motion over time of the system:

<div style="text-align: center;">'''Velocity Update Formula''':

<math>{\vec{v}_{f} = \vec{v}_{i} + \frac{\vec{F}_{net}}{m}}{Δt}</math>

'''Position Update Formula''':

<math>{\vec{r}_{f} = \vec{r}_{i} + \vec{v}_{avg}{Δt}}</math></div>

===A Visual Model===

As the momentum update formula suggests, the final momentum (shown here as <math>\vec{p}_{future}</math>) of a system after a given change in time should be the vector sum of the initial momentum (denoted <math>\vec{p}_{now}</math>) and the net force on the system multiplied by the scalar time change (<math>\vec{F}_{net}{Δt}</math>).
[[Image:MOTION.png|center|650x300px|]]

As the diagram suggests, each individual step can be analyzed through application of the momentum update form of the Momentum Principle. Once the final momentum is calculated, by utilizing the position and velocity update formula, the final velocity and position of the system can be determined at the end of the time interval of interest.

You can observe this Iterative Prediction in action by viewing this Vpython Script[http://www.glowscript.org/#/user/JBarua/folder/Public/program/ProjectilePublic] (http://www.glowscript.org/#/user/JBarua/folder/Public/program/ProjectilePublic). It shows the flight path of a ball given an initial positive x and y velocity until it hits the ground, calculated using Iterative Prediction. In order to get a visual cue on how changing the time step of an Iterative Prediction calculation, try changing the deltat value from .15 to .05. You will see that making deltat smaller will place the approximation points closer together, which will give the ball's flight path a more consistent curve.

==Example Calculations==

There are a variety of different problems that can be solved by utilizing the momentum update form of the Momentum Principle. They can vary in difficulty and require any number of iterations. It is often prudent to calculate these iterations in a program loop to save time and avoid miscalculations.

===Example #1 - Momentum Update===

'''A boy standing on level ground throws a 2 kg ball into the air at an initial velocity of <math>{<8,6,0> m/s}</math>. If the only force acting on the ball is gravity, what is the final momentum of the ball after 0.2 seconds?'''

This problem can be solved simply by solving for final momentum in the momentum update equation:

<div style="text-align: center;"><math>{\vec{p}_{f} = {m}\vec{v} + \vec{F}_{grav}{Δt}}</math></div>

<div style="text-align: center;"><math>{\vec{p}_{f} = <16,12,0> + <0,-3.92,0>}</math></div>

<div style="text-align: center;"><math>{\vec{p}_{f} = <16,8.08,0>}</math></div>

===Example #2 - Trajectory Maxima===

'''A 1 kg ball is kicked from location <math>{<9,0,0> m}</math> giving it an initial velocity of <math>{<-10,13,0> m/s}</math>. What is the maximum height that the ball will reach along its trajectory?'''

This problem is a bit more complicated than the previous one. Before calculating the maximum height of the ball, we must know the time it takes for the ball to reach its maximum height by analyzing the motion of the ball in the +y direction through the momentum update equation:

<div style="text-align: center;"><math>{t}_{f} = {v}_{i,y}{m}/{F}_{net,y}</math></div>

<div style="text-align: center;"><math>{t}_{f} = \frac{(13)(1)}{(1)(9.8)} = 1.326 s</math></div>

Now that we know how long it takes for the ball to reach its maximum height, we can solve for the final height of the ball by utilizing the position update equation:

<div style="text-align: center;"><math>{r}_{f,y} = {r}_{i,y} + {v}_{avg,y}{Δt}</math></div>

<div style="text-align: center;"><math>{r}_{f,y} = 0 + \frac{13+0}{2}(1.326) = 8.64 m</math></div>

==Connectedness==

Iterative Prediction serves as a basis for modeling complex motion throughout time. While some of the basic examples do not appear to be too trivial, iterative prediction can be used to solve more difficult problems. Having now almost completed a semester of mechanics, one topic that I became interested in almost right away was this idea of being able to solve problems in which we aren't necessarily concerned with the physics of the system at the current moment in time. Iterative Prediction of motion is just one of the many different techniques we have used to study the "physics of the future."

Iterative Prediction can have many applications to the field of Chemical Engineering, namely in the principles governing momentum transfer in chemical processes. A simple example is in studying fluid flow in a process. By making use of iterative prediction and the momentum principle, a chemical engineer can model the change in momentum of the fluid flow by analyzing the net force on the fluid (likely due to pressure) and the initial momentum of the fluid flow. In industry, this can be important so as to predict accumulation in a process and thus allow for modeling of an efficient process to maximize output.

==History==

The physics behind iterative prediction is nothing more than simple application of the momentum principle and projectile motion, which has existed in classical mechanics for quite some time. However, the practical application of iterative prediction for analyzing systems has evolved due to advances in computational methods and technology. With the computational power of a computer to iteratively calculate changes in momentum through time, one can analyze a system's motion extremely quickly without tedious and difficult mathematical calculations.

== See also ==

There are many public resources that delve further into iterative prediction examples of more complex motion.

===External links===

[http://en.wikipedia.org/wiki/Portal:Physics Physics Portal]

[http://hyperphysics.phy-astr.gsu.edu/hbase/hph.html HyperPhysics Concept Map]

[http://ocw.mit.edu/resources/res-8-002-a-wikitextbook-for-introductory-mechanics-fall-2009/index.htm MIT Open Courseware Introductory Mechanics]

[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 2015. Wednesday, Week 2 Lecture Slides. Fenton, Flavio H

Georgia Institute of Technology. Physics Department. PHYS 2211. Fall 2015. Monday, Week 3 Lecture Slides. Fenton, Flavio H

Iterative Prediction

2016-04-16T06:03:21Z

Jbarua: /* A Visual Model */

Claimed by Joshua Barua (jbarua3)

By tconnors3

==An Overview==
Iterative Prediction is a process used to approximate the effects of a force, be it constant or varying, on the motion of an object. This process does this approximation splitting the time interval up into many small intervals, finding the force on the object at each of the intervals, and then calculation the change of momentum and change of position for the object across each of the time intervals. Understanding Iterative Prediction also requires an understanding of the Momentum Principle, because you are constantly using the equation dp = F * deltat to update your momentum followed by your position.

Iterative Prediction is a necessary technique to employ when doing physics because, even though our fundamental principles are based on calculus and derivatives and you should be able to solve these problems using integrals, it is often impossible to solve the integrals required to solve complex problems. Iterative Prediction also becomes much more convenient when you implement it using a program, such as vPython, which allows you to make the approximation use as many steps as your processing power can handle, which can make your prediction much more accurate.

===A Mathematical Model===

By starting from the general equation for the Momentum Principle, a formula can derived to predict the momentum of a given system at a specified point in the future. This is often referred to as the ''momentum update form'' of the Momentum Principle:

<div style="text-align: center;"><math>{{(1)}\frac{d\vec{p}}{dt}}_{system} = \vec{F}_{net}</math></div>

<div style="text-align: center;"><math>{{(2)}\vec{p}_{f} - \vec{p}_{i} = \vec{F}_{net}{Δt}}</math></div>

<div style="text-align: center;"><math>{{(3)}\vec{p}_{f} = \vec{p}_{i} + \vec{F}_{net}{Δt}}</math></div>

Equation <math>{(3)}</math> gives the momentum update form of the momentum principle. As it shows, the momentum of a system can be predicted if the time period of the interaction and the external net force on the system are known. Momentum can be iteratively predicted like such for uniform or non-uniform time periods. Additionally, with update of momentum, velocity and position can be updated similarly to better reflect the motion over time of the system:

<div style="text-align: center;">'''Velocity Update Formula''':

<math>{\vec{v}_{f} = \vec{v}_{i} + \frac{\vec{F}_{net}}{m}}{Δt}</math>

'''Position Update Formula''':

<math>{\vec{r}_{f} = \vec{r}_{i} + \vec{v}_{avg}{Δt}}</math></div>

===A Visual Model===

As the momentum update formula suggests, the final momentum (shown here as <math>\vec{p}_{future}</math>) of a system after a given change in time should be the vector sum of the initial momentum (denoted <math>\vec{p}_{now}</math>) and the net force on the system multiplied by the scalar time change (<math>\vec{F}_{net}{Δt}</math>).
[[Image:MOTION.png|center|650x300px|]]

As the diagram suggests, each individual step can be analyzed through application of the momentum update form of the Momentum Principle. Once the final momentum is calculated, by utilizing the position and velocity update formula, the final velocity and position of the system can be determined at the end of the time interval of interest.

You can observe this Iterative Prediction in action by viewing this Vpython Script[http://www.glowscript.org/#/user/JBarua/folder/Public/program/ProjectilePublic]. It shows the flight path of a ball given an initial positive x and y velocity until it hits the ground, calculated using Iterative Prediction. In order to get a visual cue on how changing the time step of an Iterative Prediction calculation, try changing the deltat value from .15 to .05. You will see that making deltat smaller will place the approximation points closer together, which will give the ball's flight path a more consistent curve.

==Example Calculations==

There are a variety of different problems that can be solved by utilizing the momentum update form of the Momentum Principle. They can vary in difficulty and require any number of iterations. It is often prudent to calculate these iterations in a program loop to save time and avoid miscalculations.

===Example #1 - Momentum Update===

'''A boy standing on level ground throws a 2 kg ball into the air at an initial velocity of <math>{<8,6,0> m/s}</math>. If the only force acting on the ball is gravity, what is the final momentum of the ball after 0.2 seconds?'''

This problem can be solved simply by solving for final momentum in the momentum update equation:

<div style="text-align: center;"><math>{\vec{p}_{f} = {m}\vec{v} + \vec{F}_{grav}{Δt}}</math></div>

<div style="text-align: center;"><math>{\vec{p}_{f} = <16,12,0> + <0,-3.92,0>}</math></div>

<div style="text-align: center;"><math>{\vec{p}_{f} = <16,8.08,0>}</math></div>

===Example #2 - Trajectory Maxima===

'''A 1 kg ball is kicked from location <math>{<9,0,0> m}</math> giving it an initial velocity of <math>{<-10,13,0> m/s}</math>. What is the maximum height that the ball will reach along its trajectory?'''

This problem is a bit more complicated than the previous one. Before calculating the maximum height of the ball, we must know the time it takes for the ball to reach its maximum height by analyzing the motion of the ball in the +y direction through the momentum update equation:

<div style="text-align: center;"><math>{t}_{f} = {v}_{i,y}{m}/{F}_{net,y}</math></div>

<div style="text-align: center;"><math>{t}_{f} = \frac{(13)(1)}{(1)(9.8)} = 1.326 s</math></div>

Now that we know how long it takes for the ball to reach its maximum height, we can solve for the final height of the ball by utilizing the position update equation:

<div style="text-align: center;"><math>{r}_{f,y} = {r}_{i,y} + {v}_{avg,y}{Δt}</math></div>

<div style="text-align: center;"><math>{r}_{f,y} = 0 + \frac{13+0}{2}(1.326) = 8.64 m</math></div>

==Connectedness==

Iterative Prediction serves as a basis for modeling complex motion throughout time. While some of the basic examples do not appear to be too trivial, iterative prediction can be used to solve more difficult problems. Having now almost completed a semester of mechanics, one topic that I became interested in almost right away was this idea of being able to solve problems in which we aren't necessarily concerned with the physics of the system at the current moment in time. Iterative Prediction of motion is just one of the many different techniques we have used to study the "physics of the future."

Iterative Prediction can have many applications to the field of Chemical Engineering, namely in the principles governing momentum transfer in chemical processes. A simple example is in studying fluid flow in a process. By making use of iterative prediction and the momentum principle, a chemical engineer can model the change in momentum of the fluid flow by analyzing the net force on the fluid (likely due to pressure) and the initial momentum of the fluid flow. In industry, this can be important so as to predict accumulation in a process and thus allow for modeling of an efficient process to maximize output.

==History==

The physics behind iterative prediction is nothing more than simple application of the momentum principle and projectile motion, which has existed in classical mechanics for quite some time. However, the practical application of iterative prediction for analyzing systems has evolved due to advances in computational methods and technology. With the computational power of a computer to iteratively calculate changes in momentum through time, one can analyze a system's motion extremely quickly without tedious and difficult mathematical calculations.

== See also ==

There are many public resources that delve further into iterative prediction examples of more complex motion.

===External links===

[http://en.wikipedia.org/wiki/Portal:Physics Physics Portal]

[http://hyperphysics.phy-astr.gsu.edu/hbase/hph.html HyperPhysics Concept Map]

[http://ocw.mit.edu/resources/res-8-002-a-wikitextbook-for-introductory-mechanics-fall-2009/index.htm MIT Open Courseware Introductory Mechanics]

[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 2015. Wednesday, Week 2 Lecture Slides. Fenton, Flavio H

Georgia Institute of Technology. Physics Department. PHYS 2211. Fall 2015. Monday, Week 3 Lecture Slides. Fenton, Flavio H

Iterative Prediction

2016-04-16T05:54:04Z

Jbarua: /* An Overview */

Claimed by Joshua Barua (jbarua3)

By tconnors3

==An Overview==
Iterative Prediction is a process used to approximate the effects of a force, be it constant or varying, on the motion of an object. This process does this approximation splitting the time interval up into many small intervals, finding the force on the object at each of the intervals, and then calculation the change of momentum and change of position for the object across each of the time intervals. Understanding Iterative Prediction also requires an understanding of the Momentum Principle, because you are constantly using the equation dp = F * deltat to update your momentum followed by your position.

Iterative Prediction is a necessary technique to employ when doing physics because, even though our fundamental principles are based on calculus and derivatives and you should be able to solve these problems using integrals, it is often impossible to solve the integrals required to solve complex problems. Iterative Prediction also becomes much more convenient when you implement it using a program, such as vPython, which allows you to make the approximation use as many steps as your processing power can handle, which can make your prediction much more accurate.

===A Mathematical Model===

By starting from the general equation for the Momentum Principle, a formula can derived to predict the momentum of a given system at a specified point in the future. This is often referred to as the ''momentum update form'' of the Momentum Principle:

<div style="text-align: center;"><math>{{(1)}\frac{d\vec{p}}{dt}}_{system} = \vec{F}_{net}</math></div>

<div style="text-align: center;"><math>{{(2)}\vec{p}_{f} - \vec{p}_{i} = \vec{F}_{net}{Δt}}</math></div>

<div style="text-align: center;"><math>{{(3)}\vec{p}_{f} = \vec{p}_{i} + \vec{F}_{net}{Δt}}</math></div>

Equation <math>{(3)}</math> gives the momentum update form of the momentum principle. As it shows, the momentum of a system can be predicted if the time period of the interaction and the external net force on the system are known. Momentum can be iteratively predicted like such for uniform or non-uniform time periods. Additionally, with update of momentum, velocity and position can be updated similarly to better reflect the motion over time of the system:

<div style="text-align: center;">'''Velocity Update Formula''':

<math>{\vec{v}_{f} = \vec{v}_{i} + \frac{\vec{F}_{net}}{m}}{Δt}</math>

'''Position Update Formula''':

<math>{\vec{r}_{f} = \vec{r}_{i} + \vec{v}_{avg}{Δt}}</math></div>

===A Visual Model===

As the momentum update formula suggests, the final momentum (shown here as <math>\vec{p}_{future}</math>) of a system after a given change in time should be the vector sum of the initial momentum (denoted <math>\vec{p}_{now}</math>) and the net force on the system multiplied by the scalar time change (<math>\vec{F}_{net}{Δt}</math>).
[[Image:MOTION.png|center|650x300px|]]

As the diagram suggests, each individual step can be analyzed through application of the momentum update form of the Momentum Principle. Once the final momentum is calculated, by utilizing the position and velocity update formulae, the final velocity and position of the system can be determined at the end of the time interval of interest.

==Example Calculations==

There are a variety of different problems that can be solved by utilizing the momentum update form of the Momentum Principle. They can vary in difficulty and require any number of iterations. It is often prudent to calculate these iterations in a program loop to save time and avoid miscalculations.

===Example #1 - Momentum Update===

'''A boy standing on level ground throws a 2 kg ball into the air at an initial velocity of <math>{<8,6,0> m/s}</math>. If the only force acting on the ball is gravity, what is the final momentum of the ball after 0.2 seconds?'''

This problem can be solved simply by solving for final momentum in the momentum update equation:

<div style="text-align: center;"><math>{\vec{p}_{f} = {m}\vec{v} + \vec{F}_{grav}{Δt}}</math></div>

<div style="text-align: center;"><math>{\vec{p}_{f} = <16,12,0> + <0,-3.92,0>}</math></div>

<div style="text-align: center;"><math>{\vec{p}_{f} = <16,8.08,0>}</math></div>

===Example #2 - Trajectory Maxima===

'''A 1 kg ball is kicked from location <math>{<9,0,0> m}</math> giving it an initial velocity of <math>{<-10,13,0> m/s}</math>. What is the maximum height that the ball will reach along its trajectory?'''

This problem is a bit more complicated than the previous one. Before calculating the maximum height of the ball, we must know the time it takes for the ball to reach its maximum height by analyzing the motion of the ball in the +y direction through the momentum update equation:

<div style="text-align: center;"><math>{t}_{f} = {v}_{i,y}{m}/{F}_{net,y}</math></div>

<div style="text-align: center;"><math>{t}_{f} = \frac{(13)(1)}{(1)(9.8)} = 1.326 s</math></div>

Now that we know how long it takes for the ball to reach its maximum height, we can solve for the final height of the ball by utilizing the position update equation:

<div style="text-align: center;"><math>{r}_{f,y} = {r}_{i,y} + {v}_{avg,y}{Δt}</math></div>

<div style="text-align: center;"><math>{r}_{f,y} = 0 + \frac{13+0}{2}(1.326) = 8.64 m</math></div>

==Connectedness==

Iterative Prediction serves as a basis for modeling complex motion throughout time. While some of the basic examples do not appear to be too trivial, iterative prediction can be used to solve more difficult problems. Having now almost completed a semester of mechanics, one topic that I became interested in almost right away was this idea of being able to solve problems in which we aren't necessarily concerned with the physics of the system at the current moment in time. Iterative Prediction of motion is just one of the many different techniques we have used to study the "physics of the future."

Iterative Prediction can have many applications to the field of Chemical Engineering, namely in the principles governing momentum transfer in chemical processes. A simple example is in studying fluid flow in a process. By making use of iterative prediction and the momentum principle, a chemical engineer can model the change in momentum of the fluid flow by analyzing the net force on the fluid (likely due to pressure) and the initial momentum of the fluid flow. In industry, this can be important so as to predict accumulation in a process and thus allow for modeling of an efficient process to maximize output.

==History==

The physics behind iterative prediction is nothing more than simple application of the momentum principle and projectile motion, which has existed in classical mechanics for quite some time. However, the practical application of iterative prediction for analyzing systems has evolved due to advances in computational methods and technology. With the computational power of a computer to iteratively calculate changes in momentum through time, one can analyze a system's motion extremely quickly without tedious and difficult mathematical calculations.

== See also ==

There are many public resources that delve further into iterative prediction examples of more complex motion.

===External links===

[http://en.wikipedia.org/wiki/Portal:Physics Physics Portal]

[http://hyperphysics.phy-astr.gsu.edu/hbase/hph.html HyperPhysics Concept Map]

[http://ocw.mit.edu/resources/res-8-002-a-wikitextbook-for-introductory-mechanics-fall-2009/index.htm MIT Open Courseware Introductory Mechanics]

[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 2015. Wednesday, Week 2 Lecture Slides. Fenton, Flavio H

Georgia Institute of Technology. Physics Department. PHYS 2211. Fall 2015. Monday, Week 3 Lecture Slides. Fenton, Flavio H

Iterative Prediction

2016-04-12T00:11:22Z

Jbarua:

Claimed by Joshua Barua (jbarua3)

By tconnors3

==An Overview==

The Momentum Principle is one of the fundamental principles in the study of mechanics and dynamics. By applying it to real world problems, the motion of systems can be modeled at specific points in time. Additionally, analyzing the implications of the momentum principle, physicists can not only pinpoint behavior of systems, but can also predict the motion of systems at specified times in the future.

===A Mathematical Model===

By starting from the general equation for the Momentum Principle, a formula can derived to predict the momentum of a given system at a specified point in the future. This is often referred to as the ''momentum update form'' of the Momentum Principle:

<div style="text-align: center;"><math>{{(1)}\frac{d\vec{p}}{dt}}_{system} = \vec{F}_{net}</math></div>

<div style="text-align: center;"><math>{{(2)}\vec{p}_{f} - \vec{p}_{i} = \vec{F}_{net}{Δt}}</math></div>

<div style="text-align: center;"><math>{{(3)}\vec{p}_{f} = \vec{p}_{i} + \vec{F}_{net}{Δt}}</math></div>

Equation <math>{(3)}</math> gives the momentum update form of the momentum principle. As it shows, the momentum of a system can be predicted if the time period of the interaction and the external net force on the system are known. Momentum can be iteratively predicted like such for uniform or non-uniform time periods. Additionally, with update of momentum, velocity and position can be updated similarly to better reflect the motion over time of the system:

<div style="text-align: center;">'''Velocity Update Formula''':

<math>{\vec{v}_{f} = \vec{v}_{i} + \frac{\vec{F}_{net}}{m}}{Δt}</math>

'''Position Update Formula''':

<math>{\vec{r}_{f} = \vec{r}_{i} + \vec{v}_{avg}{Δt}}</math></div>

===A Visual Model===

As the momentum update formula suggests, the final momentum (shown here as <math>\vec{p}_{future}</math>) of a system after a given change in time should be the vector sum of the initial momentum (denoted <math>\vec{p}_{now}</math>) and the net force on the system multiplied by the scalar time change (<math>\vec{F}_{net}{Δt}</math>).
[[Image:MOTION.png|center|650x300px|]]

As the diagram suggests, each individual step can be analyzed through application of the momentum update form of the Momentum Principle. Once the final momentum is calculated, by utilizing the position and velocity update formulae, the final velocity and position of the system can be determined at the end of the time interval of interest.

==Example Calculations==

There are a variety of different problems that can be solved by utilizing the momentum update form of the Momentum Principle. They can vary in difficulty and require any number of iterations. It is often prudent to calculate these iterations in a program loop to save time and avoid miscalculations.

===Example #1 - Momentum Update===

'''A boy standing on level ground throws a 2 kg ball into the air at an initial velocity of <math>{<8,6,0> m/s}</math>. If the only force acting on the ball is gravity, what is the final momentum of the ball after 0.2 seconds?'''

This problem can be solved simply by solving for final momentum in the momentum update equation:

<div style="text-align: center;"><math>{\vec{p}_{f} = {m}\vec{v} + \vec{F}_{grav}{Δt}}</math></div>

<div style="text-align: center;"><math>{\vec{p}_{f} = <16,12,0> + <0,-3.92,0>}</math></div>

<div style="text-align: center;"><math>{\vec{p}_{f} = <16,8.08,0>}</math></div>

===Example #2 - Trajectory Maxima===

'''A 1 kg ball is kicked from location <math>{<9,0,0> m}</math> giving it an initial velocity of <math>{<-10,13,0> m/s}</math>. What is the maximum height that the ball will reach along its trajectory?'''

This problem is a bit more complicated than the previous one. Before calculating the maximum height of the ball, we must know the time it takes for the ball to reach its maximum height by analyzing the motion of the ball in the +y direction through the momentum update equation:

<div style="text-align: center;"><math>{t}_{f} = {v}_{i,y}{m}/{F}_{net,y}</math></div>

<div style="text-align: center;"><math>{t}_{f} = \frac{(13)(1)}{(1)(9.8)} = 1.326 s</math></div>

Now that we know how long it takes for the ball to reach its maximum height, we can solve for the final height of the ball by utilizing the position update equation:

<div style="text-align: center;"><math>{r}_{f,y} = {r}_{i,y} + {v}_{avg,y}{Δt}</math></div>

<div style="text-align: center;"><math>{r}_{f,y} = 0 + \frac{13+0}{2}(1.326) = 8.64 m</math></div>

==Connectedness==

Iterative Prediction serves as a basis for modeling complex motion throughout time. While some of the basic examples do not appear to be too trivial, iterative prediction can be used to solve more difficult problems. Having now almost completed a semester of mechanics, one topic that I became interested in almost right away was this idea of being able to solve problems in which we aren't necessarily concerned with the physics of the system at the current moment in time. Iterative Prediction of motion is just one of the many different techniques we have used to study the "physics of the future."

Iterative Prediction can have many applications to the field of Chemical Engineering, namely in the principles governing momentum transfer in chemical processes. A simple example is in studying fluid flow in a process. By making use of iterative prediction and the momentum principle, a chemical engineer can model the change in momentum of the fluid flow by analyzing the net force on the fluid (likely due to pressure) and the initial momentum of the fluid flow. In industry, this can be important so as to predict accumulation in a process and thus allow for modeling of an efficient process to maximize output.

==History==

The physics behind iterative prediction is nothing more than simple application of the momentum principle and projectile motion, which has existed in classical mechanics for quite some time. However, the practical application of iterative prediction for analyzing systems has evolved due to advances in computational methods and technology. With the computational power of a computer to iteratively calculate changes in momentum through time, one can analyze a system's motion extremely quickly without tedious and difficult mathematical calculations.

== See also ==

There are many public resources that delve further into iterative prediction examples of more complex motion.

===External links===

[http://en.wikipedia.org/wiki/Portal:Physics Physics Portal]

[http://hyperphysics.phy-astr.gsu.edu/hbase/hph.html HyperPhysics Concept Map]

[http://ocw.mit.edu/resources/res-8-002-a-wikitextbook-for-introductory-mechanics-fall-2009/index.htm MIT Open Courseware Introductory Mechanics]

[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 2015. Wednesday, Week 2 Lecture Slides. Fenton, Flavio H

Georgia Institute of Technology. Physics Department. PHYS 2211. Fall 2015. Monday, Week 3 Lecture Slides. Fenton, Flavio H