Physics Book - User contributions [en]

Iterative Prediction

2016-04-08T17:56:19Z

Jbell73:

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

Iterative Prediction

2016-04-03T18:47:52Z

Jbell73:

By tconnors3
'''CLAIMED FOR IMPROVEMENT BY jbell73, SPRING 2016'''

==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

Iterative Prediction

2016-04-03T18:46:53Z

Jbell73:

By tconnors3
'''CLAIMED FOR IMPROVEMENT BY jbell73, SPRING 2016'''

==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]

==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

2016-04-01T04:20:05Z

Jbell73:

By tconnors3
'''CLAIMED FOR IMPROVEMENT BY jbell73, SPRING 2016'''

==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]

==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

2016-04-01T04:16:22Z

Jbell73:

By tconnors3
'''CLAIMED FOR IMPROVEMENT BY JOHN BELL, SPRING 2016'''

==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]

==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

Analytical Prediction

2016-04-01T03:51:46Z

Jbell73:

By Hayden McLeod (hmcleod6)

==The Main Idea==

Analytical prediction uses a mathematical function that can describe the position or velocity of a system at any given time. In contrast to iterative prediction, this means that there is no need to make multiple calculations at small steps in order to find a solution. However, due to the method of derivation of the velocity, the analytical method is only accurate to a high degree when the force applied to the system is constant. Due to this limitation, the iterative prediction method is much more generally applicable.

===A Mathematical Model===

Unlike the iterative prediction method which is derived directly from the momentum principle, the formula for analytical prediction is based of the formula for the arithmetic mean.

[[File: 1.jpg|150px]]

Here, X bar is the arithmetic mean, sigma f x is the sum of all the values, and sigma f is the total number of terms. Since we only need two values of velocity to calculate the average velocity, the formula, for this purpose, can be simplified.

[[File: htmpng1.png|150px]]

===A Computational Model===

Below is a visualization of the analytical prediction of the average velocity. As you can see, it takes two points, (a, f(a)) and (b, f(b)), and finds the average between the two points. Also depicted in the image is the incapability to model non-linear curves by comparing the average slope (velocity) compared to the actual slope (velocity).

[[File: htmjpg2.jpg|300px]]

In addition, an example of using this method in GlowScript can be seen below.

[[File: Htmpng2.png|600px]]

Not only does this code show how to use the analytical prediction method using python/GlowScript, it also shows how you can use that average velocity to calculate the momentum and update the position. These are the typical calculations associated with prediction methods.

==Examples==
===Example 1===
A train is observed to be traveling at a speed of 24 m/s before it enters a city. While in the city, it is observed going 13 m/s. What is the average velocity of the train?

24 m/s + 13 m/s = 37 m/s

37/2 = 18.5

The average velocity of the train based on the analytical prediction method was 18.5 m/s.

===Example 2===
A car is traveling at a speed of (18, -2, 0) m/s. Later, the car is observed to be going (22, 6, 0) m/s. If the car started at the location (40, 170, 0) what is it's position after 30 seconds?

First, you have to find the average velocity:

(18, -2, 0) + (22, 6, 0) = (40, 4, 0)

(40, 4, 0) /2 = (20, 2, 0)

so the average velocity is (20, 2, 0) m/s

Second, calculate the change in position:

Change in position = average velocity x time

(20, 2, 0) * 30 = (600, 60, 0)

So the change in position is (600, 60, 0) m

Finally, update the position:

final position = initial position + change in position

(40, 170, 0) + (600, 60, 0) = (640, 230, 0)

The final position is (640, 230, 0) m

==Connectedness==
#How is this topic connected to something that you are interested in?
Since this topic is based on such a fundamental mathematical principle, my interest in it lies in how wide spread this principle can be used not only in physics but in many other applications, fields, and scenarios.

#How is it connected to your major?
As stated in the previous connection, this basic principle has ties in just about every major with any connection to math including business, sciences, and engineering. Specifically, methods like this are often used to try and model basic relationships of properties in material science.

#Is there an interesting industrial application?
In industry, methods like this, or methods very similar, are used every day to help process and analyze data in a very basic and quick way.

==History==

Historically, it is well known that basic algebra was discovered and implemented well before calculus, especially based on the fact that algebra was used as a basis to develop calculus. For this reason, it can be assumed that the analytical approach to physics problems related to velocity, or any other changing value, was used long before the iterative method. There is no information on who would have used this approach or when this approach was first used in a physics application.

== See also ==

Vectors,
Momentum Principle,
Iterative Prediction

===Further reading===

There are several easily accessible resources with more information on analytical prediction methods and arithmetic means

===External links===

==References==

http://www.platinumgmat.com/gmat_study_guide/statistics_mean (information)
http://ncalculators.com/statistics/group-arithmetic-mean.htm (image)
https://quizlet.com/24263897/mcat-physics-formulas-flash-cards/ (image)
http://mathinsight.org/approximating_nonlinear_function_by_linear (image)

[[Category:Momentum]]

Analytical Prediction

2016-04-01T03:46:36Z

Jbell73:

By Hayden McLeod (hmcleod6)

'''CLAIMED FOR IMPROVEMENT BY JOHN BELL, SPRING 2016'''

==The Main Idea==

Analytical prediction uses a mathematical function that can describe the position or velocity of a system at any given time. In contrast to iterative prediction, this means that there is no need to make multiple calculations at small steps in order to find a solution. However, due to the method of derivation of the velocity, the analytical method is only accurate to a high degree when the force applied to the system is constant. Due to this limitation, the iterative prediction method is much more generally applicable.

===A Mathematical Model===

Unlike the iterative prediction method which is derived directly from the momentum principle, the formula for analytical prediction is based of the formula for the arithmetic mean.

[[File: 1.jpg|150px]]

Here, X bar is the arithmetic mean, sigma f x is the sum of all the values, and sigma f is the total number of terms. Since we only need two values of velocity to calculate the average velocity, the formula, for this purpose, can be simplified.

[[File: htmpng1.png|150px]]

===A Computational Model===

Below is a visualization of the analytical prediction of the average velocity. As you can see, it takes two points, (a, f(a)) and (b, f(b)), and finds the average between the two points. Also depicted in the image is the incapability to model non-linear curves by comparing the average slope (velocity) compared to the actual slope (velocity).

[[File: htmjpg2.jpg|300px]]

In addition, an example of using this method in GlowScript can be seen below.

[[File: Htmpng2.png|600px]]

Not only does this code show how to use the analytical prediction method using python/GlowScript, it also shows how you can use that average velocity to calculate the momentum and update the position. These are the typical calculations associated with prediction methods.

==Examples==
===Example 1===
A train is observed to be traveling at a speed of 24 m/s before it enters a city. While in the city, it is observed going 13 m/s. What is the average velocity of the train?

24 m/s + 13 m/s = 37 m/s

37/2 = 18.5

The average velocity of the train based on the analytical prediction method was 18.5 m/s.

===Example 2===
A car is traveling at a speed of (18, -2, 0) m/s. Later, the car is observed to be going (22, 6, 0) m/s. If the car started at the location (40, 170, 0) what is it's position after 30 seconds?

First, you have to find the average velocity:

(18, -2, 0) + (22, 6, 0) = (40, 4, 0)

(40, 4, 0) /2 = (20, 2, 0)

so the average velocity is (20, 2, 0) m/s

Second, calculate the change in position:

Change in position = average velocity x time

(20, 2, 0) * 30 = (600, 60, 0)

So the change in position is (600, 60, 0) m

Finally, update the position:

final position = initial position + change in position

(40, 170, 0) + (600, 60, 0) = (640, 230, 0)

The final position is (640, 230, 0) m

==Connectedness==
#How is this topic connected to something that you are interested in?
Since this topic is based on such a fundamental mathematical principle, my interest in it lies in how wide spread this principle can be used not only in physics but in many other applications, fields, and scenarios.

#How is it connected to your major?
As stated in the previous connection, this basic principle has ties in just about every major with any connection to math including business, sciences, and engineering. Specifically, methods like this are often used to try and model basic relationships of properties in material science.

#Is there an interesting industrial application?
In industry, methods like this, or methods very similar, are used every day to help process and analyze data in a very basic and quick way.

==History==

Historically, it is well known that basic algebra was discovered and implemented well before calculus, especially based on the fact that algebra was used as a basis to develop calculus. For this reason, it can be assumed that the analytical approach to physics problems related to velocity, or any other changing value, was used long before the iterative method. There is no information on who would have used this approach or when this approach was first used in a physics application.

== See also ==

Vectors,
Momentum Principle,
Iterative Prediction

===Further reading===

There are several easily accessible resources with more information on analytical prediction methods and arithmetic means

===External links===

==References==

http://www.platinumgmat.com/gmat_study_guide/statistics_mean (information)
http://ncalculators.com/statistics/group-arithmetic-mean.htm (image)
https://quizlet.com/24263897/mcat-physics-formulas-flash-cards/ (image)
http://mathinsight.org/approximating_nonlinear_function_by_linear (image)

[[Category:Momentum]]