Iterative Prediction - Revision history

Rgovind7 at 22:27, 3 December 2022

2022-12-03T22:27:38Z

← Older revision		Revision as of 18:27, 3 December 2022
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	~~Edited by Raksha Govind (rgovind7-Fall 2022)~~

	This page describes iterative prediction, a technique used to predict the motion of particles over a period of time often used in simulations.		This page describes iterative prediction, a technique used to predict the motion of particles over a period of time often used in simulations.

Rgovind7 at 22:26, 3 December 2022

2022-12-03T22:26:19Z

← Older revision		Revision as of 18:26, 3 December 2022
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	Edited by rgovind7 (Fall 2022)		Edited by Raksha Govind (rgovind7-Fall 2022)

	This page describes iterative prediction, a technique used to predict the motion of particles over a period of time often used in simulations.		This page describes iterative prediction, a technique used to predict the motion of particles over a period of time often used in simulations.

Rgovind7 at 22:24, 3 December 2022

2022-12-03T22:24:26Z

← Older revision		Revision as of 18:24, 3 December 2022
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			Edited by rgovind7 (Fall 2022)

	This page describes iterative prediction, a technique used to predict the motion of particles over a period of time often used in simulations.		This page describes iterative prediction, a technique used to predict the motion of particles over a period of time often used in simulations.

Ilukowiecki3 at 19:20, 28 March 2020

2020-03-28T19:20:07Z

← Older revision		Revision as of 15:20, 28 March 2020
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	~~'''Claimed by ilukowiecki3 Spring 2020'''~~

Ilukowiecki3 at 18:49, 24 March 2020

2020-03-24T18:49:47Z

← Older revision		Revision as of 14:49, 24 March 2020
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	The size of the time steps used during iterative prediction is referred to as the resolution of the model. Dividing a time interval into a few large time steps is called low resolution while dividing the same time interval into many small time steps is called high resolution. Performing iterative prediction with a higher resolution requires more computations but produces more accurate results. This is because steps 2 and 3 assume constant force and velocity respectively during the duration of each time step (see the "mathematical model" section). In reality, force and velocity change continually, so this is where inaccuracy is introduced. However, if the time steps are small enough that force and velocity do not significantly change during any time step, iterative prediction is a sufficiently accurate model. In the limit where the time interval is divided into an infinite number of infinitely small time steps, iterative prediction becomes a perfectly accurate model. However, this would require an infinite number of computations, and is therefore impossible to do, although in some simple situations, analytic approaches can be used to this end instead. Deciding what time step to use requires consideration of both the computational resources available and the resolution required to accurately approximate the situation. When performing iterative prediction by hand, it is impractical to perform more than a few time steps, which cannot be used to accurately model much. If computers are available, it becomes feasible to divide the time interval into thousands of time steps, which can accurately simulate most day-to-day situations. Some of the most sensitive simulations, however, require incredibly small time steps. For example, some physicists perform atom-by-atom simulations of molecular structures interacting. A slight change in the position of an atom can cause a huge change in the forces acting on it, so tiny time steps must be used in order to ensure that force and velocity do not significantly change during any one time step. These simulations are run on the world's most powerful supercomputers and often take weeks to complete. They can simulate time intervals of several nanoseconds (<math>10^{-9}</math> s) using time steps of only a few femtoseconds. (<math>10^{-15}</math> s) [https://www.ncbi.nlm.nih.gov/pmc/articles/PMC6286305/ (Click here for an example of a paper about such a simulation.)] Most of the VPython simulations on this wiki use time steps of about .01s.		The size of the time steps used during iterative prediction is referred to as the resolution of the model. Dividing a time interval into a few large time steps is called low resolution while dividing the same time interval into many small time steps is called high resolution. Performing iterative prediction with a higher resolution requires more computations but produces more accurate results. This is because steps 2 and 3 assume constant force and velocity respectively during the duration of each time step (see the "mathematical model" section). In reality, force and velocity change continually, so this is where inaccuracy is introduced. However, if the time steps are small enough that force and velocity do not significantly change during any time step, the iterative prediction is a sufficiently accurate model. In the limit where the time interval is divided into an infinite number of infinitely small time steps, iterative prediction becomes a perfectly accurate model. However, this would require an infinite number of computations, and is therefore impossible to do, although, in some simple situations, analytic approaches can be used to this end instead. Deciding what time step to use requires consideration of both the computational resources available and the resolution required to accurately approximate the situation. When performing iterative prediction by hand, it is impractical to perform more than a few time steps, which cannot be used to accurately model much. If computers are available, it becomes feasible to divide the time interval into thousands of time steps, which can accurately simulate most day-to-day situations. Some of the most sensitive simulations, however, require incredibly small time steps. For example, some physicists perform atom-by-atom simulations of molecular structures interacting. A slight change in the position of an atom can cause a huge change in the forces acting on it, so tiny time steps must be used in order to ensure that force and velocity do not significantly change during any one time step. These simulations are run on the world's most powerful supercomputers and often take weeks to complete. They can simulate time intervals of several nanoseconds (<math>10^{-9}</math> s) using time steps of only a few femtoseconds. (<math>10^{-15}</math> s) [https://www.ncbi.nlm.nih.gov/pmc/articles/PMC6286305/ (Click here for an example of a paper about such a simulation.)] Most of the VPython simulations on this wiki use time steps of about .01s.

	Iterative prediction is only an approximation, but it is a very powerful tool because it can model complex systems that are impossible to model using known analytical approaches such as calculus and kinematics due to unsolvable differential equations and other obstacles.		Iterative prediction is only an approximation, but it is a very powerful tool because it can model complex systems that are impossible to model using known analytical approaches such as calculus and kinematics due to unsolvable differential equations and other obstacles.
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	<li>The momentum of each particle should be updated from the last time step using the newly calculated net force.</li>		<li>The momentum of each particle should be updated from the last time step using the newly calculated net force.</li>

	This step uses the [[Impulse and Momentum\|impulse-momentum theorem]]. The impulse-momentum theorem states that <math>\vec{p}_f = \vec{p}_i + \vec{J} = \vec{p}_i + \vec{F}_{net, avg} * \Delta t</math>. In iterative prediction, the initial momentum is always known; it is simply the momentum the particle had during the previous time step (or, if there is no previous time step, the initial momentum of the particle). <math>\vec{F}_{net, avg}</math> represents the average net force that has been acting on the particle since the previous time step, resulting in a new momentum this time step. To calculate it, the net force found in step 1 should be used. Note that in reality, the net force acting on the particle is constantly changing, and the net force found in step 1 is the final net force, which is not truly the average net force over the course of the previous time step (except in a scenario with constant force). As described in the introductory section, this introduces inaccuracy that can be reduced by using shorter time steps; if short time steps are used, the net force does not have much time to change over the course of the time step, meaning the average net force is very similar to the final net force.		This step uses the [[Impulse and Momentum\|impulse-momentum theorem]]. The impulse-momentum theorem states that <math>\vec{p}_f = \vec{p}_i + \vec{J} = \vec{p}_i + \vec{F}_{net, avg} * \Delta t</math>. In iterative prediction, the initial momentum is always known; it is simply the momentum the particle had during the previous time step (or if there is no previous time step, the initial momentum of the particle). <math>\vec{F}_{net, avg}</math> represents the average net force that has been acting on the particle since the previous time step, resulting in a new momentum this time step. To calculate it, the net force found in step 1 should be used. Note that in reality, the net force acting on the particle is constantly changing, and the net force found in step 1 is the final net force, which is not truly the average net force over the course of the previous time step (except in a scenario with constant force). As described in the introductory section, this introduces inaccuracy that can be reduced by using shorter time steps; if short time steps are used, the net force does not have much time to change over the course of the time step, meaning the average net force is very similar to the final net force.

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

	This step uses the kinematic equation <math>\vec{r}_f = \vec{r}_i + \vec{v}_{avg} * \Delta t</math>. In iterative prediction, the initial position is always known; it is simply the position the particle had during the previous time step (or, if there is no previous time step, the initial position of the particle). <math>\vec{v}_{avg}</math> represents the average velocity with which the particle has been moving since the previous time step, resulting in a new position this time step. To calculate it, the momentum found in step 2 should simply be divided by the mass of the particle: <math>\vec{v}_f = \frac{\vec{p}_f}{m}</math>. Note that in reality, the velocity of the particle is constantly changing, and the velocity found above is the final velocity, which is not truly the average velocity over the course of the previous time step (even in a scenario with constant force). As described in the introductory section, this introduces inaccuracy that can be reduced by using shorter time steps; if short time steps are used, the velocity does not have much time to change over the course of the time step, meaning the average velocity is very similar to the final velocity.		This step uses the kinematic equation <math>\vec{r}_f = \vec{r}_i + \vec{v}_{avg} * \Delta t</math>. In iterative prediction, the initial position is always known; it is simply the position the particle had during the previous time step (or if there is no previous time step, the initial position of the particle). <math>\vec{v}_{avg}</math> represents the average velocity with which the particle has been moving since the previous time step, resulting in a new position this time step. To calculate it, the momentum found in step 2 should simply be divided by the mass of the particle: <math>\vec{v}_f = \frac{\vec{p}_f}{m}</math>. Note that in reality, the velocity of the particle is constantly changing, and the velocity found above is the final velocity, which is not truly the average velocity over the course of the previous time step (even in a scenario with constant force). As described in the introductory section, this introduces inaccuracy that can be reduced by using shorter time steps; if short time steps are used, the velocity does not have much time to change over the course of the time step, meaning the average velocity is very similar to the final velocity.

	</ol>		</ol>
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	===A Computational Model===		===A Computational Model===

	Iterative prediction is typically done on computers because many computations are necessary in order to perform it at a meaningful resolution. VPython is a useful program to perform iterative prediction because of it can graphically display the positions of the particles each time step. Below is an example of a VPython program that uses iterative prediction to simulate a ball thrown near the surface of the earth. Be sure to read and understand the source code by clicking on "view this program" in the top left corner; there are comments for each line designed to introduce readers to their first iterative prediction program.		Iterative prediction is typically done on computers because many computations are necessary in order to perform it at a meaningful resolution. VPython is a useful program to perform iterative prediction because it can graphically display the positions of the particles each time step. Below is an example of a VPython program that uses iterative prediction to simulate a ball thrown near the surface of the earth. Be sure to read and understand the source code by clicking on "view this program" in the top left corner; there are comments for each line designed to introduce readers to their first iterative prediction program.

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

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

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

	B.) To solve this using iterative prediction, a computational tool should be used. Since an animation isn't necessary for this problem, I used Microsoft Excel; each row populates itself based on the information in the previous row using formula functions. Google Spreadsheets can also be used, as can ~~vPython~~ or any other programming language.		B.) To solve this using iterative prediction, a computational tool should be used. Since animation isn't necessary for this problem, I used Microsoft Excel; each row populates itself based on the information in the previous row using formula functions. Google Spreadsheets can also be used, as can V-Python or any other programming language.

	[[File:12steps.PNG]]		[[File:12steps.PNG]]

	Using 12 time steps of 1/3 seconds each results in a simulated displacement of 26m, which is fairly close to the actual amount of 24m.		Using 12-time steps of 1/3 seconds each results in a simulated displacement of 26m, which is fairly close to the actual amount of 24m.

	C.)		C.)
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	[[File:4steps.PNG]]		[[File:4steps.PNG]]

	using 4 time steps of 1 second each results in a simulated displacement of 30m, which is fairly different from the actual amount of 24m.		using 4-time steps of 1 second each results in a simulated displacement of 30m, which is fairly different from the actual amount of 24m.

	D.)		D.)
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	[[File:1step.PNG]]		[[File:1step.PNG]]

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

	This demonstrates that higher-resolution simulations with smaller time steps are able to predict the motion of particles more accurately than lower-resolution simulations with larger time steps. It also demonstrates that as the length of each time step approaches 0, the predicted motion of the particles approaches their actual motion. It is important to choose a resolution that keeps error within an acceptable margin.		This demonstrates that higher-resolution simulations with smaller time steps are able to predict the motion of particles more accurately than lower-resolution simulations with larger time steps. It also demonstrates that as the length of each time step approaches 0, the predicted motion of the particles approaches their actual motion. It is important to choose a resolution that keeps error within an acceptable margin.

Ilukowiecki3: /* The Main Idea */

2020-03-24T18:26:47Z

The Main Idea

← Older revision		Revision as of 14:26, 24 March 2020
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	~~<li>~~1. Chose your system~~<li>~~		'''1.''' Chose your system

	2. Draw a force diagram (no internal forces, everything depends on interactions with the environment)		'''2.''' Draw a force diagram (no internal forces, everything depends on interactions with the environment)

	3. Chose your time interval		'''3.''' Chose your time interval

	4. Substitute known values and solve for unknowns		'''4.''' Substitute known values and solve for unknowns

	5. Check your units and reasonableness of your answer		'''5.''' Check your units and reasonableness of your answer

Ilukowiecki3: /* The Main Idea */

2020-03-24T18:26:04Z

The Main Idea

← Older revision		Revision as of 14:26, 24 March 2020
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	1. Chose your system		<li>1. Chose your system<li>

	2. Draw a force diagram (no internal forces, everything depends on interactions with the environment)		2. Draw a force diagram (no internal forces, everything depends on interactions with the environment)

Ilukowiecki3: /* The Main Idea */

2020-03-24T18:25:28Z

The Main Idea

← Older revision		Revision as of 14:25, 24 March 2020
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	To apply the momentum principle:		To apply the momentum principle:


	1. Chose your system		1. Chose your system

Ilukowiecki3: /* The Main Idea */

2020-03-24T18:25:13Z

The Main Idea

← Older revision		Revision as of 14:25, 24 March 2020
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	Iterative prediction is a mathematical technique for approximating the behavior of one or more particles over an interval of time. An iteration is a repeated procedure, so iterative prediction uses repeated procedures to predict the motion of a system. Specifically, 'iterate' means to repeat; in physics, it means to perform the same calculation repeatedly using information produced by the previous calculation. To perform iterative prediction, the initial position, initial momentum, and mass of each particle must be known, and the forces acting on each particle must be known. This information is used to update the position of each particle periodically.		Iterative prediction is a mathematical technique for approximating the behavior of one or more particles over an interval of time. An iteration is a repeated procedure, so iterative prediction uses repeated procedures to predict the motion of a system. Specifically, 'iterate' means to repeat; in physics, it means to perform the same calculation repeatedly using information produced by the previous calculation. To perform iterative prediction, the initial position, initial momentum, and mass of each particle must be known, and the forces acting on each particle must be known. This information is used to update the position of each particle periodically.

			To apply the momentum principle:

			1. Chose your system

			2. Draw a force diagram (no internal forces, everything depends on interactions with the environment)

			3. Chose your time interval

			4. Substitute known values and solve for unknowns

			5. Check your units and reasonableness of your answer


	To perform iterative prediction, the time interval of interest must be divided into small sub-intervals of duration <math>\Delta t</math> called time steps. For each time step, the following steps should be performed:		To perform iterative prediction, the time interval of interest must be divided into small sub-intervals of duration <math>\Delta t</math> called time steps. For each time step, the following steps should be performed:

Ilukowiecki3 at 18:21, 24 March 2020

2020-03-24T18:21:41Z

← Older revision		Revision as of 14:21, 24 March 2020
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	'''~~Shruthi Saravanan - Fall 2019~~'''		'''Claimed by ilukowiecki3 Spring 2020'''


	This page describes iterative prediction, a technique used to predict the motion of particles over a period of time often used in simulations.		This page describes iterative prediction, a technique used to predict the motion of particles over a period of time often used in simulations.