Physics Book - User contributions [en]

Newton's Second Law: the Momentum Principle

2022-12-05T04:37:18Z

Rgovind7: /* A Computational Model */

'''Claimed by Raksha Govind, Fall 2022'''

This page describes Newton's second law of motion, also known as the momentum principle, which relates net force to the change in [[Linear Momentum]]. This principle is used to predict the effects of forces on the motion of objects.

==The Main Idea==

Newton's second law of motion, also known as the momentum principle, explains how forces cause the momentum of a system to change over time. The principle states that <math>\vec{F}_{net} = \frac{d\vec{p}_{system}}{dt}</math> where <math>\vec{p}</math> is the [[Linear Momentum]] of a system, <math>\vec{F}_{net}</math> is the external [[Net Force]] acting on the system from its surroundings, and t is time. Often, the system in question consists of a single particle whose motion we want to predict, although the law is also true for systems of particles and non-particle distributions of mass, such as disks. Note that both force and momentum are vector quantities, and that the change in momentum as a result of a force will always be in the direction of that force.

Here are some simple situations to help you build an intuition for the momentum principle:
<ol>
<li>If a particle travels in a straight line at a constant speed, its momentum is constant, so there is no net force acting on the particle.
<li>If a particle travels in a straight line while speeding up, its momentum is increasing, so there must be a net force acting in the same direction as the particle's momentum.
<li>If a particle travels in a straight line while slowing down, its momentum is decreasing, so there must be a net force acting in the opposite direction as the particle's momentum.
<li>If a particle travels along a curving path, its momentum changes direction, and a force not parallel to the object's momentum must be acting on it.
</ol>

The momentum principle has no derivation, as it is considered the definition of force. The metric unit most commonly used for force is the Newton.

===A Mathematical Model===

The momentum principle states that <math>\vec{F}_{net} = \frac{d\vec{p}}{dt}</math>. Recall that <math>\vec{p} = m\vec{v}</math>. Therefore, by product rule, <math>\frac{d\vec{p}}{dt} = m\frac{d\vec{v}}{dt} + \vec{v}\frac{dm}{dt}</math>. Usually,the mass of the particle or system is constant over time, so the second term becomes 0, and the momentum principle becomes <math>\vec{F}_{net} = m\frac{d\vec{v}}{dt}</math>, or <math>\vec{F}_{net} = m\vec{a}</math>, which may be a more familiar form of Newton's second law. The form <math>\vec{F}_{net} = \frac{d\vec{p}}{dt}</math> is preferred for several reasons:
<ol>
<li>When a particle accumulates mass that was initially at rest, such as a snowball rolling downhill, the term <math>\vec{v}\frac{dm}{dt}</math> is not 0, and <math>\vec{F}_{net} = m\vec{a}</math> is no longer accurate, while <math>\vec{F}_{net} = \frac{d\vec{p}}{dt}</math> is;</li>
<li>As a particle approaches the speed of light, <math>\vec{F}_{net} = m\vec{a}</math> is no longer accurate, while <math>\vec{F}_{net} = \frac{d\vec{p}}{dt}</math> is, as long as [[Relativistic Momentum]] is used; and</li>
<li><math>\vec{F}_{net} = \frac{d\vec{p}}{dt}</math> has a more direct rotational analogue- that is, it will be easier to accurately learn rotational physics if you learn linear physics using this form.</li>
</ol>

That said, the relation <math>\vec{F}_{net} = m\vec{a}</math> can often be of use when the particle is known to have a constant mass and travel at non-relativistic speeds, particularly in combination with [[Kinematics|kinematic equations]] that also use <math>\vec{a}</math>.

The momentum principle is responsible for the relationship between [[Impulse and Momentum]].

Note that according to the momentum principle, when the external force on a system is 0 (that is, the system is closed), the rate of change of its momentum is 0. This results in [[Conservation of Momentum]] in certain situations.

===A Computational Model===

The relationship between net force and change in momentum can be computationally modeled using the momentum update formula: <math>p_2=p_1+F_{net}*\Delta\text{t}</math>. This is a rewriting of Fnet= dp/dt, where mass is assumed to be constant. First, momentum is defined as the object’s mass times its velocity in the initial conditions of the simulations. For [[Iterative Prediction]], a while loop for the duration of the movement first updates <math>F_{net}</math> by summing all possible net forces acting on the object during the simulation. Then the momentum principle is updated using the formula described previously. Finally, using the new momentum, the position is updated using the formula: <math>\overrightarrow{r}_2=\overrightarrow{r}_1+(\overrightarrow{p}/m)*\Delta\text{t}</math>, where velocity can be substituted as <math>(\overrightarrow{p}/m)</math>.

[[File:UpdateLoopforMomentumPrinciple.png]]

Such a simulation is shown below:

The vPython computational model calculates the changing position, velocity, and momentum vs. time graphs of a free-falling ball that is only subject to the force of gravity near the surface of the Earth.

https://www.glowscript.org/#/user/rgovind/folder/MyPrograms/program/Wiki

==Examples==

===1. (Simple)===

At t=0, a 4kg particle is released from rest and a constant 12N force begins acting on it. How far has the particle moved after 5 seconds?

For this problem, it is easier to use <math>\vec{F}_{net} = m\vec{a}</math> than <math>\vec{F}_{net} = \frac{d\vec{p}}{dt}</math> because it allows us to solve for <math>\vec{a}</math> which can then be used in a kinematic equation. We know this form of Newton's second law is accurate for this problem because the particle travels at non-relativistic speeds and is of constant mass.

<math>\vec{F}_{net} = m\vec{a}</math> can be rearranged into <math>\vec{a} = \frac{\vec{F}_{net}}{m}</math>

<math> a = \frac{12}{4} = 3 </math> (direction doesn't matter for this problem.)

<math> x_f = \frac{1}{2}at^2 + v_it + x_i = \frac{1}{2}(3)(5^2) = \frac{75}{2}</math> = 37.5 m.

===2. (Middling)===

A tennis ball of mass mB = 0.8kg and an initial velocity of ViB = <7,0,0> m/s. The tennis ball collides head-on with a lump of mashed potatoes with a mass mP = 3kg that was initially at rest. The tennis ball becomes embedded in the mush. Determine the velocity of the tennis ball and mashed potatoes after the collision, assuming that this collision is totally inelastic. Calculate how much energy is lost and determine the tennis ball and mashed potatoes individual velocities after the collision if the collision was elastic.

If this collision is inelastic, then we know that kinetic energy is not conserved. However, momentum is conserved for all types of collisions (elastic, inelastic, partially elastic) due to the momentum principle.
To start, write out the momentum principle. Since the tennis ball and mashed potatoes are stuck together after the collision, the final momentum can be written with their combined mass:

[[File:momentumconserved.jpg|200px|thumb|left|alt text]]

[[File:momentumconserved.jpg]]

[[File:MCP2.jpg|200px|thumb|left|alt text]]

===3. (Difficult)===

A cylindrical rocket ship of mass <math>M</math> and speed <math>v</math> is cruising through outer space when it encounters a stationary dust cloud. The dust cloud has a number density of <math>N</math> particles per unit volume, and each dust particle has a mass of <math>m</math>. The rocket ship has a sticky surface on its front face that allows it to accumulate all of the dust that it hits, absorbing it into its mass. The radius of the rocket ship's circular face is <math>R</math>. What force does the rocket's thruster need to exert on it in order to preserve its speed as it moves through the dust cloud? (Assume that the consumption of fuel has a negligible impact on the mass of the rocket.)

[[File:Rocketdustcloudpart1.jpg]]

[[File:Rocketdustcloudpart2.jpg]]

Advanced Note: in order to use <math>\frac{dm}{dt}</math> for the accumulation of mass, the accumulated mass (i.e. the dust) must initially be at rest, like in the above problem. Otherwise, one must treat the accumulation of dust particles as a series of collisions, or change one's frame of reference so that the accumulated mass is initially at rest.

===4. (Difficult)===

A particle of mass <math>m</math> and speed <math>v</math> moves in a circular path. Its angular frequency is <math>\Omega</math>. (Angular frequency is the rate at which the angle of the particle's position is changing in radians per unit time.) Using the momentum principle, show that

a) a nonzero net force is acting on the particle,

b) the magnitude of the force is given by <math>f = mv\Omega</math>, and

c) the direction of the force is inwards towards the center of the circle.

a) The particle's momentum vector is changing over time; its magnitude is constant (mv), but its direction changes as its position along the circular path changes. It is always tangential to the circular path. Since <math>\vec{p}</math> changes over time, a nonzero net force must be acting on the particle. In fact, the only way a particle of constant mass can have a net force of 0 acting on it is if it is travelling along a straight line at a constant speed, or if it is at rest.

b)

[[File:derivationofcentripetalforce.jpg]]

c) As one can see from the diagram above, <math>d\vec{p}</math> is pointing towards the center of the circle. If the particle is exactly at the top of the circle, its momentum is to the right (or left if the particle is travelling counterclockwise), and as <math>d\theta</math> approaches 0, <math>d\vec{p}</math> approaches a downward direction, which is towards the center of the circle.

For more information, see [[Centripetal Force and Curving Motion]]. Our answer agrees with the formulas given on that page.

==Connectedness==

Newton's second law is applicable to any situation where a force is applied and a system's momentum changes. There are nearly limitless examples of such situations, as well as nearly limitless applications. A few can be found below.

===Scenario: bungee jump===

When a bungee jumper falls and their cord is pulled taut, the cord applies a force in the direction opposite to the jumper's direction of travel. When this force is greater in magnitude than the gravity force, the net force is upward, which cuses causes the jumper's momentum, which is initially downward, to change over time in the upward direction. Eventually, the jumper is pulled back to a high altitude, the cord becomes slack, and gravity is once again the dominant force. The net force is downward once more, and the jumper's momentum changes over time in the downward direction.

===Application: automobile industry===

Cars frequently undergo all sorts of acceleration; they speed up, turn, and slow down. Forces are responsible for all of those accelerations, and these forces ultimately come from the tires' contact with the road. This force is, in fact, a form of [[Static Friction]]. The magnitude of this static friction force can only be up to a certain quantity known as the maximum static friction force. This maximum is determined by the shape and materials of the tires, as well as the weight of the car. Automobile manufacturers must always be aware of this maximum friction force when designing their car, as it places a limit on the ability of the car to accelerate, turn, and brake without slipping.

==History==

In his 1687 work Principia Mathematica, Isaac Newton (1643-1727) published his three laws of motion. Although his first and third laws were inspired by other scientists' work, his second law was entirely original. Below was the law as it appeared in his book:

Original Latin:

“Lex II: Mutationem motus proportionalem esse vi motrici impressae, et fieri secundum lineam rectam qua vis illa imprimitur.”

This was translated closely in Motte's 1729:

“Law II: The alteration of motion is ever proportional to the motive force impress'd; and is made in the direction of the right line in which that force is impress'd.”

In other words, the rate of change of momentum of a body is proportional to the force impressed on the body, and happens in the direction of that force.

This law was the first time force had ever been defined, and it is a definition that is still used today. As a result of Newton's contribution to the science of forces, the unit of force most commonly still used today is named after him.

== See also ==

*[[Linear Momentum]]
*[[Impulse and Momentum]]
*[[Conservation of Momentum]]
*[[Net Force]]
*[[Mass]]
*[[Acceleration]]

===Further reading===

RANKINE, William John Macquorn, and Edward Fisher BAMBER. A Mechanical Text Book. 1873. Print.

===External links===

<ol>
<li>nhttp://homepages.engineering.auckland.ac.nz/~pkel015/SolidMechanicsBooks/Part_III/Chapter_3_Stress_Mass_Momentum/Stress_Balance_Principles_02_The_Momentum_Principles.pdf</li>
<li>http://acme.highpoint.edu/~atitus/phy221/lecture-notes/2-2-momentum-principle.pdf</li>
<li>https://www.khanacademy.org/science/physics/forces-newtons-laws/newtons-laws-of-motion/v/newton-s-second-law-of-motion</li>
<li>https://www.khanacademy.org/science/physics/forces-newtons-laws/newtons-laws-of-motion/v/more-on-newtons-second-law</li>
<li>https://science.howstuffworks.com/innovation/scientific-experiments/newton-law-of-motion3.htm</li>
<li>https://www.livescience.com/46560-newton-second-law.html</li>
</ol>

==References==

*William, Harris. "How Newton's Laws of Motion Work" 29 July 2008. HowStuffWorks.com. <https://science.howstuffworks.com/innovation/scientific-experiments/newton-law-of-motion.htm> 11 April, 2018.
*Sherwood, Bruce A. "2.1 The Momentum Principle." Matter & Interactions. By Ruth W. Chabay. 4th ed. Vol. 1. N.p.: John Wiley & Sons, 2015. 45-50. Print. Modern Mechanics. 8 April, 2018.
*http://www.physicsclassroom.com/class/newtlaws/Lesson-3/Newton-s-Second-Law
*Dugas, René (1988). A history of mechanics. Translated into English by J.R. Maddox (Dover ed.). New York: Dover Publications.
*Jennings, John (1721). Miscellanea in Usum Juventutis Academicae. Northampton: R. Aikes & G. Dicey.

[[Category: Momentum]]

File:UpdateLoopforMomentumPrinciple.png

2022-12-05T04:33:49Z

Rgovind7: Describes the update loop in a Vpython simulation of the momentum principle and Newton's 2nd Law

== Summary ==
Describes the update loop in a Vpython simulation of the momentum principle and Newton's 2nd Law

Newton's Second Law: the Momentum Principle

2022-12-05T04:32:43Z

Rgovind7: /* A Computational Model */

'''Claimed by Raksha Govind, Fall 2022'''

This page describes Newton's second law of motion, also known as the momentum principle, which relates net force to the change in [[Linear Momentum]]. This principle is used to predict the effects of forces on the motion of objects.

==The Main Idea==

Newton's second law of motion, also known as the momentum principle, explains how forces cause the momentum of a system to change over time. The principle states that <math>\vec{F}_{net} = \frac{d\vec{p}_{system}}{dt}</math> where <math>\vec{p}</math> is the [[Linear Momentum]] of a system, <math>\vec{F}_{net}</math> is the external [[Net Force]] acting on the system from its surroundings, and t is time. Often, the system in question consists of a single particle whose motion we want to predict, although the law is also true for systems of particles and non-particle distributions of mass, such as disks. Note that both force and momentum are vector quantities, and that the change in momentum as a result of a force will always be in the direction of that force.

Here are some simple situations to help you build an intuition for the momentum principle:
<ol>
<li>If a particle travels in a straight line at a constant speed, its momentum is constant, so there is no net force acting on the particle.
<li>If a particle travels in a straight line while speeding up, its momentum is increasing, so there must be a net force acting in the same direction as the particle's momentum.
<li>If a particle travels in a straight line while slowing down, its momentum is decreasing, so there must be a net force acting in the opposite direction as the particle's momentum.
<li>If a particle travels along a curving path, its momentum changes direction, and a force not parallel to the object's momentum must be acting on it.
</ol>

The momentum principle has no derivation, as it is considered the definition of force. The metric unit most commonly used for force is the Newton.

===A Mathematical Model===

The momentum principle states that <math>\vec{F}_{net} = \frac{d\vec{p}}{dt}</math>. Recall that <math>\vec{p} = m\vec{v}</math>. Therefore, by product rule, <math>\frac{d\vec{p}}{dt} = m\frac{d\vec{v}}{dt} + \vec{v}\frac{dm}{dt}</math>. Usually,the mass of the particle or system is constant over time, so the second term becomes 0, and the momentum principle becomes <math>\vec{F}_{net} = m\frac{d\vec{v}}{dt}</math>, or <math>\vec{F}_{net} = m\vec{a}</math>, which may be a more familiar form of Newton's second law. The form <math>\vec{F}_{net} = \frac{d\vec{p}}{dt}</math> is preferred for several reasons:
<ol>
<li>When a particle accumulates mass that was initially at rest, such as a snowball rolling downhill, the term <math>\vec{v}\frac{dm}{dt}</math> is not 0, and <math>\vec{F}_{net} = m\vec{a}</math> is no longer accurate, while <math>\vec{F}_{net} = \frac{d\vec{p}}{dt}</math> is;</li>
<li>As a particle approaches the speed of light, <math>\vec{F}_{net} = m\vec{a}</math> is no longer accurate, while <math>\vec{F}_{net} = \frac{d\vec{p}}{dt}</math> is, as long as [[Relativistic Momentum]] is used; and</li>
<li><math>\vec{F}_{net} = \frac{d\vec{p}}{dt}</math> has a more direct rotational analogue- that is, it will be easier to accurately learn rotational physics if you learn linear physics using this form.</li>
</ol>

That said, the relation <math>\vec{F}_{net} = m\vec{a}</math> can often be of use when the particle is known to have a constant mass and travel at non-relativistic speeds, particularly in combination with [[Kinematics|kinematic equations]] that also use <math>\vec{a}</math>.

The momentum principle is responsible for the relationship between [[Impulse and Momentum]].

Note that according to the momentum principle, when the external force on a system is 0 (that is, the system is closed), the rate of change of its momentum is 0. This results in [[Conservation of Momentum]] in certain situations.

===A Computational Model===

The relationship between net force and change in momentum can be computationally modeled using the momentum update formula: <math>p_2=p_1+F_{net}*\Delta\text{t}</math>. This is a rewriting of Fnet= dp/dt, where mass is assumed to be constant. First, momentum is defined as the object’s mass times its velocity in the initial conditions of the simulations. For [[Iterative Prediction]], a while loop for the duration of the movement first updates <math>F_{net}</math> by summing all possible net forces acting on the object during the simulation. Then the momentum principle is updated using the formula described previously. Finally, using the new momentum, the position is updated using the formula: <math>\overrightarrow{r}_2=\overrightarrow{r}_1+(\overrightarrow{p}/m)*\Delta\text{t}</math>, where velocity can be substituted as <math>(\overrightarrow{p}/m)</math>.

Such a simulation is shown below:

The vPython computational model calculates the changing position, velocity, and momentum vs. time graphs of a free-falling ball that is only subject to the force of gravity near the surface of the Earth.

https://www.glowscript.org/#/user/rgovind/folder/MyPrograms/program/Wiki

==Examples==

===1. (Simple)===

At t=0, a 4kg particle is released from rest and a constant 12N force begins acting on it. How far has the particle moved after 5 seconds?

For this problem, it is easier to use <math>\vec{F}_{net} = m\vec{a}</math> than <math>\vec{F}_{net} = \frac{d\vec{p}}{dt}</math> because it allows us to solve for <math>\vec{a}</math> which can then be used in a kinematic equation. We know this form of Newton's second law is accurate for this problem because the particle travels at non-relativistic speeds and is of constant mass.

<math>\vec{F}_{net} = m\vec{a}</math> can be rearranged into <math>\vec{a} = \frac{\vec{F}_{net}}{m}</math>

<math> a = \frac{12}{4} = 3 </math> (direction doesn't matter for this problem.)

<math> x_f = \frac{1}{2}at^2 + v_it + x_i = \frac{1}{2}(3)(5^2) = \frac{75}{2}</math> = 37.5 m.

===2. (Middling)===

A tennis ball of mass mB = 0.8kg and an initial velocity of ViB = <7,0,0> m/s. The tennis ball collides head-on with a lump of mashed potatoes with a mass mP = 3kg that was initially at rest. The tennis ball becomes embedded in the mush. Determine the velocity of the tennis ball and mashed potatoes after the collision, assuming that this collision is totally inelastic. Calculate how much energy is lost and determine the tennis ball and mashed potatoes individual velocities after the collision if the collision was elastic.

If this collision is inelastic, then we know that kinetic energy is not conserved. However, momentum is conserved for all types of collisions (elastic, inelastic, partially elastic) due to the momentum principle.
To start, write out the momentum principle. Since the tennis ball and mashed potatoes are stuck together after the collision, the final momentum can be written with their combined mass:

[[File:momentumconserved.jpg|200px|thumb|left|alt text]]

[[File:momentumconserved.jpg]]

[[File:MCP2.jpg|200px|thumb|left|alt text]]

===3. (Difficult)===

A cylindrical rocket ship of mass <math>M</math> and speed <math>v</math> is cruising through outer space when it encounters a stationary dust cloud. The dust cloud has a number density of <math>N</math> particles per unit volume, and each dust particle has a mass of <math>m</math>. The rocket ship has a sticky surface on its front face that allows it to accumulate all of the dust that it hits, absorbing it into its mass. The radius of the rocket ship's circular face is <math>R</math>. What force does the rocket's thruster need to exert on it in order to preserve its speed as it moves through the dust cloud? (Assume that the consumption of fuel has a negligible impact on the mass of the rocket.)

[[File:Rocketdustcloudpart1.jpg]]

[[File:Rocketdustcloudpart2.jpg]]

Advanced Note: in order to use <math>\frac{dm}{dt}</math> for the accumulation of mass, the accumulated mass (i.e. the dust) must initially be at rest, like in the above problem. Otherwise, one must treat the accumulation of dust particles as a series of collisions, or change one's frame of reference so that the accumulated mass is initially at rest.

===4. (Difficult)===

A particle of mass <math>m</math> and speed <math>v</math> moves in a circular path. Its angular frequency is <math>\Omega</math>. (Angular frequency is the rate at which the angle of the particle's position is changing in radians per unit time.) Using the momentum principle, show that

a) a nonzero net force is acting on the particle,

b) the magnitude of the force is given by <math>f = mv\Omega</math>, and

c) the direction of the force is inwards towards the center of the circle.

a) The particle's momentum vector is changing over time; its magnitude is constant (mv), but its direction changes as its position along the circular path changes. It is always tangential to the circular path. Since <math>\vec{p}</math> changes over time, a nonzero net force must be acting on the particle. In fact, the only way a particle of constant mass can have a net force of 0 acting on it is if it is travelling along a straight line at a constant speed, or if it is at rest.

b)

[[File:derivationofcentripetalforce.jpg]]

c) As one can see from the diagram above, <math>d\vec{p}</math> is pointing towards the center of the circle. If the particle is exactly at the top of the circle, its momentum is to the right (or left if the particle is travelling counterclockwise), and as <math>d\theta</math> approaches 0, <math>d\vec{p}</math> approaches a downward direction, which is towards the center of the circle.

For more information, see [[Centripetal Force and Curving Motion]]. Our answer agrees with the formulas given on that page.

==Connectedness==

Newton's second law is applicable to any situation where a force is applied and a system's momentum changes. There are nearly limitless examples of such situations, as well as nearly limitless applications. A few can be found below.

===Scenario: bungee jump===

When a bungee jumper falls and their cord is pulled taut, the cord applies a force in the direction opposite to the jumper's direction of travel. When this force is greater in magnitude than the gravity force, the net force is upward, which cuses causes the jumper's momentum, which is initially downward, to change over time in the upward direction. Eventually, the jumper is pulled back to a high altitude, the cord becomes slack, and gravity is once again the dominant force. The net force is downward once more, and the jumper's momentum changes over time in the downward direction.

===Application: automobile industry===

Cars frequently undergo all sorts of acceleration; they speed up, turn, and slow down. Forces are responsible for all of those accelerations, and these forces ultimately come from the tires' contact with the road. This force is, in fact, a form of [[Static Friction]]. The magnitude of this static friction force can only be up to a certain quantity known as the maximum static friction force. This maximum is determined by the shape and materials of the tires, as well as the weight of the car. Automobile manufacturers must always be aware of this maximum friction force when designing their car, as it places a limit on the ability of the car to accelerate, turn, and brake without slipping.

==History==

In his 1687 work Principia Mathematica, Isaac Newton (1643-1727) published his three laws of motion. Although his first and third laws were inspired by other scientists' work, his second law was entirely original. Below was the law as it appeared in his book:

Original Latin:

“Lex II: Mutationem motus proportionalem esse vi motrici impressae, et fieri secundum lineam rectam qua vis illa imprimitur.”

This was translated closely in Motte's 1729:

“Law II: The alteration of motion is ever proportional to the motive force impress'd; and is made in the direction of the right line in which that force is impress'd.”

In other words, the rate of change of momentum of a body is proportional to the force impressed on the body, and happens in the direction of that force.

This law was the first time force had ever been defined, and it is a definition that is still used today. As a result of Newton's contribution to the science of forces, the unit of force most commonly still used today is named after him.

== See also ==

*[[Linear Momentum]]
*[[Impulse and Momentum]]
*[[Conservation of Momentum]]
*[[Net Force]]
*[[Mass]]
*[[Acceleration]]

===Further reading===

RANKINE, William John Macquorn, and Edward Fisher BAMBER. A Mechanical Text Book. 1873. Print.

===External links===

<ol>
<li>nhttp://homepages.engineering.auckland.ac.nz/~pkel015/SolidMechanicsBooks/Part_III/Chapter_3_Stress_Mass_Momentum/Stress_Balance_Principles_02_The_Momentum_Principles.pdf</li>
<li>http://acme.highpoint.edu/~atitus/phy221/lecture-notes/2-2-momentum-principle.pdf</li>
<li>https://www.khanacademy.org/science/physics/forces-newtons-laws/newtons-laws-of-motion/v/newton-s-second-law-of-motion</li>
<li>https://www.khanacademy.org/science/physics/forces-newtons-laws/newtons-laws-of-motion/v/more-on-newtons-second-law</li>
<li>https://science.howstuffworks.com/innovation/scientific-experiments/newton-law-of-motion3.htm</li>
<li>https://www.livescience.com/46560-newton-second-law.html</li>
</ol>

==References==

*William, Harris. "How Newton's Laws of Motion Work" 29 July 2008. HowStuffWorks.com. <https://science.howstuffworks.com/innovation/scientific-experiments/newton-law-of-motion.htm> 11 April, 2018.
*Sherwood, Bruce A. "2.1 The Momentum Principle." Matter & Interactions. By Ruth W. Chabay. 4th ed. Vol. 1. N.p.: John Wiley & Sons, 2015. 45-50. Print. Modern Mechanics. 8 April, 2018.
*http://www.physicsclassroom.com/class/newtlaws/Lesson-3/Newton-s-Second-Law
*Dugas, René (1988). A history of mechanics. Translated into English by J.R. Maddox (Dover ed.). New York: Dover Publications.
*Jennings, John (1721). Miscellanea in Usum Juventutis Academicae. Northampton: R. Aikes & G. Dicey.

[[Category: Momentum]]

Newton's Second Law: the Momentum Principle

2022-12-05T04:25:58Z

Rgovind7:

'''Claimed by Raksha Govind, Fall 2022'''

This page describes Newton's second law of motion, also known as the momentum principle, which relates net force to the change in [[Linear Momentum]]. This principle is used to predict the effects of forces on the motion of objects.

==The Main Idea==

Newton's second law of motion, also known as the momentum principle, explains how forces cause the momentum of a system to change over time. The principle states that <math>\vec{F}_{net} = \frac{d\vec{p}_{system}}{dt}</math> where <math>\vec{p}</math> is the [[Linear Momentum]] of a system, <math>\vec{F}_{net}</math> is the external [[Net Force]] acting on the system from its surroundings, and t is time. Often, the system in question consists of a single particle whose motion we want to predict, although the law is also true for systems of particles and non-particle distributions of mass, such as disks. Note that both force and momentum are vector quantities, and that the change in momentum as a result of a force will always be in the direction of that force.

Here are some simple situations to help you build an intuition for the momentum principle:
<ol>
<li>If a particle travels in a straight line at a constant speed, its momentum is constant, so there is no net force acting on the particle.
<li>If a particle travels in a straight line while speeding up, its momentum is increasing, so there must be a net force acting in the same direction as the particle's momentum.
<li>If a particle travels in a straight line while slowing down, its momentum is decreasing, so there must be a net force acting in the opposite direction as the particle's momentum.
<li>If a particle travels along a curving path, its momentum changes direction, and a force not parallel to the object's momentum must be acting on it.
</ol>

The momentum principle has no derivation, as it is considered the definition of force. The metric unit most commonly used for force is the Newton.

===A Mathematical Model===

The momentum principle states that <math>\vec{F}_{net} = \frac{d\vec{p}}{dt}</math>. Recall that <math>\vec{p} = m\vec{v}</math>. Therefore, by product rule, <math>\frac{d\vec{p}}{dt} = m\frac{d\vec{v}}{dt} + \vec{v}\frac{dm}{dt}</math>. Usually,the mass of the particle or system is constant over time, so the second term becomes 0, and the momentum principle becomes <math>\vec{F}_{net} = m\frac{d\vec{v}}{dt}</math>, or <math>\vec{F}_{net} = m\vec{a}</math>, which may be a more familiar form of Newton's second law. The form <math>\vec{F}_{net} = \frac{d\vec{p}}{dt}</math> is preferred for several reasons:
<ol>
<li>When a particle accumulates mass that was initially at rest, such as a snowball rolling downhill, the term <math>\vec{v}\frac{dm}{dt}</math> is not 0, and <math>\vec{F}_{net} = m\vec{a}</math> is no longer accurate, while <math>\vec{F}_{net} = \frac{d\vec{p}}{dt}</math> is;</li>
<li>As a particle approaches the speed of light, <math>\vec{F}_{net} = m\vec{a}</math> is no longer accurate, while <math>\vec{F}_{net} = \frac{d\vec{p}}{dt}</math> is, as long as [[Relativistic Momentum]] is used; and</li>
<li><math>\vec{F}_{net} = \frac{d\vec{p}}{dt}</math> has a more direct rotational analogue- that is, it will be easier to accurately learn rotational physics if you learn linear physics using this form.</li>
</ol>

That said, the relation <math>\vec{F}_{net} = m\vec{a}</math> can often be of use when the particle is known to have a constant mass and travel at non-relativistic speeds, particularly in combination with [[Kinematics|kinematic equations]] that also use <math>\vec{a}</math>.

The momentum principle is responsible for the relationship between [[Impulse and Momentum]].

Note that according to the momentum principle, when the external force on a system is 0 (that is, the system is closed), the rate of change of its momentum is 0. This results in [[Conservation of Momentum]] in certain situations.

===A Computational Model===

The relationship between net force and change in momentum can be computationally modeled using the momentum update formula: <math>p_2=p_1+F_{net}*\Delta\text{t}</math>. This is a rewriting of Fnet= dp/dt, where mass is assumed to be constant. First, momentum is defined as the object’s mass times its velocity in the initial conditions of the simulations. For [[Iterative Prediction]], a while loop for the duration of the movement first updates <math>F_{net}</math> by summing all possible net forces acting on the object during the simulation. Then the momentum principle is updated using the formula described previously. Finally, using the new momentum, the position is updated using the formula: <math>\overrightarrow{r}_2=\overrightarrow{r}_1+(\overrightarrow{p}/m)*\Delta\text{t}</math>, where velocity can be substituted as <math>(\overrightarrow{p}/m)</math>. Such a simulation is shown below:

The vPython computational model calculates the changing position, velocity, and momentum vs. time graphs of a free-falling ball that is only subject to the force of gravity near the surface of the Earth.

https://www.glowscript.org/#/user/rgovind/folder/MyPrograms/program/Wiki

==Examples==

===1. (Simple)===

At t=0, a 4kg particle is released from rest and a constant 12N force begins acting on it. How far has the particle moved after 5 seconds?

For this problem, it is easier to use <math>\vec{F}_{net} = m\vec{a}</math> than <math>\vec{F}_{net} = \frac{d\vec{p}}{dt}</math> because it allows us to solve for <math>\vec{a}</math> which can then be used in a kinematic equation. We know this form of Newton's second law is accurate for this problem because the particle travels at non-relativistic speeds and is of constant mass.

<math>\vec{F}_{net} = m\vec{a}</math> can be rearranged into <math>\vec{a} = \frac{\vec{F}_{net}}{m}</math>

<math> a = \frac{12}{4} = 3 </math> (direction doesn't matter for this problem.)

<math> x_f = \frac{1}{2}at^2 + v_it + x_i = \frac{1}{2}(3)(5^2) = \frac{75}{2}</math> = 37.5 m.

===2. (Middling)===

A tennis ball of mass mB = 0.8kg and an initial velocity of ViB = <7,0,0> m/s. The tennis ball collides head-on with a lump of mashed potatoes with a mass mP = 3kg that was initially at rest. The tennis ball becomes embedded in the mush. Determine the velocity of the tennis ball and mashed potatoes after the collision, assuming that this collision is totally inelastic. Calculate how much energy is lost and determine the tennis ball and mashed potatoes individual velocities after the collision if the collision was elastic.

If this collision is inelastic, then we know that kinetic energy is not conserved. However, momentum is conserved for all types of collisions (elastic, inelastic, partially elastic) due to the momentum principle.
To start, write out the momentum principle. Since the tennis ball and mashed potatoes are stuck together after the collision, the final momentum can be written with their combined mass:

[[File:momentumconserved.jpg|200px|thumb|left|alt text]]

[[File:momentumconserved.jpg]]

[[File:MCP2.jpg|200px|thumb|left|alt text]]

===3. (Difficult)===

A cylindrical rocket ship of mass <math>M</math> and speed <math>v</math> is cruising through outer space when it encounters a stationary dust cloud. The dust cloud has a number density of <math>N</math> particles per unit volume, and each dust particle has a mass of <math>m</math>. The rocket ship has a sticky surface on its front face that allows it to accumulate all of the dust that it hits, absorbing it into its mass. The radius of the rocket ship's circular face is <math>R</math>. What force does the rocket's thruster need to exert on it in order to preserve its speed as it moves through the dust cloud? (Assume that the consumption of fuel has a negligible impact on the mass of the rocket.)

[[File:Rocketdustcloudpart1.jpg]]

[[File:Rocketdustcloudpart2.jpg]]

Advanced Note: in order to use <math>\frac{dm}{dt}</math> for the accumulation of mass, the accumulated mass (i.e. the dust) must initially be at rest, like in the above problem. Otherwise, one must treat the accumulation of dust particles as a series of collisions, or change one's frame of reference so that the accumulated mass is initially at rest.

===4. (Difficult)===

A particle of mass <math>m</math> and speed <math>v</math> moves in a circular path. Its angular frequency is <math>\Omega</math>. (Angular frequency is the rate at which the angle of the particle's position is changing in radians per unit time.) Using the momentum principle, show that

a) a nonzero net force is acting on the particle,

b) the magnitude of the force is given by <math>f = mv\Omega</math>, and

c) the direction of the force is inwards towards the center of the circle.

a) The particle's momentum vector is changing over time; its magnitude is constant (mv), but its direction changes as its position along the circular path changes. It is always tangential to the circular path. Since <math>\vec{p}</math> changes over time, a nonzero net force must be acting on the particle. In fact, the only way a particle of constant mass can have a net force of 0 acting on it is if it is travelling along a straight line at a constant speed, or if it is at rest.

b)

[[File:derivationofcentripetalforce.jpg]]

c) As one can see from the diagram above, <math>d\vec{p}</math> is pointing towards the center of the circle. If the particle is exactly at the top of the circle, its momentum is to the right (or left if the particle is travelling counterclockwise), and as <math>d\theta</math> approaches 0, <math>d\vec{p}</math> approaches a downward direction, which is towards the center of the circle.

For more information, see [[Centripetal Force and Curving Motion]]. Our answer agrees with the formulas given on that page.

==Connectedness==

Newton's second law is applicable to any situation where a force is applied and a system's momentum changes. There are nearly limitless examples of such situations, as well as nearly limitless applications. A few can be found below.

===Scenario: bungee jump===

When a bungee jumper falls and their cord is pulled taut, the cord applies a force in the direction opposite to the jumper's direction of travel. When this force is greater in magnitude than the gravity force, the net force is upward, which cuses causes the jumper's momentum, which is initially downward, to change over time in the upward direction. Eventually, the jumper is pulled back to a high altitude, the cord becomes slack, and gravity is once again the dominant force. The net force is downward once more, and the jumper's momentum changes over time in the downward direction.

===Application: automobile industry===

Cars frequently undergo all sorts of acceleration; they speed up, turn, and slow down. Forces are responsible for all of those accelerations, and these forces ultimately come from the tires' contact with the road. This force is, in fact, a form of [[Static Friction]]. The magnitude of this static friction force can only be up to a certain quantity known as the maximum static friction force. This maximum is determined by the shape and materials of the tires, as well as the weight of the car. Automobile manufacturers must always be aware of this maximum friction force when designing their car, as it places a limit on the ability of the car to accelerate, turn, and brake without slipping.

==History==

In his 1687 work Principia Mathematica, Isaac Newton (1643-1727) published his three laws of motion. Although his first and third laws were inspired by other scientists' work, his second law was entirely original. Below was the law as it appeared in his book:

Original Latin:

“Lex II: Mutationem motus proportionalem esse vi motrici impressae, et fieri secundum lineam rectam qua vis illa imprimitur.”

This was translated closely in Motte's 1729:

“Law II: The alteration of motion is ever proportional to the motive force impress'd; and is made in the direction of the right line in which that force is impress'd.”

In other words, the rate of change of momentum of a body is proportional to the force impressed on the body, and happens in the direction of that force.

This law was the first time force had ever been defined, and it is a definition that is still used today. As a result of Newton's contribution to the science of forces, the unit of force most commonly still used today is named after him.

== See also ==

*[[Linear Momentum]]
*[[Impulse and Momentum]]
*[[Conservation of Momentum]]
*[[Net Force]]
*[[Mass]]
*[[Acceleration]]

===Further reading===

RANKINE, William John Macquorn, and Edward Fisher BAMBER. A Mechanical Text Book. 1873. Print.

===External links===

<ol>
<li>nhttp://homepages.engineering.auckland.ac.nz/~pkel015/SolidMechanicsBooks/Part_III/Chapter_3_Stress_Mass_Momentum/Stress_Balance_Principles_02_The_Momentum_Principles.pdf</li>
<li>http://acme.highpoint.edu/~atitus/phy221/lecture-notes/2-2-momentum-principle.pdf</li>
<li>https://www.khanacademy.org/science/physics/forces-newtons-laws/newtons-laws-of-motion/v/newton-s-second-law-of-motion</li>
<li>https://www.khanacademy.org/science/physics/forces-newtons-laws/newtons-laws-of-motion/v/more-on-newtons-second-law</li>
<li>https://science.howstuffworks.com/innovation/scientific-experiments/newton-law-of-motion3.htm</li>
<li>https://www.livescience.com/46560-newton-second-law.html</li>
</ol>

==References==

*William, Harris. "How Newton's Laws of Motion Work" 29 July 2008. HowStuffWorks.com. <https://science.howstuffworks.com/innovation/scientific-experiments/newton-law-of-motion.htm> 11 April, 2018.
*Sherwood, Bruce A. "2.1 The Momentum Principle." Matter & Interactions. By Ruth W. Chabay. 4th ed. Vol. 1. N.p.: John Wiley & Sons, 2015. 45-50. Print. Modern Mechanics. 8 April, 2018.
*http://www.physicsclassroom.com/class/newtlaws/Lesson-3/Newton-s-Second-Law
*Dugas, René (1988). A history of mechanics. Translated into English by J.R. Maddox (Dover ed.). New York: Dover Publications.
*Jennings, John (1721). Miscellanea in Usum Juventutis Academicae. Northampton: R. Aikes & G. Dicey.

[[Category: Momentum]]

Newton's Second Law: the Momentum Principle

2022-12-05T04:25:30Z

Rgovind7:

'''Claimed by Raksha Govind, Fall 2022'''

This page describes Newton's second law of motion, also known as the momentum principle, which relates net force to the change in [[Linear Momentum]]. This principle is used to predict the effects of forces on the motion of objects.

==The Main Idea==

Newton's second law of motion, also known as the momentum principle, explains how forces cause the momentum of a system to change over time. The principle states that <math>\vec{F}_{net} = \frac{d\vec{p}_{system}}{dt}</math> where <math>\vec{p}</math> is the [[Linear Momentum]] of a system, <math>\vec{F}_{net}</math> is the external [[Net Force]] acting on the system from its surroundings, and t is time. Often, the system in question consists of a single particle whose motion we want to predict, although the law is also true for systems of particles and non-particle distributions of mass, such as disks. Note that both force and momentum are vector quantities, and that the change in momentum as a result of a force will always be in the direction of that force.

Here are some simple situations to help you build an intuition for the momentum principle:
<ol>
<li>If a particle travels in a straight line at a constant speed, its momentum is constant, so there is no net force acting on the particle.
<li>If a particle travels in a straight line while speeding up, its momentum is increasing, so there must be a net force acting in the same direction as the particle's momentum.
<li>If a particle travels in a straight line while slowing down, its momentum is decreasing, so there must be a net force acting in the opposite direction as the particle's momentum.
<li>If a particle travels along a curving path, its momentum changes direction, and a force not parallel to the object's momentum must be acting on it.
</ol>

The momentum principle has no derivation, as it is considered the definition of force. The metric unit most commonly used for force is the Newton.

===A Mathematical Model===

The momentum principle states that <math>\vec{F}_{net} = \frac{d\vec{p}}{dt}</math>. Recall that <math>\vec{p} = m\vec{v}</math>. Therefore, by product rule, <math>\frac{d\vec{p}}{dt} = m\frac{d\vec{v}}{dt} + \vec{v}\frac{dm}{dt}</math>. Usually,the mass of the particle or system is constant over time, so the second term becomes 0, and the momentum principle becomes <math>\vec{F}_{net} = m\frac{d\vec{v}}{dt}</math>, or <math>\vec{F}_{net} = m\vec{a}</math>, which may be a more familiar form of Newton's second law. The form <math>\vec{F}_{net} = \frac{d\vec{p}}{dt}</math> is preferred for several reasons:
<ol>
<li>When a particle accumulates mass that was initially at rest, such as a snowball rolling downhill, the term <math>\vec{v}\frac{dm}{dt}</math> is not 0, and <math>\vec{F}_{net} = m\vec{a}</math> is no longer accurate, while <math>\vec{F}_{net} = \frac{d\vec{p}}{dt}</math> is;</li>
<li>As a particle approaches the speed of light, <math>\vec{F}_{net} = m\vec{a}</math> is no longer accurate, while <math>\vec{F}_{net} = \frac{d\vec{p}}{dt}</math> is, as long as [[Relativistic Momentum]] is used; and</li>
<li><math>\vec{F}_{net} = \frac{d\vec{p}}{dt}</math> has a more direct rotational analogue- that is, it will be easier to accurately learn rotational physics if you learn linear physics using this form.</li>
</ol>

That said, the relation <math>\vec{F}_{net} = m\vec{a}</math> can often be of use when the particle is known to have a constant mass and travel at non-relativistic speeds, particularly in combination with [[Kinematics|kinematic equations]] that also use <math>\vec{a}</math>.

The momentum principle is responsible for the relationship between [[Impulse and Momentum]].

Note that according to the momentum principle, when the external force on a system is 0 (that is, the system is closed), the rate of change of its momentum is 0. This results in [[Conservation of Momentum]] in certain situations.

===A Computational Model===

The relationship between net force and change in momentum can be computationally modeled using the momentum update formula: <math>p_2=p_1+F_{net}*\Delta\text{t}</math>. This is a rewriting of Fnet= dp/dt, where mass is assumed to be constant. First, momentum is defined as the object’s mass times its velocity in the initial conditions of the simulations. For [[Iterative Motion]], a while loop for the duration of the movement first updates <math>F_{net}</math> by summing all possible net forces acting on the object during the simulation. Then the momentum principle is updated using the formula described previously. Finally, using the new momentum, the position is updated using the formula: <math>\overrightarrow{r}_2=\overrightarrow{r}_1+(\overrightarrow{p}/m)*\Delta\text{t}</math>, where velocity can be substituted as <math>(\overrightarrow{p}/m)</math>. Such a simulation is shown below:

The vPython computational model calculates the changing position, velocity, and momentum vs. time graphs of a free-falling ball that is only subject to the force of gravity near the surface of the Earth.

https://www.glowscript.org/#/user/rgovind/folder/MyPrograms/program/Wiki

==Examples==

===1. (Simple)===

At t=0, a 4kg particle is released from rest and a constant 12N force begins acting on it. How far has the particle moved after 5 seconds?

For this problem, it is easier to use <math>\vec{F}_{net} = m\vec{a}</math> than <math>\vec{F}_{net} = \frac{d\vec{p}}{dt}</math> because it allows us to solve for <math>\vec{a}</math> which can then be used in a kinematic equation. We know this form of Newton's second law is accurate for this problem because the particle travels at non-relativistic speeds and is of constant mass.

<math>\vec{F}_{net} = m\vec{a}</math> can be rearranged into <math>\vec{a} = \frac{\vec{F}_{net}}{m}</math>

<math> a = \frac{12}{4} = 3 </math> (direction doesn't matter for this problem.)

<math> x_f = \frac{1}{2}at^2 + v_it + x_i = \frac{1}{2}(3)(5^2) = \frac{75}{2}</math> = 37.5 m.

===2. (Middling)===

A tennis ball of mass mB = 0.8kg and an initial velocity of ViB = <7,0,0> m/s. The tennis ball collides head-on with a lump of mashed potatoes with a mass mP = 3kg that was initially at rest. The tennis ball becomes embedded in the mush. Determine the velocity of the tennis ball and mashed potatoes after the collision, assuming that this collision is totally inelastic. Calculate how much energy is lost and determine the tennis ball and mashed potatoes individual velocities after the collision if the collision was elastic.

If this collision is inelastic, then we know that kinetic energy is not conserved. However, momentum is conserved for all types of collisions (elastic, inelastic, partially elastic) due to the momentum principle.
To start, write out the momentum principle. Since the tennis ball and mashed potatoes are stuck together after the collision, the final momentum can be written with their combined mass:

[[File:momentumconserved.jpg|200px|thumb|left|alt text]]

[[File:momentumconserved.jpg]]

[[File:MCP2.jpg|200px|thumb|left|alt text]]

===3. (Difficult)===

A cylindrical rocket ship of mass <math>M</math> and speed <math>v</math> is cruising through outer space when it encounters a stationary dust cloud. The dust cloud has a number density of <math>N</math> particles per unit volume, and each dust particle has a mass of <math>m</math>. The rocket ship has a sticky surface on its front face that allows it to accumulate all of the dust that it hits, absorbing it into its mass. The radius of the rocket ship's circular face is <math>R</math>. What force does the rocket's thruster need to exert on it in order to preserve its speed as it moves through the dust cloud? (Assume that the consumption of fuel has a negligible impact on the mass of the rocket.)

[[File:Rocketdustcloudpart1.jpg]]

[[File:Rocketdustcloudpart2.jpg]]

Advanced Note: in order to use <math>\frac{dm}{dt}</math> for the accumulation of mass, the accumulated mass (i.e. the dust) must initially be at rest, like in the above problem. Otherwise, one must treat the accumulation of dust particles as a series of collisions, or change one's frame of reference so that the accumulated mass is initially at rest.

===4. (Difficult)===

A particle of mass <math>m</math> and speed <math>v</math> moves in a circular path. Its angular frequency is <math>\Omega</math>. (Angular frequency is the rate at which the angle of the particle's position is changing in radians per unit time.) Using the momentum principle, show that

a) a nonzero net force is acting on the particle,

b) the magnitude of the force is given by <math>f = mv\Omega</math>, and

c) the direction of the force is inwards towards the center of the circle.

a) The particle's momentum vector is changing over time; its magnitude is constant (mv), but its direction changes as its position along the circular path changes. It is always tangential to the circular path. Since <math>\vec{p}</math> changes over time, a nonzero net force must be acting on the particle. In fact, the only way a particle of constant mass can have a net force of 0 acting on it is if it is travelling along a straight line at a constant speed, or if it is at rest.

b)

[[File:derivationofcentripetalforce.jpg]]

c) As one can see from the diagram above, <math>d\vec{p}</math> is pointing towards the center of the circle. If the particle is exactly at the top of the circle, its momentum is to the right (or left if the particle is travelling counterclockwise), and as <math>d\theta</math> approaches 0, <math>d\vec{p}</math> approaches a downward direction, which is towards the center of the circle.

For more information, see [[Centripetal Force and Curving Motion]]. Our answer agrees with the formulas given on that page.

==Connectedness==

Newton's second law is applicable to any situation where a force is applied and a system's momentum changes. There are nearly limitless examples of such situations, as well as nearly limitless applications. A few can be found below.

===Scenario: bungee jump===

When a bungee jumper falls and their cord is pulled taut, the cord applies a force in the direction opposite to the jumper's direction of travel. When this force is greater in magnitude than the gravity force, the net force is upward, which cuses causes the jumper's momentum, which is initially downward, to change over time in the upward direction. Eventually, the jumper is pulled back to a high altitude, the cord becomes slack, and gravity is once again the dominant force. The net force is downward once more, and the jumper's momentum changes over time in the downward direction.

===Application: automobile industry===

Cars frequently undergo all sorts of acceleration; they speed up, turn, and slow down. Forces are responsible for all of those accelerations, and these forces ultimately come from the tires' contact with the road. This force is, in fact, a form of [[Static Friction]]. The magnitude of this static friction force can only be up to a certain quantity known as the maximum static friction force. This maximum is determined by the shape and materials of the tires, as well as the weight of the car. Automobile manufacturers must always be aware of this maximum friction force when designing their car, as it places a limit on the ability of the car to accelerate, turn, and brake without slipping.

==History==

In his 1687 work Principia Mathematica, Isaac Newton (1643-1727) published his three laws of motion. Although his first and third laws were inspired by other scientists' work, his second law was entirely original. Below was the law as it appeared in his book:

Original Latin:

“Lex II: Mutationem motus proportionalem esse vi motrici impressae, et fieri secundum lineam rectam qua vis illa imprimitur.”

This was translated closely in Motte's 1729:

“Law II: The alteration of motion is ever proportional to the motive force impress'd; and is made in the direction of the right line in which that force is impress'd.”

In other words, the rate of change of momentum of a body is proportional to the force impressed on the body, and happens in the direction of that force.

This law was the first time force had ever been defined, and it is a definition that is still used today. As a result of Newton's contribution to the science of forces, the unit of force most commonly still used today is named after him.

== See also ==

*[[Linear Momentum]]
*[[Impulse and Momentum]]
*[[Conservation of Momentum]]
*[[Net Force]]
*[[Mass]]
*[[Acceleration]]

===Further reading===

RANKINE, William John Macquorn, and Edward Fisher BAMBER. A Mechanical Text Book. 1873. Print.

===External links===

<ol>
<li>nhttp://homepages.engineering.auckland.ac.nz/~pkel015/SolidMechanicsBooks/Part_III/Chapter_3_Stress_Mass_Momentum/Stress_Balance_Principles_02_The_Momentum_Principles.pdf</li>
<li>http://acme.highpoint.edu/~atitus/phy221/lecture-notes/2-2-momentum-principle.pdf</li>
<li>https://www.khanacademy.org/science/physics/forces-newtons-laws/newtons-laws-of-motion/v/newton-s-second-law-of-motion</li>
<li>https://www.khanacademy.org/science/physics/forces-newtons-laws/newtons-laws-of-motion/v/more-on-newtons-second-law</li>
<li>https://science.howstuffworks.com/innovation/scientific-experiments/newton-law-of-motion3.htm</li>
<li>https://www.livescience.com/46560-newton-second-law.html</li>
</ol>

==References==

*William, Harris. "How Newton's Laws of Motion Work" 29 July 2008. HowStuffWorks.com. <https://science.howstuffworks.com/innovation/scientific-experiments/newton-law-of-motion.htm> 11 April, 2018.
*Sherwood, Bruce A. "2.1 The Momentum Principle." Matter & Interactions. By Ruth W. Chabay. 4th ed. Vol. 1. N.p.: John Wiley & Sons, 2015. 45-50. Print. Modern Mechanics. 8 April, 2018.
*http://www.physicsclassroom.com/class/newtlaws/Lesson-3/Newton-s-Second-Law
*Dugas, René (1988). A history of mechanics. Translated into English by J.R. Maddox (Dover ed.). New York: Dover Publications.
*Jennings, John (1721). Miscellanea in Usum Juventutis Academicae. Northampton: R. Aikes & G. Dicey.

[[Category: Momentum]]

Linear Momentum

2022-12-03T23:15:31Z

Rgovind7:

This page defines the linear momentum of particles and systems.

==The Main Idea==

Linear momentum is a vector quantity describing an object's motion. It is defined as the product of an object's [[Mass]] (<math>m</math>) and [[Velocity]] (<math>\vec{v}</math>). Note that mass is a scalar while velocity is a vector, so an object's linear momentum is always in the same direction as its velocity. Linear momentum is represented by the letter <math>\vec{p}</math> and is often referred to as simply "momentum." The most commonly used metric unit for momentum is the kilogram*meter/second. The plural of momentum is momenta or momentums.

===A Mathematical Model===

====Single Particles====
The momentum of a particle is defined as follows:

<math>\vec{p} = m\vec{v}</math>

where <math>\vec{p}</math> is the particle's linear momentum, <math>m</math> is the particle's mass, and <math>\vec{v}</math> is the particle's velocity. This formula accurately describes the momentum of particles at everyday speeds, but for particles travelling near the speed of light, the formula for [[Relativistic Momentum]] must be used for concepts such as the momentum principle to remain true.

====Multiple Particles====

The total momentum of a system of particles is defined as the vector sum of the momenta of the particles that comprise the system:

<math>\vec{p}_{system} = \sum_i \vec{p}_i</math>

Although the proof does not appear on this page, it can be shown that the total momentum of a system of particles is equal to the total mass of the system times the velocity of its [[Center of Mass]]:

<math>\vec{p}_{system} = M_{tot}\vec{v}_{COM}</math>

This formula makes it significantly easier to calculate the momentum of certain objects. For example, consider a disk rolling along the ground. The disk is comprised of an infinite number of infinitely small "mass elements," all of which have different momenta; the ones at the bottom of the disk are hardly moving while the ones at the top are moving very quickly. Without the formula above, one would have to use an integral to add the momenta of all of the mass elements to find the total momentum of the disk. However, the formula above tells us that we can simply multiply the mass of the disk by the velocity of its center of mass, which is in this case the geometric center of the disk. This reveals that the fact that the disk is rotating does not affect its momentum.

====In Relation to Other Physics Topics====

In the absence of a net force, the momentum of a particle stays constant over time, as stated by Newton's first law.

When a force is applied to a particle, its momentum evolves over time according to Newton's second law. For more information, see [[Newton's Second Law: the Momentum Principle]].

When an impulse is applied to a particle, its momentum changes in a specific way: the change in momentum <math>\Delta\vec{p}</math> is equal to the impulse <math>\vec{J}</math>. This is a consequence of the Momentum Principle. For more information, see [[Impulse and Momentum]].

When the net external force on a system of particles is 0, the system's momentum is conserved (that is, constant over time) even if the particles within the system interact with each other (exert internal forces on each other). For more information, see [[Conservation of Momentum]].

===A Computational Model===

In computational simulations of particles using [[Iterative Prediction]], a momentum vector variable is assigned to each particle. Such simulations usually in "time steps," or iterations of a loop representing a short time interval. In each time step, the particles' momenta are updated based on the forces acting on it. Then their velocities are calculated by dividing each particle's momentum by its mass. Finally, the velocities are used to update the positions of the particles. Below is an example of such a simulation:

This vPython simulation shows a cart (represented by a rectangle) whose motion is affected by a gust of wind applying a constant force.

https://trinket.io/glowscript/ce43925647

For more information, see [[Iterative Prediction]].

==Examples==

===1. (Simple)===
Find the momentum of a ball that has a mass of 69kg and is moving at <1,2,3> m/s.

[[File:momentumsimple.jpg]]

===2. (Middling)===
A car has 20,000 N of momentum.
How would the momentum of the car change if:
a) the car slowed to half of its speed?
b) the car completely stopped?
c) the car gained its original weight in luggage?

[[File:momentummiddling.jpg]]

===3. (Difficult)===
You and your friends are watching NBA highlights at home and want to practice your physics. You notice at the beginning of a clip a basketball ball is rolling down the court at 23.5 m/s to the right. At the end, it is rolling at 3.8 m/s in the same direction. The commentator tells you that the change in its momentum is 17.24 kg m/s to the left. Curious at how many basketballs you can carry, you want to find the mass of the ball.

[[File:momentumhardaf.jpg]]

===4. (Difficult)===
A system is comprised of two particles. One particle has a mass of 6kg and is travelling in a direction 30 degrees east of north at a speed of 8m/s. The total momentum of the system is 3kg*m/s west. The second particle has a mass of 3kg. What is the second particle's velocity? Give your answer in terms of a north-south component and an east-west component.

[[File:Momentumadditionalhardsmaller.jpg]]

==Connectedness==

===Scenario: runaway vehicle===

Imagine that you are standing at the bottom of a hill when a runaway vehicle comes careening down. If it is a bicycle, it would be much easier to stop than if it were a truck moving at the same speed. One explanation for this is that the truck would have a greater mass and therefore a greater momentum. In order to be brought to rest, the truck must therefore experience a large change in momentum, which means a large impulse must be exerted on it.

==History==

The oldest known attempt at quantifying motion using both an object's speed and mass was that of René Descartes (1596–1650) [https://science.jrank.org/pages/4419/Momentum.html (source)]. John Wallis (1616 – 1703) was the first to use the phrase momentum, and to define it as the product of mass and velocity (rather than speed). He recognized that this notion of momentum is conserved in closed systems [https://www.famousscientists.org/john-wallis/ (source)], a concept confirmed by experiments performed by Christian Huygens (1629-1695) [https://www2.stetson.edu/~efriedma/periodictable/html/Hg.html (source)]. Finally, Isaac Newton (1643-1727) wrote his famous three laws relating momentum to force in his book Principia Mathematica in 1687. These offered a theoretical explanation for conservation of momentum. These foundations were so logically sound and experimentally observable that they would go unquestioned for centuries, until Albert Einstein (1879-1955) had to adjust the formula for momentum to maintain its accuracy for objects moving at relativistic speeds (see [[relativistic momentum]]).

== See also ==

*[[Mass]]
*[[Velocity]]
*[[Vectors]]
*[[Newton's Second Law: the Momentum Principle]]
*[[Impulse and Momentum]]
*[[Conservation of Momentum]]

===Further reading===

Chabay, Sherwood. (2015). Matter and Interactions (4th ed., Vol. 1). Raleigh, North Carolina: Wiley.

===External links===

<ol>
<li>https://www.khanacademy.org/science/physics/linear-momentum/momentum-tutorial/v/introduction-to-momentum</li>
</ol>

==References==

[[Category: Momentum]]
*http://hyperphysics.phy-astr.gsu.edu/hbase/mom.html
*http://study.com/academy/lesson/linear-momentum-definition-equation-and-examples.html
*https://science.jrank.org/pages/4419/Momentum.html
*https://www.famousscientists.org/john-wallis/
*https://www2.stetson.edu/~efriedma/periodictable/html/Hg.html

Newton's Second Law: the Momentum Principle

2022-12-03T23:15:01Z

Rgovind7:

'''Claimed by Raksha Govind, Fall 2022'''

This page describes Newton's second law of motion, also known as the momentum principle, which relates net force to the change in [[Linear Momentum]]. This principle is used to predict the effects of forces on the motion of objects.

==The Main Idea==

Newton's second law of motion, also known as the momentum principle, explains how forces cause the momentum of a system to change over time. The principle states that <math>\vec{F}_{net} = \frac{d\vec{p}_{system}}{dt}</math> where <math>\vec{p}</math> is the [[Linear Momentum]] of a system, <math>\vec{F}_{net}</math> is the external [[Net Force]] acting on the system from its surroundings, and t is time. Often, the system in question consists of a single particle whose motion we want to predict, although the law is also true for systems of particles and non-particle distributions of mass, such as disks. Note that both force and momentum are vector quantities, and that the change in momentum as a result of a force will always be in the direction of that force.

Here are some simple situations to help you build an intuition for the momentum principle:
<ol>
<li>If a particle travels in a straight line at a constant speed, its momentum is constant, so there is no net force acting on the particle.
<li>If a particle travels in a straight line while speeding up, its momentum is increasing, so there must be a net force acting in the same direction as the particle's momentum.
<li>If a particle travels in a straight line while slowing down, its momentum is decreasing, so there must be a net force acting in the opposite direction as the particle's momentum.
<li>If a particle travels along a curving path, its momentum changes direction, and a force not parallel to the object's momentum must be acting on it.
</ol>

The momentum principle has no derivation, as it is considered the definition of force. The metric unit most commonly used for force is the Newton.

===A Mathematical Model===

The momentum principle states that <math>\vec{F}_{net} = \frac{d\vec{p}}{dt}</math>. Recall that <math>\vec{p} = m\vec{v}</math>. Therefore, by product rule, <math>\frac{d\vec{p}}{dt} = m\frac{d\vec{v}}{dt} + \vec{v}\frac{dm}{dt}</math>. Usually,the mass of the particle or system is constant over time, so the second term becomes 0, and the momentum principle becomes <math>\vec{F}_{net} = m\frac{d\vec{v}}{dt}</math>, or <math>\vec{F}_{net} = m\vec{a}</math>, which may be a more familiar form of Newton's second law. The form <math>\vec{F}_{net} = \frac{d\vec{p}}{dt}</math> is preferred for several reasons:
<ol>
<li>When a particle accumulates mass that was initially at rest, such as a snowball rolling downhill, the term <math>\vec{v}\frac{dm}{dt}</math> is not 0, and <math>\vec{F}_{net} = m\vec{a}</math> is no longer accurate, while <math>\vec{F}_{net} = \frac{d\vec{p}}{dt}</math> is;</li>
<li>As a particle approaches the speed of light, <math>\vec{F}_{net} = m\vec{a}</math> is no longer accurate, while <math>\vec{F}_{net} = \frac{d\vec{p}}{dt}</math> is, as long as [[Relativistic Momentum]] is used; and</li>
<li><math>\vec{F}_{net} = \frac{d\vec{p}}{dt}</math> has a more direct rotational analogue- that is, it will be easier to accurately learn rotational physics if you learn linear physics using this form.</li>
</ol>

That said, the relation <math>\vec{F}_{net} = m\vec{a}</math> can often be of use when the particle is known to have a constant mass and travel at non-relativistic speeds, particularly in combination with [[Kinematics|kinematic equations]] that also use <math>\vec{a}</math>.

The momentum principle is responsible for the relationship between [[Impulse and Momentum]].

Note that according to the momentum principle, when the external force on a system is 0 (that is, the system is closed), the rate of change of its momentum is 0. This results in [[Conservation of Momentum]] in certain situations.

==Examples==

===1. (Simple)===

At t=0, a 4kg particle is released from rest and a constant 12N force begins acting on it. How far has the particle moved after 5 seconds?

For this problem, it is easier to use <math>\vec{F}_{net} = m\vec{a}</math> than <math>\vec{F}_{net} = \frac{d\vec{p}}{dt}</math> because it allows us to solve for <math>\vec{a}</math> which can then be used in a kinematic equation. We know this form of Newton's second law is accurate for this problem because the particle travels at non-relativistic speeds and is of constant mass.

<math>\vec{F}_{net} = m\vec{a}</math> can be rearranged into <math>\vec{a} = \frac{\vec{F}_{net}}{m}</math>

<math> a = \frac{12}{4} = 3 </math> (direction doesn't matter for this problem.)

<math> x_f = \frac{1}{2}at^2 + v_it + x_i = \frac{1}{2}(3)(5^2) = \frac{75}{2}</math> = 37.5 m.

===2. (Middling)===

A tennis ball of mass mB = 0.8kg and an initial velocity of ViB = <7,0,0> m/s. The tennis ball collides head-on with a lump of mashed potatoes with a mass mP = 3kg that was initially at rest. The tennis ball becomes embedded in the mush. Determine the velocity of the tennis ball and mashed potatoes after the collision, assuming that this collision is totally inelastic. Calculate how much energy is lost and determine the tennis ball and mashed potatoes individual velocities after the collision if the collision was elastic.

If this collision is inelastic, then we know that kinetic energy is not conserved. However, momentum is conserved for all types of collisions (elastic, inelastic, partially elastic) due to the momentum principle.
To start, write out the momentum principle. Since the tennis ball and mashed potatoes are stuck together after the collision, the final momentum can be written with their combined mass:

[[File:momentumconserved.jpg|200px|thumb|left|alt text]]

[[File:momentumconserved.jpg]]

[[File:MCP2.jpg|200px|thumb|left|alt text]]

===3. (Difficult)===

A cylindrical rocket ship of mass <math>M</math> and speed <math>v</math> is cruising through outer space when it encounters a stationary dust cloud. The dust cloud has a number density of <math>N</math> particles per unit volume, and each dust particle has a mass of <math>m</math>. The rocket ship has a sticky surface on its front face that allows it to accumulate all of the dust that it hits, absorbing it into its mass. The radius of the rocket ship's circular face is <math>R</math>. What force does the rocket's thruster need to exert on it in order to preserve its speed as it moves through the dust cloud? (Assume that the consumption of fuel has a negligible impact on the mass of the rocket.)

[[File:Rocketdustcloudpart1.jpg]]

[[File:Rocketdustcloudpart2.jpg]]

Advanced Note: in order to use <math>\frac{dm}{dt}</math> for the accumulation of mass, the accumulated mass (i.e. the dust) must initially be at rest, like in the above problem. Otherwise, one must treat the accumulation of dust particles as a series of collisions, or change one's frame of reference so that the accumulated mass is initially at rest.

===4. (Difficult)===

A particle of mass <math>m</math> and speed <math>v</math> moves in a circular path. Its angular frequency is <math>\Omega</math>. (Angular frequency is the rate at which the angle of the particle's position is changing in radians per unit time.) Using the momentum principle, show that

a) a nonzero net force is acting on the particle,

b) the magnitude of the force is given by <math>f = mv\Omega</math>, and

c) the direction of the force is inwards towards the center of the circle.

a) The particle's momentum vector is changing over time; its magnitude is constant (mv), but its direction changes as its position along the circular path changes. It is always tangential to the circular path. Since <math>\vec{p}</math> changes over time, a nonzero net force must be acting on the particle. In fact, the only way a particle of constant mass can have a net force of 0 acting on it is if it is travelling along a straight line at a constant speed, or if it is at rest.

b)

[[File:derivationofcentripetalforce.jpg]]

c) As one can see from the diagram above, <math>d\vec{p}</math> is pointing towards the center of the circle. If the particle is exactly at the top of the circle, its momentum is to the right (or left if the particle is travelling counterclockwise), and as <math>d\theta</math> approaches 0, <math>d\vec{p}</math> approaches a downward direction, which is towards the center of the circle.

For more information, see [[Centripetal Force and Curving Motion]]. Our answer agrees with the formulas given on that page.

==Connectedness==

Newton's second law is applicable to any situation where a force is applied and a system's momentum changes. There are nearly limitless examples of such situations, as well as nearly limitless applications. A few can be found below.

===Scenario: bungee jump===

When a bungee jumper falls and their cord is pulled taut, the cord applies a force in the direction opposite to the jumper's direction of travel. When this force is greater in magnitude than the gravity force, the net force is upward, which cuses causes the jumper's momentum, which is initially downward, to change over time in the upward direction. Eventually, the jumper is pulled back to a high altitude, the cord becomes slack, and gravity is once again the dominant force. The net force is downward once more, and the jumper's momentum changes over time in the downward direction.

===Application: automobile industry===

Cars frequently undergo all sorts of acceleration; they speed up, turn, and slow down. Forces are responsible for all of those accelerations, and these forces ultimately come from the tires' contact with the road. This force is, in fact, a form of [[Static Friction]]. The magnitude of this static friction force can only be up to a certain quantity known as the maximum static friction force. This maximum is determined by the shape and materials of the tires, as well as the weight of the car. Automobile manufacturers must always be aware of this maximum friction force when designing their car, as it places a limit on the ability of the car to accelerate, turn, and brake without slipping.

==History==

In his 1687 work Principia Mathematica, Isaac Newton (1643-1727) published his three laws of motion. Although his first and third laws were inspired by other scientists' work, his second law was entirely original. Below was the law as it appeared in his book:

Original Latin:

“Lex II: Mutationem motus proportionalem esse vi motrici impressae, et fieri secundum lineam rectam qua vis illa imprimitur.”

This was translated closely in Motte's 1729:

“Law II: The alteration of motion is ever proportional to the motive force impress'd; and is made in the direction of the right line in which that force is impress'd.”

In other words, the rate of change of momentum of a body is proportional to the force impressed on the body, and happens in the direction of that force.

This law was the first time force had ever been defined, and it is a definition that is still used today. As a result of Newton's contribution to the science of forces, the unit of force most commonly still used today is named after him.

== See also ==

*[[Linear Momentum]]
*[[Impulse and Momentum]]
*[[Conservation of Momentum]]
*[[Net Force]]
*[[Mass]]
*[[Acceleration]]

===Further reading===

RANKINE, William John Macquorn, and Edward Fisher BAMBER. A Mechanical Text Book. 1873. Print.

===External links===

<ol>
<li>nhttp://homepages.engineering.auckland.ac.nz/~pkel015/SolidMechanicsBooks/Part_III/Chapter_3_Stress_Mass_Momentum/Stress_Balance_Principles_02_The_Momentum_Principles.pdf</li>
<li>http://acme.highpoint.edu/~atitus/phy221/lecture-notes/2-2-momentum-principle.pdf</li>
<li>https://www.khanacademy.org/science/physics/forces-newtons-laws/newtons-laws-of-motion/v/newton-s-second-law-of-motion</li>
<li>https://www.khanacademy.org/science/physics/forces-newtons-laws/newtons-laws-of-motion/v/more-on-newtons-second-law</li>
<li>https://science.howstuffworks.com/innovation/scientific-experiments/newton-law-of-motion3.htm</li>
<li>https://www.livescience.com/46560-newton-second-law.html</li>
</ol>

==References==

*William, Harris. "How Newton's Laws of Motion Work" 29 July 2008. HowStuffWorks.com. <https://science.howstuffworks.com/innovation/scientific-experiments/newton-law-of-motion.htm> 11 April, 2018.
*Sherwood, Bruce A. "2.1 The Momentum Principle." Matter & Interactions. By Ruth W. Chabay. 4th ed. Vol. 1. N.p.: John Wiley & Sons, 2015. 45-50. Print. Modern Mechanics. 8 April, 2018.
*http://www.physicsclassroom.com/class/newtlaws/Lesson-3/Newton-s-Second-Law
*Dugas, René (1988). A history of mechanics. Translated into English by J.R. Maddox (Dover ed.). New York: Dover Publications.
*Jennings, John (1721). Miscellanea in Usum Juventutis Academicae. Northampton: R. Aikes & G. Dicey.

[[Category: Momentum]]

Linear Momentum

2022-12-03T22:33:10Z

Rgovind7:

'''Claimed By: Raksha Govind Fall 2022'''

This page defines the linear momentum of particles and systems.

==The Main Idea==

Linear momentum is a vector quantity describing an object's motion. It is defined as the product of an object's [[Mass]] (<math>m</math>) and [[Velocity]] (<math>\vec{v}</math>). Note that mass is a scalar while velocity is a vector, so an object's linear momentum is always in the same direction as its velocity. Linear momentum is represented by the letter <math>\vec{p}</math> and is often referred to as simply "momentum." The most commonly used metric unit for momentum is the kilogram*meter/second. The plural of momentum is momenta or momentums.

===A Mathematical Model===

====Single Particles====
The momentum of a particle is defined as follows:

<math>\vec{p} = m\vec{v}</math>

where <math>\vec{p}</math> is the particle's linear momentum, <math>m</math> is the particle's mass, and <math>\vec{v}</math> is the particle's velocity. This formula accurately describes the momentum of particles at everyday speeds, but for particles travelling near the speed of light, the formula for [[Relativistic Momentum]] must be used for concepts such as the momentum principle to remain true.

====Multiple Particles====

The total momentum of a system of particles is defined as the vector sum of the momenta of the particles that comprise the system:

<math>\vec{p}_{system} = \sum_i \vec{p}_i</math>

Although the proof does not appear on this page, it can be shown that the total momentum of a system of particles is equal to the total mass of the system times the velocity of its [[Center of Mass]]:

<math>\vec{p}_{system} = M_{tot}\vec{v}_{COM}</math>

This formula makes it significantly easier to calculate the momentum of certain objects. For example, consider a disk rolling along the ground. The disk is comprised of an infinite number of infinitely small "mass elements," all of which have different momenta; the ones at the bottom of the disk are hardly moving while the ones at the top are moving very quickly. Without the formula above, one would have to use an integral to add the momenta of all of the mass elements to find the total momentum of the disk. However, the formula above tells us that we can simply multiply the mass of the disk by the velocity of its center of mass, which is in this case the geometric center of the disk. This reveals that the fact that the disk is rotating does not affect its momentum.

====In Relation to Other Physics Topics====

In the absence of a net force, the momentum of a particle stays constant over time, as stated by Newton's first law.

When a force is applied to a particle, its momentum evolves over time according to Newton's second law. For more information, see [[Newton's Second Law: the Momentum Principle]].

When an impulse is applied to a particle, its momentum changes in a specific way: the change in momentum <math>\Delta\vec{p}</math> is equal to the impulse <math>\vec{J}</math>. This is a consequence of the Momentum Principle. For more information, see [[Impulse and Momentum]].

When the net external force on a system of particles is 0, the system's momentum is conserved (that is, constant over time) even if the particles within the system interact with each other (exert internal forces on each other). For more information, see [[Conservation of Momentum]].

===A Computational Model===

In computational simulations of particles using [[Iterative Prediction]], a momentum vector variable is assigned to each particle. Such simulations usually in "time steps," or iterations of a loop representing a short time interval. In each time step, the particles' momenta are updated based on the forces acting on it. Then their velocities are calculated by dividing each particle's momentum by its mass. Finally, the velocities are used to update the positions of the particles. Below is an example of such a simulation:

This vPython simulation shows a cart (represented by a rectangle) whose motion is affected by a gust of wind applying a constant force.

https://trinket.io/glowscript/ce43925647

For more information, see [[Iterative Prediction]].

==Examples==

===1. (Simple)===
Find the momentum of a ball that has a mass of 69kg and is moving at <1,2,3> m/s.

[[File:momentumsimple.jpg]]

===2. (Middling)===
A car has 20,000 N of momentum.
How would the momentum of the car change if:
a) the car slowed to half of its speed?
b) the car completely stopped?
c) the car gained its original weight in luggage?

[[File:momentummiddling.jpg]]

===3. (Difficult)===
You and your friends are watching NBA highlights at home and want to practice your physics. You notice at the beginning of a clip a basketball ball is rolling down the court at 23.5 m/s to the right. At the end, it is rolling at 3.8 m/s in the same direction. The commentator tells you that the change in its momentum is 17.24 kg m/s to the left. Curious at how many basketballs you can carry, you want to find the mass of the ball.

[[File:momentumhardaf.jpg]]

===4. (Difficult)===
A system is comprised of two particles. One particle has a mass of 6kg and is travelling in a direction 30 degrees east of north at a speed of 8m/s. The total momentum of the system is 3kg*m/s west. The second particle has a mass of 3kg. What is the second particle's velocity? Give your answer in terms of a north-south component and an east-west component.

[[File:Momentumadditionalhardsmaller.jpg]]

==Connectedness==

===Scenario: runaway vehicle===

Imagine that you are standing at the bottom of a hill when a runaway vehicle comes careening down. If it is a bicycle, it would be much easier to stop than if it were a truck moving at the same speed. One explanation for this is that the truck would have a greater mass and therefore a greater momentum. In order to be brought to rest, the truck must therefore experience a large change in momentum, which means a large impulse must be exerted on it.

==History==

The oldest known attempt at quantifying motion using both an object's speed and mass was that of René Descartes (1596–1650) [https://science.jrank.org/pages/4419/Momentum.html (source)]. John Wallis (1616 – 1703) was the first to use the phrase momentum, and to define it as the product of mass and velocity (rather than speed). He recognized that this notion of momentum is conserved in closed systems [https://www.famousscientists.org/john-wallis/ (source)], a concept confirmed by experiments performed by Christian Huygens (1629-1695) [https://www2.stetson.edu/~efriedma/periodictable/html/Hg.html (source)]. Finally, Isaac Newton (1643-1727) wrote his famous three laws relating momentum to force in his book Principia Mathematica in 1687. These offered a theoretical explanation for conservation of momentum. These foundations were so logically sound and experimentally observable that they would go unquestioned for centuries, until Albert Einstein (1879-1955) had to adjust the formula for momentum to maintain its accuracy for objects moving at relativistic speeds (see [[relativistic momentum]]).

== See also ==

*[[Mass]]
*[[Velocity]]
*[[Vectors]]
*[[Newton's Second Law: the Momentum Principle]]
*[[Impulse and Momentum]]
*[[Conservation of Momentum]]

===Further reading===

Chabay, Sherwood. (2015). Matter and Interactions (4th ed., Vol. 1). Raleigh, North Carolina: Wiley.

===External links===

<ol>
<li>https://www.khanacademy.org/science/physics/linear-momentum/momentum-tutorial/v/introduction-to-momentum</li>
</ol>

==References==

[[Category: Momentum]]
*http://hyperphysics.phy-astr.gsu.edu/hbase/mom.html
*http://study.com/academy/lesson/linear-momentum-definition-equation-and-examples.html
*https://science.jrank.org/pages/4419/Momentum.html
*https://www.famousscientists.org/john-wallis/
*https://www2.stetson.edu/~efriedma/periodictable/html/Hg.html

Linear Momentum

2022-12-03T22:31:17Z

Rgovind7:

b/Claimed By: Raksha Govind Fall 2022

This page defines the linear momentum of particles and systems.

==The Main Idea==

Linear momentum is a vector quantity describing an object's motion. It is defined as the product of an object's [[Mass]] (<math>m</math>) and [[Velocity]] (<math>\vec{v}</math>). Note that mass is a scalar while velocity is a vector, so an object's linear momentum is always in the same direction as its velocity. Linear momentum is represented by the letter <math>\vec{p}</math> and is often referred to as simply "momentum." The most commonly used metric unit for momentum is the kilogram*meter/second. The plural of momentum is momenta or momentums.

===A Mathematical Model===

====Single Particles====
The momentum of a particle is defined as follows:

<math>\vec{p} = m\vec{v}</math>

where <math>\vec{p}</math> is the particle's linear momentum, <math>m</math> is the particle's mass, and <math>\vec{v}</math> is the particle's velocity. This formula accurately describes the momentum of particles at everyday speeds, but for particles travelling near the speed of light, the formula for [[Relativistic Momentum]] must be used for concepts such as the momentum principle to remain true.

====Multiple Particles====

The total momentum of a system of particles is defined as the vector sum of the momenta of the particles that comprise the system:

<math>\vec{p}_{system} = \sum_i \vec{p}_i</math>

Although the proof does not appear on this page, it can be shown that the total momentum of a system of particles is equal to the total mass of the system times the velocity of its [[Center of Mass]]:

<math>\vec{p}_{system} = M_{tot}\vec{v}_{COM}</math>

This formula makes it significantly easier to calculate the momentum of certain objects. For example, consider a disk rolling along the ground. The disk is comprised of an infinite number of infinitely small "mass elements," all of which have different momenta; the ones at the bottom of the disk are hardly moving while the ones at the top are moving very quickly. Without the formula above, one would have to use an integral to add the momenta of all of the mass elements to find the total momentum of the disk. However, the formula above tells us that we can simply multiply the mass of the disk by the velocity of its center of mass, which is in this case the geometric center of the disk. This reveals that the fact that the disk is rotating does not affect its momentum.

====In Relation to Other Physics Topics====

In the absence of a net force, the momentum of a particle stays constant over time, as stated by Newton's first law.

When a force is applied to a particle, its momentum evolves over time according to Newton's second law. For more information, see [[Newton's Second Law: the Momentum Principle]].

When an impulse is applied to a particle, its momentum changes in a specific way: the change in momentum <math>\Delta\vec{p}</math> is equal to the impulse <math>\vec{J}</math>. This is a consequence of the Momentum Principle. For more information, see [[Impulse and Momentum]].

When the net external force on a system of particles is 0, the system's momentum is conserved (that is, constant over time) even if the particles within the system interact with each other (exert internal forces on each other). For more information, see [[Conservation of Momentum]].

===A Computational Model===

In computational simulations of particles using [[Iterative Prediction]], a momentum vector variable is assigned to each particle. Such simulations usually in "time steps," or iterations of a loop representing a short time interval. In each time step, the particles' momenta are updated based on the forces acting on it. Then their velocities are calculated by dividing each particle's momentum by its mass. Finally, the velocities are used to update the positions of the particles. Below is an example of such a simulation:

This vPython simulation shows a cart (represented by a rectangle) whose motion is affected by a gust of wind applying a constant force.

https://trinket.io/glowscript/ce43925647

For more information, see [[Iterative Prediction]].

==Examples==

===1. (Simple)===
Find the momentum of a ball that has a mass of 69kg and is moving at <1,2,3> m/s.

[[File:momentumsimple.jpg]]

===2. (Middling)===
A car has 20,000 N of momentum.
How would the momentum of the car change if:
a) the car slowed to half of its speed?
b) the car completely stopped?
c) the car gained its original weight in luggage?

[[File:momentummiddling.jpg]]

===3. (Difficult)===
You and your friends are watching NBA highlights at home and want to practice your physics. You notice at the beginning of a clip a basketball ball is rolling down the court at 23.5 m/s to the right. At the end, it is rolling at 3.8 m/s in the same direction. The commentator tells you that the change in its momentum is 17.24 kg m/s to the left. Curious at how many basketballs you can carry, you want to find the mass of the ball.

[[File:momentumhardaf.jpg]]

===4. (Difficult)===
A system is comprised of two particles. One particle has a mass of 6kg and is travelling in a direction 30 degrees east of north at a speed of 8m/s. The total momentum of the system is 3kg*m/s west. The second particle has a mass of 3kg. What is the second particle's velocity? Give your answer in terms of a north-south component and an east-west component.

[[File:Momentumadditionalhardsmaller.jpg]]

==Connectedness==

===Scenario: runaway vehicle===

Imagine that you are standing at the bottom of a hill when a runaway vehicle comes careening down. If it is a bicycle, it would be much easier to stop than if it were a truck moving at the same speed. One explanation for this is that the truck would have a greater mass and therefore a greater momentum. In order to be brought to rest, the truck must therefore experience a large change in momentum, which means a large impulse must be exerted on it.

==History==

The oldest known attempt at quantifying motion using both an object's speed and mass was that of René Descartes (1596–1650) [https://science.jrank.org/pages/4419/Momentum.html (source)]. John Wallis (1616 – 1703) was the first to use the phrase momentum, and to define it as the product of mass and velocity (rather than speed). He recognized that this notion of momentum is conserved in closed systems [https://www.famousscientists.org/john-wallis/ (source)], a concept confirmed by experiments performed by Christian Huygens (1629-1695) [https://www2.stetson.edu/~efriedma/periodictable/html/Hg.html (source)]. Finally, Isaac Newton (1643-1727) wrote his famous three laws relating momentum to force in his book Principia Mathematica in 1687. These offered a theoretical explanation for conservation of momentum. These foundations were so logically sound and experimentally observable that they would go unquestioned for centuries, until Albert Einstein (1879-1955) had to adjust the formula for momentum to maintain its accuracy for objects moving at relativistic speeds (see [[relativistic momentum]]).

== See also ==

*[[Mass]]
*[[Velocity]]
*[[Vectors]]
*[[Newton's Second Law: the Momentum Principle]]
*[[Impulse and Momentum]]
*[[Conservation of Momentum]]

===Further reading===

Chabay, Sherwood. (2015). Matter and Interactions (4th ed., Vol. 1). Raleigh, North Carolina: Wiley.

===External links===

<ol>
<li>https://www.khanacademy.org/science/physics/linear-momentum/momentum-tutorial/v/introduction-to-momentum</li>
</ol>

==References==

[[Category: Momentum]]
*http://hyperphysics.phy-astr.gsu.edu/hbase/mom.html
*http://study.com/academy/lesson/linear-momentum-definition-equation-and-examples.html
*https://science.jrank.org/pages/4419/Momentum.html
*https://www.famousscientists.org/john-wallis/
*https://www2.stetson.edu/~efriedma/periodictable/html/Hg.html

Iterative Prediction

2022-12-03T22:27:38Z

Rgovind7:

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

==The Main Idea==

Iterative prediction is a mathematical technique for approximating the behavior of one or more particles over an interval of time. An iteration is a repeated procedure, so iterative prediction uses repeated procedures to predict the motion of a system. 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:

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

[[File:Iterative_Model.png|center]]

<center>This is a visual representation of iterative prediction being used to repeatedly update a particle's momentum. It was found on [https://p3server.pa.msu.edu/coursewiki/doku.php?id=183_notes:iterativepredict this page] of the Projects & Practices in Physics website, where it is provided under a [https://creativecommons.org/licenses/by-nc-sa/3.0/ CC Attribution-Noncommercial-Share Alike 3.0 Unported] license. It is shown here under the same license.</center>

The size of the time steps used during iterative prediction is referred to as the resolution of the model. Dividing a time interval into a few large time steps is called low resolution while dividing the same time interval into many small time steps is called high resolution. Performing iterative prediction with a higher resolution requires more computations but produces more accurate results. This is because steps 2 and 3 assume constant force and velocity respectively during the duration of each time step (see the "mathematical model" section). In reality, force and velocity change continually, so this is where inaccuracy is introduced. However, if the time steps are small enough that force and velocity do not significantly change during any time step, 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.

===Relation between Newton's 2nd Law and Momentum Principle===

The rate of change of momentum of an object is directly proportional to the net force applied and is in the direction of the net force; the net force is equal to the rate of change of momentum.

[[File:Fma_.jpg]]

- F = net force in newtons (N)

- m = mass in kilograms (kg)

- a = acceleration in meters per second per second (ms-2)

[[File:rate-of-change-of-momentum-2.jpg]]

- F = net force in newtons (N)

- <math>\Delta mv</math> = change in momentum in kilogram metres per second (kg ms-1)

- t = change in time in seconds (s)

===A Mathematical Model===

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

<ol>

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

This step varies widely from program to program, depending on what system is being modeled. Sometimes, the forces acting on each particle are constant, such as when modeling the motion of a ball with a specific weight thrown near the surface of the earth. Other times, the forces are functions of properties such as the particles' positions, such as when modeling the motion of celestial bodies whose gravitational attraction is greater when they are closer together. Sometimes, there is only one force acting on each particle, and other times, the net force must be found by adding together the individual force vectors.

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

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

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

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

</ol>

===A Computational Model===

Iterative prediction is typically done on computers because many computations are necessary in order to perform it at a meaningful resolution. VPython is a useful program to perform iterative prediction 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]

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

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

Note that the first program is more realistic; the more frequent position updates result in a smoother curve that more closely approximates the true trajectory of a ball. On the other hand, the second program has fewer calculations to do. Both programs are artificially slowed using the "rate" command in order to simulate a realistic travel time, but if both programs were allowed to run as quickly as possible, the second program would finish first. For larger, more complicated simulations, time can be a significant factor.

Here is another more complicated simulation that uses iterative prediction to simulate the motion of a mass swinging on a spring. This is an example of iterative prediction with a varying force; during each time step, the forces acting on the mass must be calculated depending on its position using [[Hooke's Law]].

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

Note: both projectile motion and spring-mass systems can be analytically modeled; that is, in both of the above systems, analytical techniques such as calculus and kinematics can be used to find the position of the particles as a function of time. However, for some other, more complicated systems, analytical approaches may not exist, while iterative prediction often still works. For example, it would be easy to modify the projectile motion simulation to include the effects of [[Air Resistance]], which is difficult to take into account analytically.

==Examples==

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

===1. (Middling)===

A 2kg particle is released from rest at time t=0. A constant force of 6N is applied to it. How far has the particle traveled after 4 seconds?

Solve this question

A.) analytically (using kinematic equations)

B.) using iterative prediction with 12 equal time steps

C.) using iterative prediction with 4 equal time steps

D.) using iterative prediction with a single time step

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

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

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

Using [[Newton's Second Law: the Momentum Principle]], we know that <math>a = \frac{f}{m} = 3</math>m/s^2. We are also given that <math>v_0 = 0</math>, and we are interested in the time t=4. Substituting these values into the kinematic equation yields

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

B.) To solve this using iterative prediction, a computational tool should be used. Since 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]]

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

C.)

[[File:4steps.PNG]]

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

D.)

[[File:1step.PNG]]

using 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.

==Connectedness==

===Application: video game industry===

One popular application of iterative prediction is in physics-based video games. These are often little more than physics simulations with player input. Iterative prediction is used in most major video games to predict the motion of in-game objects. It can be used even if the laws of physics within the video game are different from those of the real world.

==History==

The impulse-momentum theorem, which is used in iterative prediction to update the momenta of particles, is derived from Newton's Second Law, which Isaac Newton (1643-1727) publish in his 1687 book Principia Mathematica. From this point onward, all of the math necessary for iterative prediction was known, but it was not useful to perform iterative prediction until significant computational advances in the twentieth century allowed for high resolutions.

== See also ==

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

===External links===

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

==References==

*Georgia Institute of Technology. Physics Department. PHYS 2211. Fall 2016. '''Wednesday, Week 2 Lecture Slides. Fenton, Flavio H'''
*Georgia Institute of Technology. Physics Department. PHYS 2211. Fall 2016. '''Monday, Week 3 Lecture Slides. Fenton, Flavio H'''
*Georgia Institute of Technology. Physics Department. PHYS 2211. Fall 2016. Lab 07 Fancart Energy & Spring-Mass Instructions, Greco, Edwin
*http://p3server.pa.msu.edu/coursewiki/doku.php?id=183_notes:iterativepredict
*http://physicsnet.co.uk/a-level-physics-as-a2/mechanics/newtons-laws-of-motion/
*Bañó-Polo, M., Baeza-Delgado, C., Tamborero, S., Hazel, A., Grau, B., Nilsson, I., … Mingarro, I. (2018). Transmembrane but not soluble helices fold inside the ribosome tunnel. Nature communications, 9(1), 5246. doi:10.1038/s41467-018-07554-7

Iterative Prediction

2022-12-03T22:26:19Z

Rgovind7:

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.

==The Main Idea==

Iterative prediction is a mathematical technique for approximating the behavior of one or more particles over an interval of time. An iteration is a repeated procedure, so iterative prediction uses repeated procedures to predict the motion of a system. 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:

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

[[File:Iterative_Model.png|center]]

<center>This is a visual representation of iterative prediction being used to repeatedly update a particle's momentum. It was found on [https://p3server.pa.msu.edu/coursewiki/doku.php?id=183_notes:iterativepredict this page] of the Projects & Practices in Physics website, where it is provided under a [https://creativecommons.org/licenses/by-nc-sa/3.0/ CC Attribution-Noncommercial-Share Alike 3.0 Unported] license. It is shown here under the same license.</center>

The size of the time steps used during iterative prediction is referred to as the resolution of the model. Dividing a time interval into a few large time steps is called low resolution while dividing the same time interval into many small time steps is called high resolution. Performing iterative prediction with a higher resolution requires more computations but produces more accurate results. This is because steps 2 and 3 assume constant force and velocity respectively during the duration of each time step (see the "mathematical model" section). In reality, force and velocity change continually, so this is where inaccuracy is introduced. However, if the time steps are small enough that force and velocity do not significantly change during any time step, 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.

===Relation between Newton's 2nd Law and Momentum Principle===

The rate of change of momentum of an object is directly proportional to the net force applied and is in the direction of the net force; the net force is equal to the rate of change of momentum.

[[File:Fma_.jpg]]

- F = net force in newtons (N)

- m = mass in kilograms (kg)

- a = acceleration in meters per second per second (ms-2)

[[File:rate-of-change-of-momentum-2.jpg]]

- F = net force in newtons (N)

- <math>\Delta mv</math> = change in momentum in kilogram metres per second (kg ms-1)

- t = change in time in seconds (s)

===A Mathematical Model===

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

<ol>

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

This step varies widely from program to program, depending on what system is being modeled. Sometimes, the forces acting on each particle are constant, such as when modeling the motion of a ball with a specific weight thrown near the surface of the earth. Other times, the forces are functions of properties such as the particles' positions, such as when modeling the motion of celestial bodies whose gravitational attraction is greater when they are closer together. Sometimes, there is only one force acting on each particle, and other times, the net force must be found by adding together the individual force vectors.

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

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

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

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

</ol>

===A Computational Model===

Iterative prediction is typically done on computers because many computations are necessary in order to perform it at a meaningful resolution. VPython is a useful program to perform iterative prediction 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]

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

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

Note that the first program is more realistic; the more frequent position updates result in a smoother curve that more closely approximates the true trajectory of a ball. On the other hand, the second program has fewer calculations to do. Both programs are artificially slowed using the "rate" command in order to simulate a realistic travel time, but if both programs were allowed to run as quickly as possible, the second program would finish first. For larger, more complicated simulations, time can be a significant factor.

Here is another more complicated simulation that uses iterative prediction to simulate the motion of a mass swinging on a spring. This is an example of iterative prediction with a varying force; during each time step, the forces acting on the mass must be calculated depending on its position using [[Hooke's Law]].

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

Note: both projectile motion and spring-mass systems can be analytically modeled; that is, in both of the above systems, analytical techniques such as calculus and kinematics can be used to find the position of the particles as a function of time. However, for some other, more complicated systems, analytical approaches may not exist, while iterative prediction often still works. For example, it would be easy to modify the projectile motion simulation to include the effects of [[Air Resistance]], which is difficult to take into account analytically.

==Examples==

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

===1. (Middling)===

A 2kg particle is released from rest at time t=0. A constant force of 6N is applied to it. How far has the particle traveled after 4 seconds?

Solve this question

A.) analytically (using kinematic equations)

B.) using iterative prediction with 12 equal time steps

C.) using iterative prediction with 4 equal time steps

D.) using iterative prediction with a single time step

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

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

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

Using [[Newton's Second Law: the Momentum Principle]], we know that <math>a = \frac{f}{m} = 3</math>m/s^2. We are also given that <math>v_0 = 0</math>, and we are interested in the time t=4. Substituting these values into the kinematic equation yields

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

B.) To solve this using iterative prediction, a computational tool should be used. Since 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]]

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

C.)

[[File:4steps.PNG]]

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

D.)

[[File:1step.PNG]]

using 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.

==Connectedness==

===Application: video game industry===

One popular application of iterative prediction is in physics-based video games. These are often little more than physics simulations with player input. Iterative prediction is used in most major video games to predict the motion of in-game objects. It can be used even if the laws of physics within the video game are different from those of the real world.

==History==

The impulse-momentum theorem, which is used in iterative prediction to update the momenta of particles, is derived from Newton's Second Law, which Isaac Newton (1643-1727) publish in his 1687 book Principia Mathematica. From this point onward, all of the math necessary for iterative prediction was known, but it was not useful to perform iterative prediction until significant computational advances in the twentieth century allowed for high resolutions.

== See also ==

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

===External links===

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

==References==

*Georgia Institute of Technology. Physics Department. PHYS 2211. Fall 2016. '''Wednesday, Week 2 Lecture Slides. Fenton, Flavio H'''
*Georgia Institute of Technology. Physics Department. PHYS 2211. Fall 2016. '''Monday, Week 3 Lecture Slides. Fenton, Flavio H'''
*Georgia Institute of Technology. Physics Department. PHYS 2211. Fall 2016. Lab 07 Fancart Energy & Spring-Mass Instructions, Greco, Edwin
*http://p3server.pa.msu.edu/coursewiki/doku.php?id=183_notes:iterativepredict
*http://physicsnet.co.uk/a-level-physics-as-a2/mechanics/newtons-laws-of-motion/
*Bañó-Polo, M., Baeza-Delgado, C., Tamborero, S., Hazel, A., Grau, B., Nilsson, I., … Mingarro, I. (2018). Transmembrane but not soluble helices fold inside the ribosome tunnel. Nature communications, 9(1), 5246. doi:10.1038/s41467-018-07554-7

Iterative Prediction

2022-12-03T22:24:26Z

Rgovind7:

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.

==The Main Idea==

Iterative prediction is a mathematical technique for approximating the behavior of one or more particles over an interval of time. An iteration is a repeated procedure, so iterative prediction uses repeated procedures to predict the motion of a system. 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:

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

[[File:Iterative_Model.png|center]]

<center>This is a visual representation of iterative prediction being used to repeatedly update a particle's momentum. It was found on [https://p3server.pa.msu.edu/coursewiki/doku.php?id=183_notes:iterativepredict this page] of the Projects & Practices in Physics website, where it is provided under a [https://creativecommons.org/licenses/by-nc-sa/3.0/ CC Attribution-Noncommercial-Share Alike 3.0 Unported] license. It is shown here under the same license.</center>

The size of the time steps used during iterative prediction is referred to as the resolution of the model. Dividing a time interval into a few large time steps is called low resolution while dividing the same time interval into many small time steps is called high resolution. Performing iterative prediction with a higher resolution requires more computations but produces more accurate results. This is because steps 2 and 3 assume constant force and velocity respectively during the duration of each time step (see the "mathematical model" section). In reality, force and velocity change continually, so this is where inaccuracy is introduced. However, if the time steps are small enough that force and velocity do not significantly change during any time step, 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.

===Relation between Newton's 2nd Law and Momentum Principle===

The rate of change of momentum of an object is directly proportional to the net force applied and is in the direction of the net force; the net force is equal to the rate of change of momentum.

[[File:Fma_.jpg]]

- F = net force in newtons (N)

- m = mass in kilograms (kg)

- a = acceleration in meters per second per second (ms-2)

[[File:rate-of-change-of-momentum-2.jpg]]

- F = net force in newtons (N)

- <math>\Delta mv</math> = change in momentum in kilogram metres per second (kg ms-1)

- t = change in time in seconds (s)

===A Mathematical Model===

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

<ol>

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

This step varies widely from program to program, depending on what system is being modeled. Sometimes, the forces acting on each particle are constant, such as when modeling the motion of a ball with a specific weight thrown near the surface of the earth. Other times, the forces are functions of properties such as the particles' positions, such as when modeling the motion of celestial bodies whose gravitational attraction is greater when they are closer together. Sometimes, there is only one force acting on each particle, and other times, the net force must be found by adding together the individual force vectors.

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

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

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

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

</ol>

===A Computational Model===

Iterative prediction is typically done on computers because many computations are necessary in order to perform it at a meaningful resolution. VPython is a useful program to perform iterative prediction 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]

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

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

Note that the first program is more realistic; the more frequent position updates result in a smoother curve that more closely approximates the true trajectory of a ball. On the other hand, the second program has fewer calculations to do. Both programs are artificially slowed using the "rate" command in order to simulate a realistic travel time, but if both programs were allowed to run as quickly as possible, the second program would finish first. For larger, more complicated simulations, time can be a significant factor.

Here is another more complicated simulation that uses iterative prediction to simulate the motion of a mass swinging on a spring. This is an example of iterative prediction with a varying force; during each time step, the forces acting on the mass must be calculated depending on its position using [[Hooke's Law]].

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

Note: both projectile motion and spring-mass systems can be analytically modeled; that is, in both of the above systems, analytical techniques such as calculus and kinematics can be used to find the position of the particles as a function of time. However, for some other, more complicated systems, analytical approaches may not exist, while iterative prediction often still works. For example, it would be easy to modify the projectile motion simulation to include the effects of [[Air Resistance]], which is difficult to take into account analytically.

==Examples==

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

===1. (Middling)===

A 2kg particle is released from rest at time t=0. A constant force of 6N is applied to it. How far has the particle traveled after 4 seconds?

Solve this question

A.) analytically (using kinematic equations)

B.) using iterative prediction with 12 equal time steps

C.) using iterative prediction with 4 equal time steps

D.) using iterative prediction with a single time step

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

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

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

Using [[Newton's Second Law: the Momentum Principle]], we know that <math>a = \frac{f}{m} = 3</math>m/s^2. We are also given that <math>v_0 = 0</math>, and we are interested in the time t=4. Substituting these values into the kinematic equation yields

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

B.) To solve this using iterative prediction, a computational tool should be used. Since 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]]

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

C.)

[[File:4steps.PNG]]

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

D.)

[[File:1step.PNG]]

using 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.

==Connectedness==

===Application: video game industry===

One popular application of iterative prediction is in physics-based video games. These are often little more than physics simulations with player input. Iterative prediction is used in most major video games to predict the motion of in-game objects. It can be used even if the laws of physics within the video game are different from those of the real world.

==History==

The impulse-momentum theorem, which is used in iterative prediction to update the momenta of particles, is derived from Newton's Second Law, which Isaac Newton (1643-1727) publish in his 1687 book Principia Mathematica. From this point onward, all of the math necessary for iterative prediction was known, but it was not useful to perform iterative prediction until significant computational advances in the twentieth century allowed for high resolutions.

== See also ==

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

===External links===

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

==References==

*Georgia Institute of Technology. Physics Department. PHYS 2211. Fall 2016. '''Wednesday, Week 2 Lecture Slides. Fenton, Flavio H'''
*Georgia Institute of Technology. Physics Department. PHYS 2211. Fall 2016. '''Monday, Week 3 Lecture Slides. Fenton, Flavio H'''
*Georgia Institute of Technology. Physics Department. PHYS 2211. Fall 2016. Lab 07 Fancart Energy & Spring-Mass Instructions, Greco, Edwin
*http://p3server.pa.msu.edu/coursewiki/doku.php?id=183_notes:iterativepredict
*http://physicsnet.co.uk/a-level-physics-as-a2/mechanics/newtons-laws-of-motion/
*Bañó-Polo, M., Baeza-Delgado, C., Tamborero, S., Hazel, A., Grau, B., Nilsson, I., … Mingarro, I. (2018). Transmembrane but not soluble helices fold inside the ribosome tunnel. Nature communications, 9(1), 5246. doi:10.1038/s41467-018-07554-7