Multi-particle analysis of Momentum - Revision history

Acrutcher6: /* Difficult */

2025-11-14T22:55:27Z

Difficult

← Older revision		Revision as of 18:55, 14 November 2025
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	A woman, Katherine, and a man, Robert, each have a mass of <math> {m_{people}} </math>, and decide to have some fun in the theatre before putting away curtains. They stand on top of a curtain bin with mass <math> {m_{bin}} </math>. Katherine and Robert jump off of one side of the bin at a velocity of v relative to the bin, and when they do so, the bin rolls in the direction opposite of which they jumped. Symbolically, what is the final velocity of the bin if both Katherine and Robert jump at the ''same time''?		A woman, Katherine, and a man, Robert, each have a mass of <math> {m_{people}} </math>, and decide to have some fun in the theatre before putting away curtains. They stand on top of a curtain bin with mass <math> {m_{bin}} </math>. Katherine and Robert jump off of one side of the bin at a velocity of v relative to the bin, and when they do so, the bin rolls in the direction opposite of which they jumped. Symbolically, what is the final velocity of the bin if both Katherine and Robert jump at the ''same time''?

	'''Answer: <math> v_f = \frac{2m * v}{m_{people} + m_{bin}} </math>'''		'''Answer: <math> v_f = \frac{2m_{people} * v}{m_{people} + m_{bin}} </math>'''

	'''Explanation:'''		'''Explanation:'''

Acrutcher6: /* Difficult */

2025-11-14T22:54:29Z

Difficult

← Older revision		Revision as of 18:54, 14 November 2025
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	===Difficult===		===Difficult===

	A woman, Katherine, and a man, Robert, each have a mass of m_{people}, and decide to have some fun in the theatre before putting away curtains. They stand on top of a curtain bin with mass m_{bin}. Katherine and Robert jump off of one side of the bin at a velocity of v relative to the bin, and when they do so, the bin rolls in the direction opposite of which they jumped. Symbolically, what is the final velocity of the bin if both Katherine and Robert jump at the ''same time''?		A woman, Katherine, and a man, Robert, each have a mass of <math> {m_{people}} </math>, and decide to have some fun in the theatre before putting away curtains. They stand on top of a curtain bin with mass <math> {m_{bin}} </math>. Katherine and Robert jump off of one side of the bin at a velocity of v relative to the bin, and when they do so, the bin rolls in the direction opposite of which they jumped. Symbolically, what is the final velocity of the bin if both Katherine and Robert jump at the ''same time''?

	'''Answer: <math> v_f = \frac{2m * v}{m_{people} + m_{bin}} </math>'''		'''Answer: <math> v_f = \frac{2m * v}{m_{people} + m_{bin}} </math>'''

Acrutcher6: /* Difficult */

2025-11-14T22:52:06Z

Difficult

← Older revision		Revision as of 18:52, 14 November 2025
Line 179:		Line 179:
	===Difficult===		===Difficult===

	A woman, Katherine, and a man, Robert, each have a mass of m, and decide to have some fun in the theatre before putting away curtains. They stand on top of a curtain bin. Katherine and Robert jump off of one side of the bin at a velocity of v relative to the bin, and when they do so, the bin rolls in the direction opposite of which they jumped. Symbolically, what is the final velocity of the bin if both Katherine and Robert jump at the ''same time''?		A woman, Katherine, and a man, Robert, each have a mass of m_{people}, and decide to have some fun in the theatre before putting away curtains. They stand on top of a curtain bin with mass m_{bin}. Katherine and Robert jump off of one side of the bin at a velocity of v relative to the bin, and when they do so, the bin rolls in the direction opposite of which they jumped. Symbolically, what is the final velocity of the bin if both Katherine and Robert jump at the ''same time''?

	'''Answer: <math> v_f = \frac{2m * v}{m_{people} + m_{bin}} </math>'''		'''Answer: <math> v_f = \frac{2m * v}{m_{people} + m_{bin}} </math>'''

Parkercoye: /* The Main Idea */

2019-07-29T15:18:45Z

The Main Idea

← Older revision		Revision as of 11:18, 29 July 2019
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	<math>\vec{F}_{net} = \vec{F}_{1} + \vec{F}_{2} + \vec{F}_{3} + ...</math>		<math>\vec{F}_{net} = \vec{F}_{1} + \vec{F}_{2} + \vec{F}_{3} + ...</math>

	[[File:blue_interact.png\|350px\|center]]		[[File:blue_interact.png\|350px\|center\|thumb\|Figure 1 <br> Note: Figure created by author]]

	From ~~Figure 1~~ we can see that there are internal forces within the system since each object contributes a force to its surrounding objects. However, when applying the multi-particle momentum principle, we only look at the external forces. So in this case, to sum the forces in order to find the change in momentum we would add F1,surr + F2,surr + F3,surr + F4,surr to sum up to an Ftotal that we would multiply by the time elapsed in order to determine the change in momentum.		From the figure above, we can see that there are internal forces within the system since each object contributes a force to its surrounding objects. However, when applying the multi-particle momentum principle, we only look at the external forces. So in this case, to sum the forces in order to find the change in momentum we would add F1,surr + F2,surr + F3,surr + F4,surr to sum up to an Ftotal that we would multiply by the time elapsed in order to determine the change in momentum.

	The reason that the internal forces are not used within the momentum principle is because the internal forces would cancel due to the principle of reciprocity. Since each object exerts an equal and opposite force on each other, all of the internal forces would cancel, leaving only the external forces. This makes solving the momentum for the system, which would have been a complicated to solve for with a multitude of forces, much simpler.		The reason that the internal forces are not used within the momentum principle is because the internal forces would cancel due to the principle of reciprocity. Since each object exerts an equal and opposite force on each other, all of the internal forces would cancel, leaving only the external forces. This makes solving the momentum for the system, which would have been a complicated to solve for with a multitude of forces, much simpler.

Parkercoye at 17:10, 28 July 2019

2019-07-28T17:10:37Z

← Older revision		Revision as of 13:10, 28 July 2019
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	~~'''Written by Madeline Helmstadter; mhelmstadter3 (Fall 2016)''' '''Claimed by Simran Dhal (sdhal3) (Spring 2018)'''~~

	==The Main Idea==		==The Main Idea==

Parkercoye: /* Computational Models */

2019-07-28T16:26:07Z

Computational Models

← Older revision		Revision as of 12:26, 28 July 2019
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	[https://trinket.io/embed/glowscript/8271b15824 GlowScript]		[https://trinket.io/embed/glowscript/8271b15824 GlowScript]

			# GlowScript 1.1 VPython

			#people
			#note that the people has a trail
			human1=sphere(pos=vec(-40,0,0), radius=0.5, color=color.red, make_trail=true)
			human2=sphere(pos=vec(-20,0,0), radius=0.5, color=color.green, make_trail=true)
			human3=sphere(pos=vec(20,0,0), radius=0.5, color=color.blue, make_trail=true)
			human4=sphere(pos=vec(40,0,0), radius=0.5, color=color.white, make_trail=true)

			human1.m=0.1 #mass of the people in kg
			human2.m=0.1
			human3.m=0.1
			human4.m=0.1

			human1.p=human1.m*vec(0,0,0) #starting momentum mass times velocity
			human2.p=human2.m*vec(-2,0,0)
			human3.p=human3.m*vec(0,2,0)
			human4.p=human4.m*vec(-2,2,0)

			t=0
			dt=0.01 #size of the time step

			#putting the loop as "while True" means it runs forever
			#you could change this to while t< 10: or something
			while t<0.02:
			rate(100) #this says 100 calculations per second

			#given net force
			Fnet=vec(1,-6,-1)

			#update the momentum
			human1.p=human1.p+Fnet*dt
			human2.p=human2.p+Fnet*dt
			human3.p=human3.p+Fnet*dt
			human4.p=human4.p+Fnet*dt

			#update the position of the ball
			human1.pos= human1.pos +human1.p*dt/human1.m
			human2.pos= human2.pos +human2.p*dt/human2.m
			human3.pos= human3.pos +human3.p*dt/human3.m
			human4.pos= human4.pos +human4.p*dt/human4.m

			#this next line updates the spring
			#spring.axis=ball.pos-holder.pos

			print("Final Momentum of Human 1 = ",human1.p," kg*m/s")
			print("Final Momentum of Human 2 = ",human2.p," kg*m/s")
			print("Final Momentum of Human 3 = ",human3.p," kg*m/s")
			print("Final Momentum of Human 4 = ",human4.p," kg*m/s")


	or check out this collision simulation to see a multi-particle analysis of it:		or check out this collision simulation to see a multi-particle analysis of it:

Parkercoye: /* The Main Idea */

2019-07-28T16:23:15Z

The Main Idea

← Older revision		Revision as of 12:23, 28 July 2019
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	<math>\vec{F}_{net} = \vec{F}_{1} + \vec{F}_{2} + \vec{F}_{3} + ...</math>		<math>\vec{F}_{net} = \vec{F}_{1} + \vec{F}_{2} + \vec{F}_{3} + ...</math>

	[[File:~~ExternalForces~~.png\|~~300px~~\|~~thumb\|left\|Figure 1~~]]		[[File:blue_interact.png\|350px\|center]]

	From Figure 1 we can see that there are internal forces within the system since each object contributes a force to its surrounding objects. However, when applying the multi-particle momentum principle, we only look at the external forces. So in this case, to sum the forces in order to find the change in momentum we would add F1,surr + F2,surr + F3,surr + F4,surr to sum up to an Ftotal that we would multiply by the time elapsed in order to determine the change in momentum.		From Figure 1 we can see that there are internal forces within the system since each object contributes a force to its surrounding objects. However, when applying the multi-particle momentum principle, we only look at the external forces. So in this case, to sum the forces in order to find the change in momentum we would add F1,surr + F2,surr + F3,surr + F4,surr to sum up to an Ftotal that we would multiply by the time elapsed in order to determine the change in momentum.

Sdhal3: /* The Main Idea */

2018-04-18T20:28:34Z

The Main Idea

@@ Line 29: / Line 29: @@
 <math>\vec{F}_{1,4} = \vec{-F}_{1,1}</math>
 And this would be the case for every object in the system (objects 2,3 and 4).
-===A Mathematical Model===
+'''More on Proving the Cancellation of Internal Forces:'''
 Where '''p''' is momentum, '''F''' is the net force from the surroundings, and '''t''' is time, measured in seconds, The Momentum Principle is formulaically defined as
@@ Line 70: / Line 56: @@
 <math> \vec{F}_{12} = m_2\vec{a}_2 </math>.
 The total momentum does not change despite the particles' change in position, velocity, and acceleration. This does not mean that each individual particle will retain the same momentum, but that the total momentum of the system is conserved.
@@ Line 87: / Line 72: @@
 See more on that at: [http://www.physicsbook.gatech.edu/Momentum_with_respect_to_external_Forces]
 ===Computational Models===

Sdhal3: /* Middling */

2018-04-18T20:24:58Z

Middling

← Older revision		Revision as of 16:24, 18 April 2018
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	As explained through the mathematical and computational models above, Newton's Third Law allows us to assume that momentum is conserved within the system before and after a collision, meaning that the initial and final momentum of the system are the same. The initial momentum here is Mcart + Mbrick, therefore the final momentum is the same.		As explained through the mathematical and computational models above, Newton's Third Law allows us to assume that momentum is conserved within the system before and after a collision, meaning that the initial and final momentum of the system are the same. The initial momentum here is Mcart + Mbrick, therefore the final momentum is the same.

	===~~Middling~~===		===Middle===

	Given a car with a mass of 200kg and an initial velocity of 90m/s and a truck with a mass of 950kg and an initial velocity of 80m/s, find the TOTAL, FINAL momentum of the system after the car and truck collide. Include both the car and truck in the system.		Given a car with a mass of 200kg and an initial velocity of 90m/s and a truck with a mass of 950kg and an initial velocity of 80m/s, find the TOTAL, FINAL momentum of the system after the car and truck collide. Include both the car and truck in the system.
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	Initial momentum = Final momentum, therefore		Initial momentum = Final momentum, therefore
	total final momentum = <math> p_1 + p_2 = m_1v_1 + m_2v_2</math>,		total final momentum = <math> p_1 + p_2 = m_1v_1 + m_2v_2</math>,
	which is equal to 9.4e4N.		which is equal to 9.4e4N.

	===Difficult===		===Difficult===

Sdhal3: /* A Mathematical Model */

2018-04-18T20:22:57Z

A Mathematical Model

@@ Line 33: / Line 33: @@
 ===A Mathematical Model===
-Mathematically, momentum is the product of a body's (in our case, a particle's) mass and velocity. It is also defined as the product of mass, net force, and change in time.
+Mathematically, momentum is the product of a system's mass and the change in its velocity. It is also defined as the product of the net force on the system and the time in which the force was applied. This principle stays the same for a multi-particle system and for a multi-particle system we would define the momentum as this:
 Where '''p''' is momentum, '''F''' is the net force from the surroundings, and '''t''' is time, measured in seconds, The Momentum Principle is formulaically defined as