Head-on Collision of Unequal Masses - Revision history

Sydney: /* Difficult */

2019-07-08T20:56:05Z

Difficult

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	'''Question'''		'''Question'''

	In an orbiting spacecraft a ping-pong ball of mass m (object 1) traveling in the +x direction with initial momentum <math>~~p_1i~~</math> hits a stationary bowling ball of mass M (object 2) head on. What are the '''a) momentum''', '''b) speed''', and '''c) kinetic energy''' of each object after the collision? Assume little change in the speed of the ping-pong ball, and assume that the collision is elastic.		In an orbiting spacecraft a ping-pong ball of mass m (object 1) traveling in the +x direction with initial momentum <math>p_{1i}</math> hits a stationary bowling ball of mass M (object 2) head on. What are the '''a) momentum''', '''b) speed''', and '''c) kinetic energy''' of each object after the collision? Assume little change in the speed of the ping-pong ball, and assume that the collision is elastic.

	'''Answer'''		'''Answer'''

Sydney: /* Middling */

2019-07-08T20:55:15Z

Middling

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	===Middling===		===Middling===

	~~'''Question'''~~

	Here is another example that analyzes the recoil of the earth from a collision with a ball. This example applies the concepts discussed above; the low-mass object undergoes minimal change in kinetic energy and the stationary large mass has zero momentum, showing a detailed calculations.
	[[File:Collisionexample.jpg]]		[[File:Collisionexample.jpg]]

	Let's analyze the recoill of Earth when a ball bounces off a wall embedded in the Earth. A baseball with a mass of 0.155 kg is hurled with a speed of 44.4 m/s at a wall, and then bounces back with little loss of kinetic energy.

	~~a. What is the recoil speed of Earth?(Earth mass = 6 x 10^24 kg)~~

	~~'''Answer'''~~



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

Sydney: /* A Mathematical Model */

2019-07-08T20:54:09Z

A Mathematical Model

Sydney: /* Connectedness */

2019-07-07T01:46:43Z

Connectedness

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	Collisions in general have a multitude of applications in everyday life and in other fields. We encounter collisions almost all the time, especially collisions of unequal mass. It is more common outside of a lab setting to encounter collisions between objects of different masses. For example; a drumstick hitting a drum, a baseball hitting a bat, or a foot stepping on the ground.		Collisions in general have a multitude of applications in everyday life and in other fields. We encounter collisions almost all the time, especially collisions of unequal mass. It is more common outside of a lab setting to encounter collisions between objects of different masses. For example; a drumstick hitting a drum, a baseball hitting a bat, or a foot stepping on the ground.

	Collisions of unequal masses are also vital in other branches of science and engineering. For example, in astronomy, these collisions are useful in studying		Collisions of unequal masses are also vital in other branches of science and engineering. For example, in astronomy, these collisions are useful in studying asteroid interactions. If an asteroid is massive enough or has a great enough velocity, it can cause a noticeable change in momentum or energy in a more massive object. For example, one theory of the Moon's creation is that a very massive asteroid collided with a relatively young Earth and split into two large chunks, which were smoothed and condensed by gravity to become the Earth and the Moon.

			In mechanical engineering, it is important to consider collisions of unequal mass to design for any intended or unintended impacts. Preparing for collisions with objects of different masses helps to create a safer and more durable product. This is also a consideration in marketing and for safety engineers.

	==History==		==History==

Sydney: /* Connectedness */

2019-07-06T21:52:52Z

Connectedness

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	==Connectedness==		==Connectedness==
			Collisions in general have a multitude of applications in everyday life and in other fields. We encounter collisions almost all the time, especially collisions of unequal mass. It is more common outside of a lab setting to encounter collisions between objects of different masses. For example; a drumstick hitting a drum, a baseball hitting a bat, or a foot stepping on the ground.

			Collisions of unequal masses are also vital in other branches of science and engineering. For example, in astronomy, these collisions are useful in studying

	==History==		==History==

Sydney: /* Difficult */

2019-07-06T21:20:49Z

Difficult

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	3. <math>K_{2,f} = {(2p_{1,i})^2 \over 2M}</math> and <math>K_{1,f} = {(p_{1,i})^2 \over 2m}</math>		3. <math>K_{2,f} = {(2p_{1,i})^2 \over 2M}</math> and <math>K_{1,f} = {(p_{1,i})^2 \over 2m}</math>

	Because the mass of the bowling ball (M) is much larger than the mass of the ping-pong ball, the kinetic energy of the bowling ball is much smaller than the kinetic energy of the ping-pong ball.'		Because the mass of the bowling ball (M) is much larger than the mass of the ping-pong ball, the kinetic energy of the bowling ball is much smaller than the kinetic energy of the ping-pong ball.

	==Connectedness==		==Connectedness==

Sydney: /* Simple */

2019-07-06T21:19:39Z

Simple

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	'''Question'''		'''Question'''

	You just got a 100 on your last physics test and are so psyched you throw your calculator down on your bed. It hits your bed at a velocity of 6 meters per second and bounces off at a velocity of 3 meters per second.		You just got a 100 on your last physics test and are so psyched you throw your .15 kilogram calculator down on your bed. It hits your bed at a velocity of 6 meters per second and bounces off at a velocity of 1.5 meters per second. How much momentum and kinetic energy did your calculator lose from the collision?

	'''Answer'''		'''Answer'''

			Momentum <math>= m_{calc,i}v_{calc,i}</math> before the collision and momentum <math>= m_{calc,f}v_{calc,f}</math> after the collision.

			Therefore, change in momentum <math>= \|m_{calc,f}v_{calc,f} - m_{calc,i}v_{calc,i}\|</math>.

			Substituting the given information in, we get <math>= \|(.15)(1.5) - (.15)(6)\| = 0.675 kgm/s</math>.


			We can use a similar method to solve for the change in kinetic energy using the formula <math>K = \frac{1}{2}mv^2</math>, so that change in kinetic energy is <math>= \|\frac{1}{2}m_{calc}v_{calc,f}^2 - \frac{1}{2}m_{calc}v_{calc,i}^2\|</math>.

			Substituting the given information in again, we get <math>= \|\frac{1}{2}(.15)(1.5^2) - \frac{1}{2}(.15)(6^2)\| = 2.53 J</math>

	===Middling===		===Middling===

Sydney: /* Simple */

2019-07-06T20:49:56Z

Simple

← Older revision		Revision as of 16:49, 6 July 2019
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	'''Question'''		'''Question'''

			You just got a 100 on your last physics test and are so psyched you throw your calculator down on your bed. It hits your bed at a velocity of 6 meters per second and bounces off at a velocity of 3 meters per second.

	'''Answer'''		'''Answer'''


	===Middling===		===Middling===

Sydney: /* Examples */

2019-07-06T20:41:46Z

Examples

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	===Simple===		===Simple===

			'''Question'''


			'''Answer'''


			===Middling===

	'''Question'''		'''Question'''
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	a. What is the recoil speed of Earth?(Earth mass = 6 x 10^24 kg)		a. What is the recoil speed of Earth?(Earth mass = 6 x 10^24 kg)

			'''Answer'''

	~~'''Answer'''~~


	===~~Middling~~===		===Difficult===

	'''Question'''		'''Question'''
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	3. <math>K_{2,f} = {(2p_{1,i})^2 \over 2M}</math> and <math>K_{1,f} = {(p_{1,i})^2 \over 2m}</math>		3. <math>K_{2,f} = {(2p_{1,i})^2 \over 2M}</math> and <math>K_{1,f} = {(p_{1,i})^2 \over 2m}</math>

	Because the mass of the bowling ball (M) is much larger than the mass of the ping-pong ball, the kinetic energy of the bowling ball is much smaller than the kinetic energy of the ping-pong ball.		Because the mass of the bowling ball (M) is much larger than the mass of the ping-pong ball, the kinetic energy of the bowling ball is much smaller than the kinetic energy of the ping-pong ball.'

	~~===Difficult===~~

	~~'''Question'''~~


	~~'''Answer''~~'

	==Connectedness==		==Connectedness==

Sydney: /* A Computational Model */

2019-07-06T17:35:55Z

A Computational Model

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	===A Computational Model===		===A Computational Model===
			[https://trinket.io/glowscript/0fc8b587eb Glowscript model of a head-on collision of unequal masses]

	==Examples==		==Examples==

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	Most commonly, head-on ~~collision~~ of unequal masses are studied as elastic collisions, in which case, kinetic energy is conserved. What this means is that the total final kinetic energy of the system is equal to the total initial kinetic energy of the system. In equations, it looks like this: <math>{1 \over 2}~~m_1v_1i~~^2 + {1 \over 2}~~m_2v_2i~~^2 = {1 \over 2}~~m_1v_1f~~^2+ {1 \over 2}~~m_2v_2f~~^2</math>.		Most commonly, head-on collisions of unequal masses are studied as elastic collisions, in which case, kinetic energy is conserved. What this means is that the total final kinetic energy of the system is equal to the total initial kinetic energy of the system. In equations, it looks like this: <math>{1 \over 2}m_1v_{1i}^2 + {1 \over 2}m_2v_{2i}^2 = {1 \over 2}m_1v_{1f}^2+ {1 \over 2}m_2v_{2f}^2</math>.

	[[File:Trkc3.gif ]]		[[File:Trkc3.gif ]]
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	<math>m_1v_{1,i}^2 + m_2v_{2,i}^2 = m_1v_{1,f}^2 + m_2v_{2,f}^2</math>		<math>m_1v_{1,i}^2 + m_2v_{2,i}^2 = m_1v_{1,f}^2 + m_2v_{2,f}^2</math>

	As referenced to before, the ping-pong and bowling ball example clearly demonstrates what occurs in a head on collision of unequal masses. Say the system consisting of the ping-pong and bowling balls is in outer space. The ping-pong ball of mass m is traveling in the +x direction with initial momentum p_{1,i}, and hits a stationary bowling ball of mass M head on.		As referenced to before, the ping-pong and bowling ball example clearly demonstrates what occurs in a head on collision of unequal masses. Say the system consisting of the ping-pong and bowling balls is in outer space. The ping-pong ball of mass m is traveling in the +x direction with initial momentum <math>p_{1,i}</math>, and hits a stationary bowling ball of mass M head on.

	[[File:Head-on collision ping pong bowling ball.png\|850 px]]		[[File:Head-on collision ping pong bowling ball.png\|850 px]]
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	<math>m_1v_{1,i}^2 + m_2v_{2,i}^2 = m_1v_{1,f}^2 + m_2v_{2,f}^2</math> where p=mv.		<math>m_1v_{1,i}^2 + m_2v_{2,i}^2 = m_1v_{1,f}^2 + m_2v_{2,f}^2</math> where p=mv.

	Assume the speed of the ping-pong ball doesn't change significantly, so if ~~p_1f~~ ≈ -~~p_1i~~, then:		Assume the speed of the ping-pong ball doesn't change significantly, so if <math>p_{1f} ≈ -p_{1i}</math>, then:
	<math>-p_{1,i} + p_{2,f} = p_{1,i}</math>		<math>-p_{1,i} + p_{2,f} = p_{1,i}</math>
	and		and
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	In summary, when the ping-pong ball hits the stationary bowling ball, head-on and elastically:		In summary, when the ping-pong ball hits the stationary bowling ball, head-on and elastically:
	#The ping-pong ball bounces back with nearly the same speed, but has a large change in momentum -~~2p_1i~~.		#The ping-pong ball bounces back with nearly the same speed, but has a large change in momentum <math>-2p_{1i}</math>.
	#The bowling ball also has a large change of momentum +~~2p_1i~~, even though its final speed is very small due to a large mass.		#The bowling ball also has a large change of momentum <math>+2p_{1i}</math>, even though its final speed is very small due to a large mass.
	#The bowling ball's final kinetic energy is very small due to a low speed.		#The bowling ball's final kinetic energy is very small due to a low speed.