Sbhardwaj70: Improved physics accuracy and clarity of the page. Added coefficient of restitution (e=0) for maximally inelastic collisions, clarified system choice and external impulse assumptions in the momentum principle, corrected misuse of “adiabatic,” and fixed the ΔU energy expression. Added multi-dimensional (vector) momentum conservation, included a center-of-mass frame explanation, clarified when relativistic KE is needed, and added notes on physical realism of “sticking.” Cleaned up language and imp

2025-12-01T03:17:22Z

Improved physics accuracy and clarity of the page. Added coefficient of restitution (e=0) for maximally inelastic collisions, clarified system choice and external impulse assumptions in the momentum principle, corrected misuse of “adiabatic,” and fixed the ΔU energy expression. Added multi-dimensional (vector) momentum conservation, included a center-of-mass frame explanation, clarified when relativistic KE is needed, and added notes on physical realism of “sticking.” Cleaned up language and imp

← Older revision		Revision as of 23:17, 30 November 2025
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	claimed by dwinegarden3 (spring 2017)		claimed by dwinegarden3 (spring 2017)
	Edited by Alexander Hom (spring 2022)		Edited by Alexander Hom (spring 2022)
			Edited by Suyash Bhardwaj (Fall 2025)

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

	A collision is a brief interaction between large forces. Collisions can include such phenomena as a ball bouncing off a wall, two people running into each other on their way to class, or atoms bouncing around within a star. There are two primary types of collisions: elastic and inelastic. Elastic collisions occur when the total kinetic energy of the system is conserved while inelastic collisions occur when the objects' kinetic energies are not conserved. Maximally inelastic collisions are a subset of inelastic collisions in which the objects in the system collide and stick together to form one object. The new object has a new velocity and the mass of the object equals the total mass of all the objects that have become one. An example of a maximally inelastic collision is a car crash or meteor hitting a planet.		A collision is a brief interaction between large forces. Collisions can include such phenomena as a ball bouncing off a wall, two people running into each other on their way to class, or atoms bouncing around within a star. There are two primary types of collisions: elastic and inelastic. Elastic collisions occur when the total kinetic energy of the system is conserved while inelastic collisions occur when the objects' kinetic energies are not conserved. Maximally inelastic collisions are a subset of inelastic collisions in which the objects in the system collide and stick together to form one object. The new object has a new velocity and the mass of the object equals the total mass of all the objects that have become one. In terms of the coefficient of restitution, a maximally inelastic collision corresponds to <math>e = 0</math>, meaning the objects have no relative speed of separation after impact and move together with the same final velocity. An example of a maximally inelastic collision is a car crash or meteor hitting a planet.

			Perfectly “sticking” collisions are an idealization; in real materials, whether objects actually stick depends on deformation, bonding, and surface properties, but the maximally inelastic model is a good approximation whenever the objects move together after impact.

	[[File:Maximally inelastic collision.png\|600px]]		[[File:Maximally inelastic collision.png\|600px]]
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	===A Mathematical Model===		===A Mathematical Model===

	Maximally inelastic collisions are first and foremost momentum problems, so modeling such a collision usually begins with the fundamental momentum principle, ~~modeled~~ in ~~the equation:~~		Maximally inelastic collisions are first and foremost momentum problems, so modeling such a collision usually begins with the fundamental momentum principle, written in impulse form as
	::<math>{\~~frac{d~~\vec{p}}{~~dt}}_~~{system} = \vec{F}_{net}~~{Δt~~}</math>		::<math>\Delta \vec{p}_{\text{system}} = \vec{F}_{\text{net,ext}}\,\Delta t</math>
	where '''p''' is the momentum of the system, '''F''' is the net force ~~from~~ the ~~surroundings~~, '''Δt''' is the ~~change in~~ time ~~for~~ the process.		where '''\vec{p}''' is the momentum of the system, '''\vec{F}_{\text{net,ext}}''' is the net external force on the system, and '''Δt''' is the time interval of the process.

	~~Because the momentum principle states that~~ the ~~change in momentum, also called impulse, within a~~ system ~~in which~~ the objects ~~stick together is not changed~~, the time ~~of interaction~~ is negligible, ~~so <math>{Δt} ≈ {0}</math>. Substituting '''Δt''' with zero and rewriting the equation yields:~~		For a collision, we choose the system to be just the colliding objects, and the interaction typically occurs over a very short time. External forces (like gravity or friction) act for such a brief interval that the external impulse is negligible. In that case,
	::<math>{~~ΔP_~~{system}} ~~= {~~0}</math> ~~where~~ we can break the change in momentum of the system to its initial and final components to get		::<math>\Delta \vec{p}_{\text{system}} \approx 0</math>
	::<math>{P_{final}} - {P_{initial}} ~~= {~~0}</math> which becomes		so we can break the change in momentum of the system into its initial and final components to get
	::<math>{P_{final}} = {P_{initial}}</math>.		::<math>\vec{p}_{\text{final}} - \vec{p}_{\text{initial}} \approx 0</math> which becomes
			::<math>\vec{p}_{\text{final}} \approx \vec{p}_{\text{initial}}</math>.

	If we substitute the mass and velocity of multiple objects into the above equation, we see that:		If we substitute the mass and velocity of multiple objects into the above equation, we see that:
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	where '''M''' is sum of the masses of all collided objects and '''V''' is the final velocity of the amalgamated object		where '''M''' is sum of the masses of all collided objects and '''V''' is the final velocity of the amalgamated object
	::<math> V=\frac{\sum {m v}}{M}</math>		::<math> V=\frac{\sum {m v}}{M}</math>

			In maximally inelastic collisions, kinetic energy is not conserved between the initial and final states of the objects.


			In more than one spatial dimension, momentum is a vector, so the conservation equation must be applied separately to each component. For example, in two dimensions we have
			::<math>m_1 v_{1x} + m_2 v_{2x} + \dots + m_n v_{nx} = M V_x</math>
			::<math>m_1 v_{1y} + m_2 v_{2y} + \dots + m_n v_{ny} = M V_y</math>
			and similarly for three-dimensional collisions.

	In maximally inelastic collisions, kinetic energy is not conserved between the initial and final states of the objects. However, total energy is conserved. This is possible because some of the energy is lost to heat and changes in shape, accounted for in the change of internal energy called '''ΔU'''. Thus, we can use the energy principle:		In maximally inelastic collisions, kinetic energy is not conserved between the initial and final states of the objects. However, total energy is conserved. This is possible because some of the energy is lost to heat and changes in shape, accounted for in the change of internal energy called '''ΔU'''. Thus, we can use the energy principle:
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	:: <math>{ΔE_k}+{ΔE_p}+{ΔU}= {Q} + {W} </math> where '''<math>ΔE_k</math>''' is the change in kinetic energy of the system,'''<math>ΔE_p</math>''' is the change in the potential energy of the system, and '''<math>ΔU</math>''' is the change in internal energy.		:: <math>{ΔE_k}+{ΔE_p}+{ΔU}= {Q} + {W} </math> where '''<math>ΔE_k</math>''' is the change in kinetic energy of the system,'''<math>ΔE_p</math>''' is the change in the potential energy of the system, and '''<math>ΔU</math>''' is the change in internal energy.

	Since we assume that the objects all end up as one combined object with no significant loss of mass, we can also assume that the work done ~~is negligible and~~ the ~~process~~ is ~~adiabatic, or a process that produces~~ negligible ~~heat~~, and ~~the~~ change in ~~overall~~ potential energy of the ~~system~~ is ~~negligible~~. With these assumptions in place, the equation above simplifies to		Since we assume that the objects all end up as one combined object with no significant loss of mass, we can also assume that the work done on the system by the surroundings during the short collision is negligible, and that any change in potential energy over this brief interval is negligible. The collision itself can generate significant internal (thermal and deformation) energy within the objects, but because the interaction is so rapid we typically neglect any heat exchanged with the surroundings. With these assumptions in place, the equation above simplifies to
	:: <math> {ΔE_k} + {ΔU} = 0 </math> which we can break into the initial and final components to get		:: <math> {ΔE_k} + {ΔU} = 0 </math> which we can break into the initial and final components to get
	:: <math> {ΔU} = -({K_{final}} - ({K_{initial,1}} + {K_{initial,2}} +... {K_{initial,n}})) </math>		:: <math> {ΔU} = -({K_{final}} - ({K_{initial,1}} + {K_{initial,2}} +... {K_{initial,n}})) </math>
	Kinetic Energy for objects that have a velocity significantly smaller than the speed of light is defined as <math> \frac{1}{2}{mv^2} </math>, so putting in the values for mass and velocity we get that:		Kinetic Energy for objects that have a velocity significantly smaller than the speed of light is defined as <math> \frac{1}{2}{mv^2} </math>, so putting in the values for mass and velocity we get that:
	:: <math> ~~{ΔU}~~ = \frac{1}{2}{(v_1 ~~+ v_2 +... v_n)~~^2~~}{(M)} - (~~\frac{1}{2}~~{m_1 (v_1)~~^2} + \frac{1}{2}~~{m_2 (v_2)~~^2~~}+...~~\frac{1}{2}~~{m_n (v_n)~~^2)} </math>		:: <math>\Delta U = \left(\frac{1}{2} m_1 v_1^2 + \frac{1}{2} m_2 v_2^2 + \dots + \frac{1}{2} m_n v_n^2\right) - \frac{1}{2} M V^2</math>
	However, when the kinetic energy of an object is close to the speed of light, relativistic equations are required. In this case, the kinetic energy is equal instead to		However, when the kinetic energy of an object is close to the speed of light, relativistic equations are required. In this case, the kinetic energy is equal instead to
	:: <math> {K} = \frac{mc^2}{\sqrt{1-(\frac{v}{c})^2}} - mc^2 </math>		:: <math> {K} = \frac{mc^2}{\sqrt{1-(\frac{v}{c})^2}} - mc^2 </math>
	where '''m''' is the mass of the object, '''V''' is the velocity of the object, '''c''' is the speed of light (~3e8 m/s). The first element on the left of the above equation accounts for all of the energy of an object and the second element removes the rest energy of the system from the total energy, leaving the relativistic kinetic energy. When this is put into the overall equation with the known values, we get		where '''m''' is the mass of the object, '''V''' is the velocity of the object, '''c''' is the speed of light (~3e8 m/s). The first element on the left of the above equation accounts for all of the energy of an object and the second element removes the rest energy of the system from the total energy, leaving the relativistic kinetic energy. Relativistic expressions are only necessary when the speed <math>v</math> is a significant fraction of the speed of light (typically <math>v \gtrsim 0.1c</math>); for most introductory collision problems, the classical expression <math>\tfrac{1}{2}mv^2</math> is entirely sufficient.
			When this is put into the overall equation with the known values, we get
	:: <math> {ΔU} = (\frac{Mc^2}{\sqrt{1-(\frac{V}{c})^2}} - Mc^2) - (\sum_{n=1}^N \frac{m_N c^2}{\sqrt{1-(\frac{v_N}{c})^2}} - m_N c^2)</math>		:: <math> {ΔU} = (\frac{Mc^2}{\sqrt{1-(\frac{V}{c})^2}} - Mc^2) - (\sum_{n=1}^N \frac{m_N c^2}{\sqrt{1-(\frac{v_N}{c})^2}} - m_N c^2)</math>

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2022-04-18T02:44:38Z

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	'''Solution'''		'''Solution'''
	~~efrefe~~
	Using the conservation of momentum principle, we can find the final velocity of the object: <math>m_1 v_1 = \left( m_1 + m_2 \right) v \,</math> which is 1.961 m/s.		Using the conservation of momentum principle, we can find the final velocity of the object: <math>m_1 v_1 = \left( m_1 + m_2 \right) v \,</math> which is 1.961 m/s.
	Now we can use the conservation of Energy principle to solve for the thermal energy: <math> {ΔE_k} + {ΔU} = 0. </math> Then we break the change of kinetic energy into it's initial and final conditions to get :		Now we can use the conservation of Energy principle to solve for the thermal energy: <math> {ΔE_k} + {ΔU} = 0. </math> Then we break the change of kinetic energy into it's initial and final conditions to get :

Ahom6: /* Middling */

2022-04-18T02:44:02Z

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	'''Solution'''		'''Solution'''
			efrefe
	Using the conservation of momentum principle, we can find the final velocity of the object: <math>m_1 v_1 = \left( m_1 + m_2 \right) v \,</math> which is 1.961 m/s.		Using the conservation of momentum principle, we can find the final velocity of the object: <math>m_1 v_1 = \left( m_1 + m_2 \right) v \,</math> which is 1.961 m/s.
	Now we can use the conservation of Energy principle to solve for the thermal energy: <math> {ΔE_k} + {ΔU} = 0. </math> Then we break the change of kinetic energy into it's initial and final conditions to get :		Now we can use the conservation of Energy principle to solve for the thermal energy: <math> {ΔE_k} + {ΔU} = 0. </math> Then we break the change of kinetic energy into it's initial and final conditions to get :

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2022-04-18T02:38:20Z

Simple

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	'''Hint'''		'''Hint'''


	[[File:hint1resized.jpg\|hint1updated]]		[[File:hint1resized.jpg\|hint1updated]]

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2022-04-18T02:38:06Z

Simple

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	'''Hint'''		'''Hint'''
	[[File:~~hint1updated~~.jpg~~\|500px\|thumb\|left~~\|hint1updated]]		[[File:hint1resized.jpg\|hint1updated]]

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2022-04-18T02:35:37Z

Simple

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	'''Hint'''		'''Hint'''
	[[File:hint1updated.jpg\|500px\|hint1updated]]		[[File:hint1updated.jpg\|500px\|thumb\|left\|hint1updated]]

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2022-04-18T02:34:59Z

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	'''Hint'''		'''Hint'''
	[[File:hint1updated.jpg\|hint1updated~~\|1000px~~]]		[[File:hint1updated.jpg\|500px\|hint1updated]]

Ahom6: /* Simple */

2022-04-18T02:34:29Z

Simple

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	'''Hint'''		'''Hint'''
	[[File:hint1updated.jpg\|hint1updated\|~~12px~~]]		[[File:hint1updated.jpg\|hint1updated\|1000px]]

Ahom6: /* Simple */

2022-04-18T02:34:17Z

Simple

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	'''Hint'''		'''Hint'''
	[[File:hint1updated.jpg\|hint1updated\|~~756px~~]]		[[File:hint1updated.jpg\|hint1updated\|12px]]

Ahom6: /* Simple */

2022-04-18T02:34:02Z

Simple

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	'''Hint'''		'''Hint'''
	[[File:hint1updated.jpg\|hint1updated\|~~700px~~]]		[[File:hint1updated.jpg\|hint1updated\|756px]]

Maximally Inelastic Collision - Revision history

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