Completely Inelastic Collision

  • Thread starter MathewsMD
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I'm currently trying to make a proof to convince myself that when two object collide and stick afterwards, there is maximum energy loss. I've been thinking about it and trying to come up with a mathematical proof to solidify the idea in my head.
Please tell me if there's any errors in my explanation or if there's anything that should be added.

Case 1: object 1 is moving and object 2 is stationary (with no external forces, a frame of reference can always be used in which the motion is 0 m/s, and I realize this is a proof in itself, but I want to come up with something mathematically instead of intuitively)

Ki = (1/2)m1vi2 [1]

Taking the derivative and solving for 0 will give me an extreme value for the kinetic energy.

Kf = (1/2)(m1+m2)vf2 [2]

K'f=p=(m1+m2)vf
and if vf=0 m/s, then this system will have 0 J (which, using the right frame of reference, is possible in any situation where the velocity of the two "stuck" objects is constant"

pi = pf since there is no net external force
m1vi=(m1+m2)vf
vf=m1vi/(m1+m2) [3]

Plugging [3] into [2] and dividing by [1], to see the ratio between Kf and Ki

=[(1/2)(m1)[m12vi2/(m1+m2)2]/(1/2)m1vi2

=m12/(m1+m2)2

I'm just confused now since this ratio doesn't seem to tell me much about two kinetic energies, does it? What else should I do now?
 
Last edited:

Nugatory

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I'm just confused now since this ratio doesn't seem to tell me much about two kinetic energies, does it? What else should I do now?
Try working the problem in a frame in which the total momentum is zero. Then, before you go through the work of finding the extreme value, look carefully at the pre-collision and post-collision state.
 
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Firstly, where K is kinetic energy, m is mass, v is velocity, and p is momentum, it could be technically debated that, if:
Kf = (1/2)(m1+m2)vf2 [2]
then:
K'fp = (m1+m2)vf
What has K been differentiated with respect to?

Another more easily addressed problem appears to be;
pi = pf
This is true if momentum is conserved, as is the case with ideal elastic collisions. In an inelastic collision, KE from equation [1] in the OP is lost to heat and other processes involved in the coalescing of the two objects, regardless of whether the final velocity; has been fixed to be zero, is measured to be zero, or otherwise. So pi ≠ pf , and therefore equation [3] is incorrect.

Lastly if K is dependent on v, and the final velocity of the coalesced objects is 0m.s-1, it becomes clear that the ratio Kf / Ki provides no useful information.
 

Nugatory

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This is true if momentum is conserved, as is the case with ideal elastic collisions.
Momentum is always conserved, in both elastic and inelastic collisions, whether ideal or not.
 
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Use the center of mass reference frame. In this frame the objects are at rest after the collision (by momentum conservation) which means the energy is a minimum (since its zero, and cannot be negative).
 

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