# Completely Inelastic Collision

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?

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## Answers and Replies

Nugatory
Mentor
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.

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
Mentor
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.

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).