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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)

K

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

K

K'

and if v

p

m

v

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

=[(1/2)(m

=m

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?

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)

K

_{i}= (1/2)m_{1}v_{i}^{2}[1]Taking the derivative and solving for 0 will give me an extreme value for the kinetic energy.

K

_{f}= (1/2)(m_{1}+m_{2})v_{f}^{2}[2]K'

_{f}=p=(m_{1}+m_{2})v_{f}and if v

_{f}=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"p

_{i}= p_{f}since there is no net external forcem

_{1}v_{i}=(m_{1}+m_{2})v_{f}v

_{f}=m_{1}v_{i}/(m_{1}+m_{2}) [3]Plugging [3] into [2] and dividing by [1], to see the ratio between K

_{f}and K_{i}=[(1/2)(m

_{1})[m_{1}^{2}v_{i}^{2}/(m_{1}+m_{2})^{2}]/(1/2)m_{1}v_{i}^{2}=m

_{1}^{2}/(m_{1}+m_{2})^{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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