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  Taking the derivative and solving for 0 will give me an extreme value for the kinetic energy. Kf = (1/2)(m1+m2)vf2  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)  Plugging  into  and dividing by , 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?