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In this question I am ignoring relativistic effects. The following statements I think are true:

  1. change in Kinetic energy is not invariant under change of frame
  2. Force is invariant under change in frame
  3. Change in velocity and relative velocity between two particles is invariant under change of frame
  4. The coefficient of restitution is given by $e=|\frac{v_1-v_2}{u_1-u_2}|=\sqrt{1-\frac{\Delta E}{T'}}$ where $\Delta E$ is the change in kinetic energy in the frame we are in and $T'$ is the initial kinetic energy in the centre of mass frame.

So here is my problem. If 2. is correct work done is invariant and therefore from the work energy theorem 1. must be wrong (since w.d=change in kinetic energy). If both 1 and 3 are right then 4. must be wrong as the first equation for e would stay the same under change of frame whilst the second would change. Please can you explain which of these expressions is wrong, thanks.

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The first statement is wrong. The total energy of a body changes with frame but the change in kinetic energy does not. It is constant. This statement can also be viewed as the fact that conservation of energy is valid on all inertial frames.

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  • $\begingroup$ I think change of kinetic energy is invariant under change in frame. Force remain invariant under change in frame. Again displacement means the difference between final and initial position of the body in a frame. That's why displacement also remain invariant under change in frame. Hence the work done also remain invariant under change in frame, i.e. change in KE also remains invariant.For details see $ link_italic_bold'click"$ $\endgroup$ – Rajesh Sardar Nov 15 '14 at 15:26
  • $\begingroup$ please see the link. click here. $\endgroup$ – Rajesh Sardar Nov 15 '14 at 15:34
  • $\begingroup$ I see the link that you have gave. I think there relativistic effects has consider. I that case obviously displacement is not invariant. But here relativistic effects have been ignored. And I have taking the displacement there as $(x_2-x_1)$, where $x_1$ is the initial position of the body and $x_2$ is the final position. $\endgroup$ – Rajesh Sardar Nov 15 '14 at 15:53
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Your first question is answered quite easily - the force may be the same in both frames, but the distance traveled is not. In the frame where the change in KE is greater, the object will also cover a greater distance. So the change in KE is not frame invariant, and neither is the work done.

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