# Is the coefficient of restitution frame independent and energy conservation?

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.

• 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 $link_italic_bold'click"$ – Rajesh Sardar Nov 15 '14 at 15:26
• 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. – Rajesh Sardar Nov 15 '14 at 15:53