A New Look at Energy — Question 5

Question 5

A symmetric inelastic collision stores kinetic energy as system mass

Two particles of equal invariant mass m move at $\pm0.60c$ in one frame and stick together. No radiation or other energy escapes. Use $E^2=p^2c^2+m^2c^4$, $E=\gamma mc^2$, $K=E-mc^2$, and $\mathbf v=c^2\mathbf p/E$.

1. Find the final composite velocity and invariant mass.

2. Find the increase in rest energy relative to two separated stationary particles.

Question 5 - Solution

A symmetric inelastic collision stores kinetic energy as system mass

(a) Method and calculation

$$P_{\mathrm{total}}=\gamma m(.60c)+\gamma m(-.60c)=0\quad\Rightarrow\quad\boxed{V_f=0}.$$$$Mc^2=2\gamma mc^2\quad\Rightarrow\quad\boxed{M=2.50m}.$$

(b) Interpretation and further calculation

$$\boxed{(M-2m)c^2=0.500mc^2}.$$$$K_{\mathrm{initial}}=2(\gamma-1)mc^2=0.500mc^2.$$

Physical check

The final rest mass includes retained internal energy; adding only the initial particle rest masses would violate energy conservation.