A New Look at Energy — Question 4

Question 4

Symmetric radiation removes invariant mass without recoil

A body initially at rest emits two equal, oppositely directed light pulses with total energy $Q=9.00\times10^{12}\,\mathrm J$ in that rest frame. For each pulse use $p=E/c$. No other energy enters or leaves. Take $c=3.00\times10^8\,\mathrm{m/s}$.

1. Use momentum conservation to identify the final body motion.

2. Find the body mass decrease and explain why energy loss does not require a change in its speed.

Question 4 - Solution

Symmetric radiation removes invariant mass without recoil

(a) Method and calculation

$$\mathbf p_{\mathrm{radiation}}=\frac{Q}{2c}\hat{\mathbf x}-\frac{Q}{2c}\hat{\mathbf x}=0.$$$$\mathbf p_{\mathrm{body,f}}=0\quad\Rightarrow\quad\boxed{v_f=0}.$$

(b) Interpretation and further calculation

$$M_ic^2=M_fc^2+Q\quad\Rightarrow\quad\boxed{M_i-M_f=Q/c^2}.$$$$\boxed{\Delta M=1.00\times10^{-4}\,\mathrm{kg}=0.100\,\mathrm g}.$$

Physical check

The energy comes from the body internal energy. Its invariant mass decreases even while its translational kinetic energy stays zero.