GATE 2022 PH – Question 61
Two particles of rest mass $m_0$ approach at velocities $\pm0.5c$. One has energy $\alpha m_0c^2$ in the other rest frame. Find alpha to two decimals.
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Correct answer: 1.66 to 1.68
Explanation
Two particles of rest mass $m_0$ approach each other with velocities $+0.5c$ and $-0.5c$ in the laboratory frame. In the rest frame of one of them, the other has the relative velocity given by the **relativistic velocity addition** formula:
$$u=\frac{0.5c+0.5c}{1+(0.5)(0.5)}=\frac{c}{1.25}=0.8c .$$
**Energy in the other particle's rest frame:**
$$E=\gamma m_0c^2,\qquad\gamma=\frac1{\sqrt{1-0.8^2}}=\frac1{0.6}=\frac53 .$$
So $\alpha=\gamma=\mathbf{1.67}$.