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GATE 2024 PH – Question 25

Classical Mechanics · canonical transformations: Poisson bracket. Special theory of relativity: Lorentz transformations, relativistic kinematics, mass-energy equivalence. · 1 mark · Multiple choice

Two spacecraft fly apart along x with individual speeds $0.5c$ in an inertial frame. The speed of T with respect to S is

  1. $4c/5$
  2. $4c/3$
  3. $c$
  4. $2c/3$

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Correct answer: (A) $4c/5$

Explanation

Use the relativistic velocity-addition formula. In the inertial frame the two spacecraft move apart with speeds $0.5c$ each, so S moves at $-0.5c$ and T at $+0.5c$.

The speed of T as seen from S is
$$u=\frac{v_1+v_2}{1+v_1v_2/c^2}=\frac{0.5c+0.5c}{1+(0.5)(0.5)}=\frac{c}{1.25}=\frac{4c}{5}.$$

The result is $0.8c$, which is less than $c$ (the Galilean answer $c$ would be wrong). Answer **$4c/5$** (A).