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QPage 458: Q15: HC Verma: Concepts of Physics - Volume 2 The Special Theory of Relativity Page 458: Q15 Solution: At what speed the volume of an object shrinks to half its rest…

⚠️ Common Exam Traps & Conceptual Pitfalls

Students often incorrectly assume that since three dimensions are involved, the volume scales with the cube of the Lorentz factor ($V = V_0(1-v^2/c^2)^{3/2}$). However, only the dimension parallel to the motion contracts, so the volume scales linearly with the Lorentz factor ($V = V_0\sqrt{1-v^2/c^2}$), leading to the correct velocity $v = \frac{\sqrt{3}}{2}c$.

At what speed the volume of an object shrinks to half its rest value?

Step-by-Step Verified Solution

**Answer:**

v = \frac{\sqrt{3}}{2}c \approx 0.866c

. **Step-by-Step Explanation:** Volume

V = L_x L_y L_z

. Only the length along motion contracts:

V = V_0\sqrt{1 - v^2/c^2}

.

V = V_0/2 \Rightarrow \sqrt{1 - v^2/c^2} = 0.5 \Rightarrow v = \frac{\sqrt{3}}{2}c

.

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