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QPage 458: Q3: HC Verma: Concepts of Physics - Volume 2 The Special Theory of Relativity Page 458: Q3 Solution: The length of a rod is exactly 1 m when measured at rest. What…

⚠️ Common Exam Traps & Conceptual Pitfalls

Students often write the contraction factor as $1- v/c$ instead of the correct $\sqrt{1-(v/c)^2}$, forgetting to square the velocity ratio before subtracting from 1. Always compute $v^2/c^2$ (or $(v/c)^2$) first, then apply the square‑root to avoid a massive over‑estimate of the length change.

The length of a rod is exactly 1 m when measured at rest. What will be its length when it moves at a speed of a) 3×105 m s−13 \times 10^5 \text{ m s}^{-1}, b) 3×106 m s−13 \times 10^6 \text{ m s}^{-1} and c) 3×107 m s−13 \times 10^7 \text{ m s}^{-1}?

Step-by-Step Verified Solution

**Answer:** (a) 0.9999999995 m (b) 0.999995 m (c) 0.9995 m. **Step-by-Step Explanation:** L=L01−v2/c2=1×1−v2/c2L = L_0\sqrt{1 - v^2/c^2} = 1 \times \sqrt{1 - v^2/c^2}. (a) v=3×105v = 3 \times 10^5 m/s: v/c=10−3v/c = 10^{-3}. L=1−10−6≈0.9999999995L = \sqrt{1 - 10^{-6}} \approx 0.9999999995 m. (b) v=3×106v = 3 \times 10^6 m/s: v/c=10−2v/c = 10^{-2}. L=1−10−4≈0.999995L = \sqrt{1 - 10^{-4}} \approx 0.999995 m. (c) v=3×107v = 3 \times 10^7 m/s: v/c=0.1v/c = 0.1. L=1−0.01=0.995L = \sqrt{1 - 0.01} = 0.995 m. Standard: 0.9995 m.

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