Question

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⁻¹, (b) 3 × 10⁶ m s⁻¹ and (c) 3 × 10⁷ m s⁻¹?

Answer 1

Given:-

Proper length of the rod, L = 1 m
If v is the velocity of the rod, then the moving length of the rod is given by

\[L' = L\sqrt{1 - \frac{v^2}{c^2}}\]

(a) Here,
v = 3 × 10⁵ m/s

\[L' = 1 \times \sqrt{1 - \frac{9 \times {10}^{10}}{9 \times {10}^{16}}}\]

\[ = \sqrt{1 - {10}^{- 6}} = 0 . 9999995 m\]

(b) Here,
v = 3 × 10⁶ m/s

\[L' = 1 \times \sqrt{1 - \frac{9 \times {10}^{12}}{9 \times {10}^{16}}}\]

\[ = \sqrt{1 - {10}^{- 4}}\]

\[ = 0 . 99995 m\]

(c) Here,
v = 3 × 10⁷ m/s

\[L' = 1 \times \sqrt{1 - \frac{9 \times {10}^{14}}{9 \times {10}^{16}}}\]

\[ = 1\sqrt{1 - {10}^{- 2}}\]

\[ = 0 . 995 m\]