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Question
A person standing on a platform finds that a train moving with velocity 0.6c takes one second to pass by him. Find (a) the length of the train as seen by the person and (b) the rest length of the train.
Solution
Given:-
Velocity of train, v = 0.6c
Time taken, t = 1 s
(a) Length observed by the observer, L= vt
⇒ L = 0.6 × 3 × 108
= 1.8 × 108 m
(b) Let the rest length of train be L0. Then,
\[L = L_0 \sqrt{1 - v^2 / c^2}\]
\[\Rightarrow 1 . 8 \times {10}^8 = L_0 \sqrt{1 - \frac{\left( 0 . 6 \right)^2 c^2}{c^2}}\]
\[ \Rightarrow 1 . 8 \times {10}^8 = L_0 \times 0 . 8\]
\[ \Rightarrow L_0 = \frac{1 . 8 \times {10}^8}{0 . 8}\]
\[ \Rightarrow L_0 = 2 . 25 \times {10}^8 m\]
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