Multiple choice

A train of length $100 m$ is moving in a hilly region. At what speed must it approach a tunnel of length $80 m$ so that a person at rest with respect to the tunnel will see that the entire train is in the tunnel at one time?

  1. $1.25 c$
  2. $0.8 c$
  3. $0.64 c$
  4. $0.6 c$
  5. $0.36 c$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

This is a relativistic length contraction problem. The train (100m) must fit inside the tunnel (80m). Since the train is longer than the tunnel, it cannot fit in the tunnel in the rest frame. However, in a frame moving at speed v, the train's length contracts to L' = L * sqrt(1 - v^2/c^2). We need L' <= 80. 100 * sqrt(1 - v^2/c^2) <= 80 => sqrt(1 - v^2/c^2) <= 0.8 => 1 - v^2/c^2 <= 0.64 => v^2/c^2 >= 0.36 => v >= 0.6c.

AI explanation

For the entire train to be observed inside the tunnel, we apply the relativistic length contraction formula L = L0 multiplied by the square root of (1 - v squared divided by c squared). Setting the contracted length L to 80 m and the proper length L0 to 100 m, we get 80 = 100 multiplied by the square root of (1 - v squared divided by c squared). Dividing by 100 and squaring yields 0.64 = 1 - v squared divided by c squared, and solving this gives v squared divided by c squared = 0.36, so the required speed is 0.6c.

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