Multiple choice

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

  1. 1.25c

  2. 0.8c

  3. 0.64c

  4. 0.6c

  5. 0.36c
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D Correct answer
Explanation

For the train to be entirely in the tunnel, the tunnel must be at least as long as the train. However, the tunnel is 80 m and the train is 100 m. In classical physics, this is impossible. In special relativity, length contraction occurs: L = L0 * sqrt(1 - v^2/c^2). We need the contracted length of the train to be 80 m. 80 = 100 * sqrt(1 - v^2/c^2). 0.8 = sqrt(1 - v^2/c^2). 0.64 = 1 - v^2/c^2. v^2/c^2 = 0.36, so v = 0.6c.

AI explanation

To observe the entire train inside the tunnel, the moving train must undergo length contraction so its contracted length equals the tunnel length of 80 m. Using the relativistic length contraction formula L = L0 * square root of (1 - v squared / c squared), we write 80 = 100 * square root of (1 - v squared / c squared). Dividing by 100 gives 0.8, and squaring both sides yields 0.64 = 1 - v squared / c squared. Solving this equation results in v squared / c squared = 0.36, meaning the required speed v is 0.6c.

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