Answer to Question #156669 in Mechanics | Relativity for Hawa Ichika

Question

3. A 300 m long train passed through a 500 m long tunnel. Both lengths are measured at rest. The train speeds at 4/5c from west to east through the tunnel.

In the train's reference frame:

i. What would be the length of the tunnel?

There are two passengers on the train, Ahmad who sits at the very front of train and Siti, who sits at the very rear. At the instant that the train enters the tunnel completely, Ahmad and Siti both stick out their heads up. Siti should be able to see the west end of the tunnel and Ahmad should be able see the east end of the tunnel.

ii. Will they see the ends of the tunnel simultaneously? Explain and sketch the situation.

In the tunnel's reference frame:

iii.At the instant that the train enters the tunnel completely, Ahmad and Siti stick out their heads up, will they simultaneously see the ends of the tunnel?

iv.Explain If the answer in (iii) is no, who will see the end of tunnel first? Sketch the situation.

Answer 1

i) We can calculate the length of the tunnel from the length contraction formula:

$$l=l_0\sqrt{1-\dfrac{v^2}{c^2}},$$ $$l=500\ m\cdot\sqrt{1-\dfrac{(0.8c)^2}{c^2}}=300\ m.$$

ii) In the train's reference frame, the length of the tunnel equals the length of the train (due to the relativistic length contraction). Therefore, at the instant that the train enters the tunnel completely, the passengers will see the ends of the tunnel simultaneously.

iii)-iv) In the tunnel reference frame, the length of the tunnel is longer then the length of the train, therefore, at the instant that the train enters the tunnel completely, Siti at the very rear of the train will see the end of the tunnel first.

Here, the appropriate diagrams: