Question

A train of length 100m is moving in a hilly region. At what speed must it approach the tunnel of length 80m so that a person at rest with respect to the tunnel will see that the entire train is in the tunnel at one time?
A. $1.25c$
B. $0.8c$
C. $0.64c$
D. $0.6c$
E. $0.36c$

Answer 1

Hint: The person at rest with respect to the 80m tunnel will see that the entire train is in the tunnel at one time only when the length of the train is 80 m. Thus, the apparent length of the train must be reduced with respect to the person. Use the formula for length contraction to calculate the speed of the train.

Formula used:
\[L = {L_0}\sqrt {1 - \dfrac{{{v^2}}}{{{c^2}}}} \]
Here, \[{L_0}\] is the original length, L is the contracted length of the object when it is moving at a speed v and c is the speed of light.

Complete step by step answer:
We can see the question is based on the length contraction phenomenon of relativity theory. The person at rest with respect to the 80m tunnel will see that the entire train is in the tunnel at one time only when the length of the train is 80 m. The length of the train will contract to 80m only when it is travelling at very high speed.

We have the expression for length contraction,
\[L = {L_0}\sqrt {1 - \dfrac{{{v^2}}}{{{c^2}}}} \]
Here, \[{L_0}\] is the original length of the train, L is the contracted length of the train when it is moving at a speed v and c is the speed of light.

Substituting 80m for L and 100m for \[{L_0}\] in the above equation, we get,
\[80 = 100\sqrt {1 - \dfrac{{{v^2}}}{{{c^2}}}} \]
\[ \Rightarrow \dfrac{4}{5} = \sqrt {1 - \dfrac{{{v^2}}}{{{c^2}}}} \]
\[ \Rightarrow 1 - \dfrac{{{v^2}}}{{{c^2}}} = \dfrac{{16}}{{25}}\]
\[ \Rightarrow \dfrac{{{v^2}}}{{{c^2}}} = 1 - \dfrac{{16}}{{25}}\]
\[ \Rightarrow \dfrac{{{v^2}}}{{{c^2}}} = \dfrac{9}{{25}}\]

Taking square root of the above equation, we get,
\[\dfrac{v}{c} = \dfrac{3}{5}\]
\[ \therefore v = 0.6c\]
Therefore, if the speed of the train is 0.6 times the speed of light, the entire train will appear to be in the tunnel at one time. However, we cannot travel at this speed and therefore, it is practically impossible.

So, the correct answer is option D.

Note: When the object travels with the speed of light, the length of that object will become zero that means the mass of that object will also be zero. That is why we cannot see photons and their rest mass is zero. The length contraction phenomenon applies only when the speed of the object is in the fraction of speed of light.