A train has a rest length of 100 m. Traveling at a very high velocity, it goes through a tunnel of length 80 m. Observers located at both ends of the tunnel note that at one instant the train appears to exactly fit within the tunnel. What is the velocity of the train expressed in units of c?
A) 0.333c
B) 0.50c
C) 0.60c
D) 0.80c
E) 0.866c

Question

Quizplus · Accepted Answer

Let the velocity of the train be v and the length contraction factor be γ. Then, according to the observers, the length of the train as seen from outside the tunnel would be γ×100 m and the total length of the train and the tunnel together would be (γ×100 + 80) m. However, since the train exactly fits within the tunnel at that instant, we have:
γ×100 + 80 = 100
Solving for γ, we get γ = 4/5. Therefore, the velocity of the train can be found using:
v = √(1 - γ^2) × c ≈ 0.6c (since γ is very close to 1)
So the best choice is (C).

A Train Has a Rest Length of 100 M