Introduction to Classical Mechanics · Problem 11.6
Relativity (Kinematics) — Morin
Physics — Mechanics · train-in-a-tunnel
Problem
A train and a tunnel both have proper lengths . The train moves toward the tunnel at speed . A bomb is located at the front of the train. The bomb is designed to explode when the front of the train passes the far end of the tunnel. A deactivation sensor is located at the back of the train. When the back of the train passes the near end of the tunnel, the sensor tells the bomb to disarm itself. Does the bomb explode?Intuitive explanation
This is a relativity paradox that hinges on length contraction and the relativity of simultaneity. In the train's frame, the tunnel contracts to length , so the train (still length ) won't fit—the bomb reaches the far end before the back enters, so it explodes. In the tunnel's frame, the train contracts to length , so it fits entirely inside—the back passes the near end before the front reaches the far end, so it disarms. The key is that "simultaneous" events in one frame aren't simultaneous in the other; you must pick a frame and track the actual sequence of events there, not mix perspectives.
Intuitive explanation
This is a relativity paradox that hinges on length contraction and the relativity of simultaneity. In the train's frame, the tunnel contracts to length , so the train (still length ) won't fit—the bomb reaches the far end before the back enters, so it explodes. In the tunnel's frame, the train contracts to length , so it fits entirely inside—the back passes the near end before the front reaches the far end, so it disarms. The key is that "simultaneous" events in one frame aren't simultaneous in the other; you must pick a frame and track the actual sequence of events there, not mix perspectives.