Introduction to Classical Mechanics · Problem 11.61
Relativity (Kinematics) — Morin
Physics — Mechanics · invariant-interval
Problem
Throwing on a train ** A train with proper length moves at speed with respect to the ground. A ball is thrown from the back to the front, at speed with respect to the train. How much time does this take, and what distance does the ball cover, in: (a) The train frame? (b) The ground frame? Solve this by (i.) Using a velocity-addition argument. (ii.) Using the Lorentz transformations to go from the train frame to the ground frame. (c) The ball frame? (d) Verify that the invariant interval is indeed the same in all three frames. (e) Show that the times in the ball frame and ground frame are related by the relevant factor.Intuitive explanation
This problem tests whether you can track the same physical event—a ball's journey—across different reference frames using special relativity. The key insight is that while time and distance change between frames, the spacetime interval stays constant, and velocity addition must account for relativistic effects rather than simple arithmetic. Start by finding the ball's motion in the train frame (the easiest), then use either velocity addition or Lorentz transformations to shift to the ground frame, and finally push to the ball's rest frame to see how dramatically lengths contract and times dilate.
Intuitive explanation
This problem tests whether you can track the same physical event—a ball's journey—across different reference frames using special relativity. The key insight is that while time and distance change between frames, the spacetime interval stays constant, and velocity addition must account for relativistic effects rather than simple arithmetic. Start by finding the ball's motion in the train frame (the easiest), then use either velocity addition or Lorentz transformations to shift to the ground frame, and finally push to the ball's rest frame to see how dramatically lengths contract and times dilate.