Introduction to Classical Mechanics · Problem 11.55
Relativity (Kinematics) — Morin
Physics — Mechanics · velocity-addition-derivation
Problem
Velocity addition ** The fact that the previous exercise can be solved in two different ways suggests a method of deriving the velocity-addition formula: A train of proper length moves at speed with respect to the ground. A ball is thrown from the back of the train to the front at speed with respect to the train. Let the speed of the ball with respect to the ground be . Calculate the time of the ball's journey, as measured by an observer on the ground, in the following two different ways, and then set them equal to solve for in terms of and . (This gets a bit messy. And yes, you have to solve a quadratic.) (a) First way: Find the relative speed of the ball and the train (as viewed by someone on the ground), and then find the time it takes for the ball to erase the initial ``head start'' that the front of the train had. (b) Second way: Find the time elapsed on the ball's clock (by working in whatever frame you want), and then use time dilation to get the time elapsed on a ground clock.Intuitive explanation
The core idea is that time intervals measured in different reference frames are related by time dilation, and this constraint—combined with a kinematic analysis in the ground frame—uniquely determines how velocities must add in relativity. By calculating the same physical duration (the ball's journey) two independent ways and equating them, you'll derive the relativistic velocity-addition formula. The "messiness" comes from the fact that relativistic addition isn't simply ; the formula must respect the invariance of the speed of light.
Intuitive explanation
The core idea is that time intervals measured in different reference frames are related by time dilation, and this constraint—combined with a kinematic analysis in the ground frame—uniquely determines how velocities must add in relativity. By calculating the same physical duration (the ball's journey) two independent ways and equating them, you'll derive the relativistic velocity-addition formula. The "messiness" comes from the fact that relativistic addition isn't simply ; the formula must respect the invariance of the speed of light.