Chapter 2: Velocity Addition Formula
In Galilean physics, velocities add simply: if you walk at 3 mph on a train moving at 60 mph, you're moving at 63 mph relative to the ground. But this breaks down at relativistic speeds. The correct formula ensures that no combination of velocities ever exceeds the speed of light.
The Relativistic Velocity Addition Formula
\( u = \frac{u' + v}{1 + \frac{u'v}{c^2}} \)
u' = velocity in frame S' | v = velocity of S' relative to S | u = velocity in frame S
Galilean (Wrong at High Speeds)
\( u = u' + v \)
Simple addition—can exceed c!
Relativistic (Correct)
\( u = \frac{u' + v}{1 + u'v/c^2} \)
Always gives u ≤ c!
Derivation
From the Lorentz transformations: \( x = \gamma(x' + vt') \) and \( t = \gamma(t' + vx'/c^2) \)
Velocity in S: \( u = \frac{dx}{dt} = \frac{dx'/dt' + v}{1 + (v/c^2)(dx'/dt')} = \frac{u' + v}{1 + u'v/c^2} \)
Worked Examples
Example 1: Low Speeds
Train at v = 60 mph, walking at u' = 3 mph. Then u'v/c² ≈ 0, so u ≈ 63 mph (Galilean result).
Example 2: High Speeds
Spaceship at v = 0.9c, fires missile at u' = 0.9c forward.
\( u = \frac{0.9c + 0.9c}{1 + (0.9)(0.9)} = \frac{1.8c}{1.81} = 0.994c \) (not 1.8c!)
Example 3: Adding Light Speed
Spaceship at v = 0.9c, shines a flashlight (u' = c).
\( u = \frac{c + 0.9c}{1 + (0.9c)(c)/c^2} = \frac{1.9c}{1.9} = c \) (light speed is invariant!)
Rapidity: The Natural Velocity Parameter
Define rapidity φ by: \( \tanh\phi = v/c \), then velocities add as:
\( \phi_{total} = \phi_1 + \phi_2 \)
Rapidities add linearly like angles in Euclidean rotations!
This is why rapidity is often more convenient than velocity in relativistic calculations. As v → c, φ → ∞, but the addition remains simple.