Relativistic Velocity Addition

Relativistic Velocity Addition

In classical mechanics, velocities simply add: if a train moves at $v$ relative to the ground, and you throw a ball at $u'$ relative to the train, the ball moves at $u = u' + v$ relative to the ground. This breaks down at relativistic speeds.

The Relativistic Formula

Suppose frame $S'$ moves at velocity $v$ relative to frame $S$ (along the same axis). An object moves at velocity $u'$ in $S'$. Its velocity in $S$ is:

$u = \frac{u' + v}{1 + \dfrac{u'v}{c^2}}$

The denominator $1 + u'v/c^2$ is what prevents the result from exceeding $c$.

Key Properties

Low-speed limit: When $u' \ll c$ and $v \ll c$, the denominator $\approx 1$ and we recover Galilean addition $u \approx u' + v$.

Light speed is invariant: If $u' = c$:

$u = \frac{c + v}{1 + v/c} = \frac{c(1 + v/c)}{1 + v/c} = c$

No matter how fast the source moves, light still travels at $c$.

No velocity exceeds $c$: Combining any two sub-luminal velocities always gives a sub-luminal result.

Example: 0.5c + 0.5c

Classically: $0.5c + 0.5c = c$. Relativistically:

$u = \frac{0.5c + 0.5c}{1 + (0.5)(0.5)} = \frac{c}{1.25} = 0.8c$

Your Task

Implement velocity_addition(u_prime, v) that returns the velocity $u$ in frame $S$ when an object moves at $u'$ in frame $S'$, and $S'$ moves at $v$ relative to $S$. All velocities in m/s. Use $c = 299792458.0$ m/s inside the function.