Relative Velocity Calculator

Classical relative velocity is simple subtraction: if A moves at v₁ and B moves at v₂ (both relative to the ground, same direction positive), then A’s velocity relative to B is v₁ - v₂.

A car at 80 km/h and another at 50 km/h in the same direction: relative velocity = 30 km/h. In opposite directions: 80 + 50 = 130 km/h.

This works perfectly for everyday speeds. It breaks down at speeds approaching the speed of light.

Relativistic velocity addition

Einstein showed that velocities do not add linearly. If A moves at v₁ and B moves at v₂ relative to the ground:

v_rel = (v₁ - v₂) / (1 - v₁v₂/c²)

c = 2.998 × 10⁸ m/s (speed of light)

This formula has important consequences:

No combination of subluminal velocities can exceed c. If A moves at 0.9c and B moves at -0.9c (opposite directions), the Newtonian answer would be 1.8c. The relativistic result: (0.9c + 0.9c) / (1 + 0.81) = 1.8c/1.81 ≈ 0.994c.

Adding c to anything gives c: if A moves at c (a photon) and B moves at 0.99c in the same direction, A’s speed relative to B is (c - 0.99c)/(1 - 0.99c²/c²) = 0.01c/0.01 = c. The photon still moves at c relative to every observer.

When to use which formula

For speeds below ~1% of c (3,000 km/s), the relativistic correction is less than 0.01% and Newtonian is fine. Particle accelerators, cosmic ray physics, and GPS satellite corrections all require relativistic velocity addition.

Enter both velocities in m/s. Positive = one direction, negative = opposite.