Magnitude of Relative Velocity

Theorem

Let $A$ and $B$ be bodies in space, both moving in the same direction.

Let $\mathbf v_A$ and $\mathbf v_B$ denote the velocities of $A$ and $B$.

Let $\mathbf v_{AB}$ denote the velocity of $A$ relative to $B$.

Then:

$v_{AB} = \dfrac {\size {v_A - v_B} } {1 - \dfrac {v_A v_B} {c^2} }$

where:

$v_{AB}$, $v_A$ and $v_B$ are the magnitudes of $\mathbf v_{AB}$, $\mathbf v_A$ and $\mathbf v_B$ respectively

$c$ denotes the speed of light.

Proof

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Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): relative velocity
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): relative velocity