Velocity with respect to Relative Velocity

Theorem

Let $A$ and $B$ be bodies in space.

Let $\mathbf v_A$ and $\mathbf v_B$ denote the velocities of $A$ and $B$ such that $\mathbf v_A$ and $\mathbf v_B$ are very much smaller than the speed of light.

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

Then:

$\mathbf v_A = \mathbf v_{AB} + \mathbf v_B$

Proof

Recall definition $2$ of Relative Velocity:

Let $\mathbf v_A$ and $\mathbf v_B$ be the velocities of $A$ and $B$ in some frame of reference $\RR$.

The velocity $\mathbf v_{AB}$ of $A$ relative to $B$ is defined as:

$\mathbf v_{AB} := \mathbf v_A - \mathbf v_B$

The result follows directly.

$\blacksquare$

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