A.5.4 (HL)—Galilean velocity addition

Use the classical addition rule

If an object has velocity $u$ in frame S and frame S′ moves at velocity $v$ relative to S, the velocity measured in S′ is $u′=u-v$. Both $u$ and $v$ are signed velocities along the chosen axis.

Calculate in two steps

Choose the positive direction, write the velocity of the object and the relative frame velocity with signs, then subtract. If the object and S′ move in the same direction, the relative speed is reduced; if they move in opposite directions, the signed relative velocity has greater magnitude.

Check the model

Galilean addition assumes inertial frames, common time, and non-relativistic speeds. It can predict a resultant speed greater than $c$, which signals that special relativity—not the arithmetic—is required for a high-speed light problem.

Worked example from local Question Bank row 31297

In the classical model, two spacecraft moving in opposite directions at $0.80c$ have signed velocities $u=-0.80c$ and $v=+0.80c$.

$$u'=u-v=-0.80c-0.80c=-1.60c$$

The magnitude $1.60c$ shows why Galilean addition cannot describe relativistic relative velocity; the sign only gives direction in S′.

Common trap

Do not subtract speed magnitudes before deciding the direction. The sign of $u′$ tells you the object’s direction in S′; dropping signs can reverse the physical interpretation.