A.5.7 (HL)—Relativistic velocity addition

Use the relativistic rule

For an object with velocity $u$ in S and a frame S′ moving at velocity $v$ relative to S, use $u′=(u-v)/(1-uv/c^2)$. The numerator is the classical relative velocity; the denominator is the correction required by special relativity.

Substitute signed velocities

Choose one positive direction, write signed values for $u$ and $v$, calculate $uv/c^2$, and evaluate the full fraction. Report the magnitude if the question asks for speed; retain the sign if it asks for velocity or direction.

Check the limiting cases

When $u\ll c$ and $v\ll c$, the denominator is close to 1 and the result approaches $u-v$. If $u=c$, the equation gives $u′=c$ for any sub-light $v$, so light does not gain or lose speed between inertial frames.

Worked example from local Question Bank row 31056

A train has $u=-0.70c$ in the ground frame and observer P moves at $v=+0.60c$.

$$u'=\frac{-0.70c-0.60c}{1-(-0.70)(0.60)}=\frac{-1.30c}{1.42}=-0.915c\approx-0.92c$$

The negative sign means the train moves opposite P's positive direction; its speed remains below $c$.

Common trap

Do not use $u-v$ for a high-speed signal. Also do not report a result above $c$; a sign error or an omitted denominator usually caused it.