A.5.11 (HL)—Length contraction

Use the contraction equation

If $L_0$ is the proper length measured in the object’s rest frame, an observer who sees the object moving at speed $v$ measures $L=L_0/\gamma$, with $\gamma=1/\sqrt{1-v^2/c^2}$.

Select the proper length

Find the frame in which the object is at rest; that frame measures $L_0$. Calculate $\gamma$, then divide the proper length by $\gamma$. The endpoints must be measured simultaneously in the observer’s frame.

Check the result

Since $\gamma\ge1$, a moving observer measures $L\le L_0$. At low speed the contraction is negligible; as $v$ approaches $c$, the measured length along the direction of motion becomes substantially smaller.

Worked example from local Question Bank row 30450

A rocket's proper length is $L_0=450\,\mathrm{m}$ and $\gamma=5/3$. An observer who sees it moving measures

$$L=\frac{L_0}{\gamma}=\frac{450}{5/3}=270\,\mathrm{m}$$

Only the dimension parallel to the relative motion is contracted.

Common trap

Do not contract a length perpendicular to the motion, and do not multiply by $\gamma$ when the requested quantity is the moving-frame length.