A.5.14 (HL)—World lines and speed

Use the angle relation

On a space–time diagram with equal scales, the angle $\theta$ between a particle’s world line and the time axis satisfies $\tan\theta=v/c$. Therefore $v=c\tan\theta$.

Read the line

A vertical line has $\theta=0$ and represents rest. As the line tilts toward the x-axis, $\theta$ and the speed increase. The light line has $\theta=45^\circ$ on equal scales and represents $v=c$.

Calculate from a diagram

Measure or read the angle from the time axis, evaluate $\tan\theta$, and multiply by $c$. If the diagram gives a rise/run ratio, use that ratio as $\tan\theta$ only after confirming the axes and angle definition.

Worked example from local Question Bank row 31477

For a world line representing $v=0.80c$ on equal-scale axes,

$$\theta=\tan^{-1}(v/c)=\tan^{-1}(0.80)=38.7^\circ\approx39^\circ$$

The angle is measured from the $ct$ axis, not the $x$ axis.

Common trap

Do not measure the angle from the x-axis, and do not assume every diagram uses equal visual scales. The syllabus relation is tied to the stated world-line angle and labelled axes.