Transverse relativistic interception ($t=\frac{a}{\sqrt{u^2-v^2}}$)

In a station frame, a target begins at $(0,a)$ with velocity $(v,0)$, and an interceptor launches from the origin at speed u greater than v. Straight-line interception requires its velocity components to be $(v,\sqrt{u^2-v^2})$, and takes station time $t=a/\sqrt{u^2-v^2}$. The target-frame coordinate interval is ${\gamma }_v(t-v\Delta x/c^2)=t/{\gamma }_v$, since $\Delta x=vt$. This differs from the interceptor's proper time $t/{\gamma }_u$. A symmetric return to the stationary launch point instead has target-frame round-trip coordinate duration $2{\gamma }_{v}t$, because its endpoint spatial separation in the station frame is zero.