Past exam of the physics course of the year 1 of the natural sciences course of the University of Cambridge / 2024 / Paper 1 / Section B / 7 / d / Solution

Set the launch event to $(t,x)=(0,0)$ and measure $x$ toward Earth in light-years and $t$ in years, so light has coordinate slope magnitude one. Earth's worldline is $x=2$, and the lander's is $x=(35/37)t$. The outgoing signal is $x=t$ and reaches Earth at $B=(2,2)$. Earth's reply is sent at $C=(25/12,2)$.

The return signal has $x=2-(t-25/12)$. Its intersection with the lander gives $D=(1813/864,1715/864)$. Both signal lines are null; the lander's worldline lies inside the light cone. The drawing exaggerates the small gap near Earth in an inset so that B, C and D remain distinguishable.