Past exam of the mathematical methods II course of the University of Oxford / 2021 resit / 11 / d / Solution

For $q=2$, the common coordinate has $X(0)=(x_1(0)+2x_2(0))/3=0$ and $\dot{X}(0)=\pi \omega r/20$. The relative coordinate has $\delta (0)=3\pi r/20$ and $\dot{\delta }(0)=0$. ConsequentlySince $x_1=X+2\delta /3$ and $x_2=X-\delta /3$, the full superposition isThe heavier particle's velocity isIt first vanishes when $\sin \omega t=-1$, soThis velocity is nonnegative and only touches zero: the heavier particle does not reverse its direction at that instant. Both the uniform zero mode and the oscillatory normal mode are needed to obtain this result.