Past exam of the mathematics course of the University of Cambridge / 2018 / ia / Paper 4 / 9A / a / Solution

For rolling without slipping, $\dot{s}=a\Omega$. Conservation of energy givesand differentiation yields

For a sphere with density proportional to $r^c$,so $I/(M{a}^2)=2(c+3)/(3(c+5))$. A uniform disc has $I/(M{a}^2)=1/2$. Equal launch speeds give equal return times exactly when the accelerations, and hence these ratios, agree. Solving gives $c=3$.