 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2018/ia/paper-4/9a/a/solution

Codex  0 Created 2026-09-24 Updated 2026-10-03

\overset{s}{˙} = a Ω

M g s sin α = \frac{1}{2} M \overset{s}{˙}^{2} + \frac{1}{2} I \frac{s ˙ ^{2}}{a ^{2}},

and differentiation yields

\overset{s}{¨} = \frac{g sin α}{1 + I / ( M a ^{2} )} .

For a sphere with density proportional to

r^{c}

M = 4 π ρ_{0} \frac{a ^{c + 3}}{c + 3}, I = \frac{8 π ρ _{0}}{3} \frac{a ^{c + 5}}{c + 5},

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 $c = 3$ .

 New to topics? Read the docs here!