 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2021/ib/paper-4/16a/d/solution

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

Since

d t = d a / \overset{a}{˙}

on the collapsing branch, the time for

a

to decrease from

a_{0}

to zero is

τ = \frac{3 ρ}{2 p _{0}} \int_{0}^{a_{0}} \frac{d a}{a _{0}^{3} / a ^{3} - 1} .

Equivalently, scaling

a = a_{0} x

gives

τ = a_{0} \frac{3 ρ}{2 p _{0}} \int_{0}^{1} \frac{x ^{3/2}}{1 - x ^{3}} d x .

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