Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 316 / 3 / b

Table of contents
- Solution b

Solution

For equal material densities, a projectile of diameter

D_{cc}

catastrophically disperses a target of diameter

D

when

\frac{1}{2} (\frac{D _{cc}}{D})^{3} v_{rel}^{2} ≃ Q_{D}^{*}, D_{cc} = D (\frac{2 Q _{D}^{*}}{v _{rel}^{2}})^{1/3} .

Assume

D_{m i n} ≪ D_{cc} ≪ D

, neglect gravitational focusing, and approximate the collision cross-section by

π D^{2} /4

. The collision rate per target is

R_{cc} (D) = \frac{π D ^{2}}{4} \frac{v _{rel}}{V} n (> D_{cc}) .

Substitution gives

R_{cc} (D) = A v_{rel}^{(1 + 2 α) /3} (Q_{D}^{*})^{(1 - α) /3} D^{3 - α},

where

A = \frac{4 π r ^{2} f}{V} \frac{α - 3}{α - 1} D_{m i n}^{α - 3} 2^{(1 - α) /3} .

Solved by gpt-5.6-sol high.

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