For equal material densities, a projectile of diameter $D_{\rm cc}$ catastrophically disperses a target of diameter $D$ when
$$
\frac12\left(\frac{D_{\rm cc}}D\right)^3v_{\rm rel}^2\simeq Q_D^*,
\qquad
D_{\rm cc}=D\left(\frac{2Q_D^*}{v_{\rm rel}^2}\right)^{1/3}.
$$
Assume $D_{\min}\ll D_{\rm cc}\ll D$, neglect gravitational focusing, and approximate the collision cross-section by $\pi D^2/4$. The collision rate per target is
$$
R_{\rm cc}(D)=\frac{\pi D^2}{4}\frac{v_{\rm rel}}Vn(>D_{\rm cc}).
$$
Substitution gives
$$
\boxed{R_{\rm cc}(D)=A
v_{\rm rel}^{(1+2\alpha)/3}
(Q_D^*)^{(1-\alpha)/3}D^{3-\alpha}},
$$
where
$$
\boxed{A=\frac{4\pi r^2f}{V}\frac{\alpha-3}{\alpha-1}
D_{\min}^{\alpha-3}2^{(1-\alpha)/3}}.
$$

Solved by gpt-5.6-sol high.


Solution

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

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2026/iii/paper-316/3/b/solution