 Solution

Neglect gravitational focusing and assume a fixed well-mixed volume $V$, $\alpha >1$, and $D_{\min }\ll D_{\mathrm{cc}}\ll D\ll D_{\max }$. For the small impactors that dominate the rate, the geometric collision cross-section is approximately $\pi D^2/4$. ThusEvaluating and neglecting the upper-end term gives the catastrophic planetesimal collision rateUsing part (ii), its size dependence isIf the destructive cutoff is near an endpoint or comparable to $D$, retain the full integral and the cross-section $\pi (D+d{)}^2/4$. The simple power law then need not apply.