The impactors' number density per diameter is $KD_{\rm im}^{-\alpha}/V$. With gravitational focusing neglected, the <geometric collision cross-section> is $\pi(D+D_{\rm im})^2/4$. The <planetesimal collision rate> is consequently
$$
R_{\rm col}(D,D_{\rm im})dD_{\rm im}
=\frac{K D_{\rm im}^{-\alpha}}{V}
\frac{\pi(D+D_{\rm im})^2}{4}v_{\rm rel}\,dD_{\rm im}
=A D_{\rm im}^{2-\alpha}\left(1+\frac D{D_{\rm im}}\right)^2dD_{\rm im},
$$
where
$$
\boxed{A=\frac{\pi K v_{\rm rel}}{4V}}.
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 316 / 1 / ii / Solution

For an impactor number density per size $n(D_{\rm im})$, relative speed $v_{\rm rel}$, and collision cross-section $\sigma(D,D_{\rm im})$, the differential collision rate of one target is $n(D_{\rm im})v_{\rm rel}\sigma(D,D_{\rm im})dD_{\rm im}$.


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 Incoming links: Planetesimal collision rate