Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 316 / 2 / vi

Combining constant mass flux in a collisional cascade with the rate in part (iv) givesSet the exponent to zero and multiply by three:or $21+b+p-\alpha (6+b-2p)=0$. Therefore the velocity-dependent collisional cascade hasFor $b=p=0$, this recovers the Dohnanyi collisional cascade slope $\alpha =7/2$. The denominator must be nonzero, and the resulting slope and destructive cutoff must satisfy the assumptions used for the collision-rate integral. A size-dependent vertical belt thickness would supply an additional volume factor and change this result.