= Solution
Combining <constant mass flux in a collisional cascade> with the rate in part (iv) gives
$$
\dot M_{\rm loss}(D)\propto
D^{6+p-\alpha+(1-\alpha)(3+b-2p)/3}.
$$
Set the exponent to zero and multiply by three:
$$
3(6+p-\alpha)+(1-\alpha)(3+b-2p)=0,
$$
or $21+b+p-\alpha(6+b-2p)=0$. Therefore the <velocity-dependent collisional cascade> has
$$
\boxed{\alpha=\frac{21+b+p}{6+b-2p}.}
$$
For $b=p=0$, 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.
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