If $Q_D^*\propto D^b$, part (b) gives
$$
R_{\rm cc}(D)\propto
D^{3-\alpha+b(1-\alpha)/3}.
$$
The mass in one logarithmic size interval scales as $D^3[D n(D)]\propto D^{4-\alpha}$. A steady <collisional cascade> requires the mass destroyed per unit time, and hence the mass flux through every logarithmic interval, to be independent of $D$. Its exponent is therefore
$$
4-\alpha+3-\alpha+\frac{b(1-\alpha)}3=0.
$$
Solving gives
$$
\boxed{\alpha=\frac{21+b}{6+b}}.
$$

Solved by gpt-5.6-sol high.

Solution

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

 Solution

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