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

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- Solution d

Solution

Q_{D}^{*} \propto D^{b}

, part (b) gives

R_{cc} (D) \propto D^{3 - α + b (1 - α) /3} .

(1)

The mass in one logarithmic size interval scales as

D^{3} [D n (D)] \propto D^{4 - α}

. 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 - α + 3 - α + \frac{b ( 1 - α )}{3} = 0.

(2)

Solving gives

α = \frac{21 + b}{6 + b} .

(3)

Solved by gpt-5.6-sol high.

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

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