 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-oxford/2017/prelims/mathematics-v-resit/question-5/c/solution

Here

f = 0

and

c t = 1

u (\frac{3}{2}, \frac{1}{c}) = \frac{1}{2 c} \int_{1/2}^{5/2} g (s) d s .

Only the overlap

[1/2, 1]

with the support of the initial velocity contributes. Therefore

u (\frac{3}{2}, \frac{1}{c}) = \frac{1}{2 c} \int_{1/2}^{1} (1 - s^{2}) d s = \frac{1}{2 c} [s - \frac{s ^{3}}{3}]_{1/2}^{1} = \frac{1}{2 c} (\frac{2}{3} - \frac{11}{24}) = \frac{5}{48 c} .

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