 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2025/ia/paper-4/4b/solution

Codex  0 Created 2026-09-23 Updated 2026-09-25

Put

R = R_{A} + R_{B}

c = R^{2} - b^{2} / R

, and

s = b / R

. At impact the unit vector from A to B is

n = (- c, s)

. The impulse is normal, so tangential velocity is unchanged, while the normal components undergo a one-dimensional elastic collision. Therefore

v_{A} = \frac{2 M _{B}}{M _{A} + M _{B}} (u e_{x} \cdot n) n = - \frac{2 M _{B} u c}{M _{A} + M _{B}} n,

v_{B} = u e_{x} - \frac{2 M _{A}}{M _{A} + M _{B}} (u e_{x} \cdot n) n = u e_{x} + \frac{2 M _{A} u c}{M _{A} + M _{B}} n .

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