Past exam of the mathematics course of the University of Cambridge / 2026 / ii / Paper 2 / 14A / c

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- Solution c

Solution

The radial effective potential is

V_{eff} = m g r cot α + 5 J^{2} / (14 m r^{2})

. Initially

r = 8 a

,

J = (7 m /5) (8 a) v_{0}

. Comparing the initial energy with

V_{eff} (2 a)

gives

\frac{7}{10} m v_{0}^{2} + 8 m g a cot α \leq \frac{56}{5} m v_{0}^{2} + 2 m g a cot α,

which is exactly

v_{0}^{2} \geq (4/7) g a cot α

. Larger speed raises the centrifugal barrier at the neck faster than it raises the initial kinetic energy.

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