Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 316 / 1 / a

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- Solution a

Solution

The force is central force, so its torque about the star vanishes. The specific angular momentum is therefore constant:

h = ∣ r \times v ∣ = r^{2} \dot{θ} .

Taking the scalar product of

\ddot{r} = - μ r / r^{3}

with

\dot{r}

gives

\frac{d}{d t} (\frac{v ^{2}}{2} - \frac{μ}{r}) = \dot{r} \cdot \ddot{r} + \frac{μ}{r ^{3}} r \cdot \dot{r} = 0.

Thus conservation of energy gives the second constant

C = \frac{v ^{2}}{2} - \frac{μ}{r} .

Solved by gpt-5.6-sol high.

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