Past exam of the physics B course of the University of Cambridge / 2021 / Paper 1 / Section B / B6 / b

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- Solution b

Solution

Choose zero gravitational potential energy at infinite separation. The spacecraft's conserved total orbital energy is

E = \frac{1}{2} m v_{p}^{2} - \frac{GM m}{r _{p}} = \frac{1}{2} m v_{a}^{2} - \frac{GM m}{r _{a}} .

At periapsis and apoapsis, the radial velocity vanishes, so the velocity is perpendicular to the radius. The conserved angular momentum magnitude is therefore

J = m r_{p} v_{p} = m r_{a} v_{a} .

The angular momentum vector is normal to the fixed orbital plane. The respective units are joules and

kg m^{2} s^{- 1}

. These expressions use Newtonian gravity and the approximation

M ≫ m

.

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