Discussion
The velocities of a satellite (orbiting around earth) at perigee and apogee points are 25 km/s and 10 km/s respectively. If the perigee distance from the centre of the earth is 10000 km, what is the apogee distance ?
Answer: A
The apogee distance is 25000 km.
At perigee and apogee the velocity is perpendicular to the radius, so conservation of angular momentum gives \(v_{p}r_{p} = v_{a}r_{a}\):
\[ \begin{aligned} r_{a} &= \frac{v_{p}r_{p}}{v_{a}} \\ &= \frac{25 \times 10000}{10} \\ &= 25000\ \text{km} \end{aligned} \]
The satellite moves slowest when farthest away, so the apogee distance must exceed 10000 km; 2500 km and 2000 km get the ratio upside down, and 20000 km does not satisfy \(v_{p}r_{p} = v_{a}r_{a}\).
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