GATE 2024 AE – Question 60

Flight Mechanics & Space Dynamics · Space dynamics: central force motion, Keplerian orbits, Kepler's laws and escape velocity · 2 marks · Numerical answer

Consider an artificial satellite moving around the Moon in an elliptic orbit. The altitude of the satellite from the Moon's surface at the perigee is 25 km and that at the apogee is 134 km. Assume the Moon to be spherical with a radius of 1737 km. The trajectory is considered with reference to a coordinate system fixed to the center of mass of the Moon. The ratio of the speed of the satellite at the perigee to that at the apogee is ___________ (rounded off to 2 decimal places).

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Correct answer: 1.05 to 1.07

Explanation

The perigee and apogee radii are $r_p = 1737 + 25 = 1762$ km and $r_a = 1737 + 134 = 1871$ km. At these two points the velocity is perpendicular to the radius, so conservation of angular momentum gives $v_pr_p = v_ar_a$. The ratio is $\frac{v_p}{v_a} = \frac{1871}{1762} = 1.06$.