A satellite having orbital speed V orbits a planet of mass M. If the planet had half as much mass, the orbital speed of the satellite at the same distance from the center would be
A) V $\sqrt { 2 }$ .
B) 2V.
C) V.
D) V/ $\sqrt { 2 }$ .
E) V/2.

Question

Quizplus · Accepted Answer

According to Kepler's Third Law, $T^2 \propto a^3$, where $T$ is the period of the orbit and $a$ is the semi-major axis of the ellipse. Since the distance from the center of the planet is the same in both cases, the semi-major axis is not changing, and therefore the period remains the same. The centripetal force keeping the satellite in orbit is given by $F = \frac{GMm}{r^2}$, where $r$ is the distance from the center of the planet, $M$ is the mass of the planet, and $m$ is the mass of the satellite. Therefore, the orbital speed of the satellite is proportional to $\sqrt{\frac{M}{r}}$. If $M$ is halved, the orbital speed becomes $\sqrt{\frac{M/2}{r}} = \frac{1}{\sqrt{2}}\sqrt{\frac{M}{r}}$, or $\frac{1}{\sqrt{2}}$ times the original speed. Therefore, the correct answer is D, because the orbital speed is halved.

A Satellite Having Orbital Speed V Orbits a Planet of Mass