First slide

Motion of planets and satellites

Question

The orbital velocity of an artificial satellite in a circular orbit just above the earth's surface is v. For a satellite orbiting at an altitude of half of the earth's radius, the orbital velocity is

Easy

A

$\frac{3}{2} v$
B

$\sqrt{\frac{3}{2}} v$
C

$\sqrt{\frac{2}{3}} v$
D

$\frac{2}{3} v$

Solution

$v = \sqrt{\frac{G M}{R + h}} F o r f i r s t s a t e l l i t e h = 0, v_{1} = \sqrt{\frac{G M}{R}} F o r s e c o n d s a t e l l i t e h = \frac{R}{2}, v_{2} = \sqrt{\frac{G M}{R + h}} \Rightarrow v = \sqrt{\frac{G M}{R + \frac{R}{2}}} = \sqrt{\frac{2 G M}{3 R}} v_{2} = \sqrt{\frac{2}{3}} v_{1} = \sqrt{\frac{2}{3}} v$

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