First slide

Gravitational potential and potential energy

Question

A satellite is orbiting with areal velocity v_a. At what height from the surface of the earth, it is rotating, if the radius of earth is R?

Moderate

A

$\frac{4 v_{a}^{2}}{{gR}^{2}} - R$
B

$\frac{2 v_{a}^{2}}{{gR}^{2}} - R$
C

$\frac{v_{a}^{2}}{{gR}^{2}} - R$
D

$\frac{3 v_{a}^{2}}{{gR}^{2}} - R$

Solution

$\begin{array}{l} v_{a} = \frac{d A}{d t} = \frac{1}{2} r v_{0} = \frac{1}{2} r \sqrt{\frac{G M}{r}} \\ \Rightarrow 4 v_{a}^{2} = G M r = G M (R + h) \\ \Rightarrow 4 v_{a}^{2} = g R^{2} (R + h) \\ \Rightarrow h = \frac{4 v_{a}^{2}}{g R^{2}} - R \end{array}$

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