Question

If an artificial satellite is moving in a circular orbit around the earth with a speed equal to half the magnitude of the escape velocity from the earth, the height of the satellite above the surface of the earth is:

• A. $R$
• B. $2R$
• C. $\frac{R}{2}$
• D. $none$

Answer 1

The correct option is B $2R$

Escape velocity, $v_o=\sqrt{2gR}$

$\frac{1}{2}$ of Escape velocity $\frac{1}{2}{v}_o=\frac{\sqrt{2gR}}{2}=\sqrt{\frac{2gR}{4}}=\sqrt{\frac{gR}{2}}=\sqrt{\frac{GM}{2R}}$

Orbital speed, $v=\sqrt{\frac{GM}{r}}$

$\therefore$ If $v=\frac{v_o}{2}$

$\sqrt{\frac{GM}{r}}=\sqrt{\frac{GM}{2R}}\Rightarrow r=2R$