A satellite orbits Earth at constant speed v. A mass m is ejected from it so that it just barely escapes Earth's gravitational pull. What additional kinetic energy must be given to mass m?

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EEJ MAIN Physics QUESTION #7003

Question 1

A satellite orbits Earth at constant speed $v$. A mass $m$ is ejected from it so that it just barely escapes Earth's gravitational pull. What additional kinetic energy must be given to mass $m$?

$2mv^2$
$mv^2$
$\dfrac{1}{2}mv^2$✔️
$\dfrac{3}{2}mv^2$

Correct Answer Explanation

The satellite is in circular orbit at some radius $r$: $v = \sqrt{GM/r}$, so $v^2 = GM/r$.

Escape speed from radius $r$: $v_e = \sqrt{2GM/r} = v\sqrt{2}$

Extra KE needed $= \frac{1}{2}mv_e^2 - \frac{1}{2}mv^2 = \frac{1}{2}m(2v^2) - \frac{1}{2}mv^2 = \frac{1}{2}mv^2$

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