A satellite orbits at height h above Earth's surface (radius R; h ll R). What is the minimum increase in orbital speed needed for the satellite to escape Earth's gravitational field? (Neglect atmosphere)

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EEJ MAIN Physics QUESTION #7000

Question 1

A satellite orbits at height $h$ above Earth's surface (radius $R$; $h \ll R$). What is the minimum increase in orbital speed needed for the satellite to escape Earth's gravitational field? (Neglect atmosphere)

$\sqrt{gR}$
$\sqrt{gR/2}$
$\sqrt{gR}(\sqrt{2}-1)$✔️
$\sqrt{2gR}$

Correct Answer Explanation

For $h \ll R$, orbital radius $\approx R$.

Orbital velocity: $v_o = \sqrt{\frac{GM}{R}} = \sqrt{gR}$

Escape velocity from orbit at $R$: $v_e = \sqrt{\frac{2GM}{R}} = \sqrt{2gR}$

Minimum increase in speed:

$\Delta v = v_e - v_o = \sqrt{2gR} - \sqrt{gR} = \sqrt{gR}(\sqrt{2}-1)$

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