satellite motion

Question

A satellite is revolving in a circular orbit at a height h from the earth's surface (radius of earth R, h << R). The minimum increase in its orbital velocity required so that the satellite could escape from the earth's gravitational field is close to (neglect the effect of atmosphere):

RankGuru · Accepted Answer

1. **Orbital Velocity ($v_o$)**: For a satellite very close to Earth ($h \approx 0$), $v_o = \sqrt{\frac{GM}{R}} = \sqrt{gR}$ (since $g = \frac{GM}{R^2}$).
2. **Escape Velocity ($v_e$)**: The velocity required to leave Earth's gravity completely is $v_e = \sqrt{\frac{2GM}{R}} = \sqrt{2gR}$.
3. **Required Increase**: $\Delta v = v_e - v_o = \sqrt{2gR} - \sqrt{gR}$.
4. **Factorisation**: $\Delta v = \sqrt{gR}(\sqrt{2} - 1)$.

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satellite motion

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