Question 7

A satellite is revolving in a circular orbit at a height h from the earth surface, such that $$h \ll R$$ where R is the radius of the earth. Assuming that the effect of earth's atmosphere can be neglected the minimum increase in the speed required so that the satellite could escape from the gravitational field of earth is

$$h \ll R \implies r = R + h \approx R$$

Orbital speed of the satellite: $$v_o = \sqrt{\frac{GM}{r}} \approx \sqrt{\frac{GM}{R}}$$

Escape speed from that orbit: $$v_e = \sqrt{\frac{2GM}{r}} \approx \sqrt{\frac{2GM}{R}}$$

Acceleration due to gravity at the surface: $$g = \frac{GM}{R^2}$$

$$v_o = \sqrt{gR}$$

$$v_e = \sqrt{2gR}$$

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

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