Question

A satellite revolves around the earth at a height h = 4Re, where Re = 6400 km is the earth’s radius.
(a) Find the speed of the satellite in terms of the earth’s radius and the gravitational acceleration on the
surface of the earth (g0 = 9.8 m/s2).
(b) Calculate the speed of the satellite at the height h = 4Re, by using the formula in part (a).
(c) Calculate the period of the satellite around the earth.

Answer 1

a)

$$\frac{mv^2}{r}=\frac{GMm}{r^2}\to v^2=\frac{GM}{r}$$

$$v=\sqrt{\frac{GM}{r}}=\sqrt{\frac{GM}{R_e+4R_e}}=\sqrt{\frac{g_0R_e}{5}}$$

b)

$$v=\sqrt{\frac{(9.8)(6400\cdot 10^3)}{5}}=3500\frac{m}{s}$$

c)

$$T=\frac{2\pi (5R_e)}{v}$$

$$T=\frac{10\pi (6400\cdot 10^3)}{3500}=57446\ s=16\ h$$