Question

A satellite moves in an elliptical orbit with the Earth at one focus. At the perigee its
speed is v and its distance from the centre of the Earth is R . If the eccentricity of its
orbit is 0.5, calculate its speed at the apogee.

Answer 1

$$R_p=R$$

$$e=0.5=\frac{R_a-R}{R_a+R}$$

$$e=0.5=\frac{Rv^2}{GM}-1$$

$$GM=\frac{2}{3}Rv^2$$

$$R_a=3R$$

$$v_a=\sqrt{\frac{2GMR}{R_a(R_a+R)}}$$

$$v_a=\sqrt{\frac{2\frac{2}{3}Rv^2R}{3R(3R+R)}}=\frac{v}{3}$$