Question

A satellite going around Earth in an elliptic orbit has a speed of 10 km s−1 at the perigee which is at a distance of 227 km from the surface of the earth. Calculate the apogee distance and its speed at that point

Answer 1

$\text we \text know ,$
$r_p = H + R_{Earth}$
$r_p = 227 + 6371$
$r_p = 6598 km$
Now,
$v_p = \sqrt{GM( \frac{2}{r_p}- \frac{1}{a}) }$
$\Rightarrow \frac{GM}{a} = \frac{2GM}{r_p}-v^2_p$
$\Rightarrow a = \frac{GMr_p}{2GM-r_pv^2_p}$
$\Rightarrow 18724km$
So,
$r_a = 2a-r_p = \frac{GMr_p}{2GM-r_pv^2_p} = 30850km$
$v_a = \sqrt{GM( \frac{2}{r_p}- \frac{1}{a}) } = 2.138km/s$