Question

A spaceship approaches the Moon along a parabolic trajectory which is almost tangent to the Moon's surface. At the moment of the maximum approach the brake rocket was fired for a short time interval, and the spaceship was transferred into a circular orbit of a Moon satellite. Find how the spaceship velocity modulus increased in the process of braking.

Answer 1

In a parabolic orbit, $E=0$
So $\frac{1}{2}m{v}_i^2-\frac{\gamma Mm}{R}=0$ or, $v_i=\sqrt{2}\sqrt{\frac{\gamma M}{R}}$
where $M=$ mass of the Moon, $R=$ its radius. (This is just the escape velocity.)
On the other hand in orbit
$m{v}_f^{2}R=\frac{\gamma Mm}{R^2}$ or $v_f=\sqrt{\frac{\gamma M}{R}}$
Thus $\Delta v=(1-\sqrt{2})\sqrt{\frac{\gamma M}{R}}=-0.70km/s$