Physics

Problem

A space vehicle approaching a planet has a speed

v

, very long way out and is on a trajectory which would miss the centre of the planet by a distance

R

if it continued in a straight line. If the planet has a mass

M

and radius

r

, what is the smallest value of

R

in order that the resulting orbit will just miss the surface: a.

R = \frac{2GMv}{r}

R = vr \left[ 1 + \frac{2GM}{r} \right]

R = \frac{r}{v} \left[ v^2 + \frac{2GM}{r} \right]

R = \frac{r}{v} \left[ v^2 + \frac{2GM}{r} \right]^{1/2}