A space vehicle approaching a planet has a speed v when it is very far from the planet At that moment tangent of its trajectory would miss the centre of the planet by distance R if the planet has mass M and radius r what is the smallest value of R in order that the resulting orbit of the space vehicle will just miss the surface of the planet?

Options

A. `R = r/v[v^(2)+(2GM)/r]^(1//2)`
B. `R = vr[1+(2GM)/r]`
C. `R = v/r[v^(2)+(2GM)/r]`
D. `R = (2GMv)/r`

Answer

A

Explanation

Conserving angular momentum we have mv'r…….(1) [v' = speed when spaceship is just touching the planet]
`1/2mv^(2) = 1/2mv'^(2) - (GMm)/r` …….(2)
soving (1)&(2)we get `R=r/v[v^(2)+(2GM)/r] ^(1//2)`