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

Answer

A

Explanation

a. From the principle of conserving angular momentum, we have
`mvR=mv'r`……..i
`[v'=`speed when spaceship is just touching the planet]
From conserving of energy, we have
`1/2mv^(2)=1/2mv^('2)-(GMm)/r`…………ii
Solving eqn i and ii we get
`R=r/v[v^(2)+(2GM)/r]^(1/2)`