Question

A spaceship is sent to investigate a planet of mass `M` and radius `R`. While hanging motionless in space at a distance `5R` from the centre of the planet, the spaceship fires an instrument package of mass `m`, which is much smaller than the mass of the spaceship. For what angle `theta` will the package just graze the surface of the planet?

Answer 1

Correct Answer - `sin^(-1)(1/5sqrt(1+(8GM)/5v_(0)^(2)R)))`
Let the speed of the instrument package be `v` when it grazes the surface of the planet.

Conserving angular momentum of the package about the centre of the planet
`mv_(0)xx5Rsinteta=mvR`
`impliesv=5v_(0)sintheta`
Conserving mechanical energy, we get ...........i
`-(GMm)/(5R)+1/2mv_(0)^(2)=-(GMm)/R+1/wmv^(2)`
`implies 1/2m(v^(2)-v_(0)^(2))=(4GMm)/(5R)`
`implies v^(2)=v_(0)^(2)=(8GM)/(5R)`..............ii
Substituting the value of `v` from eqn i in Eqn ii we have
`25v_(0)^(2)sintheta-v_(0)^(2)=(8GM)/(5R)`
`implies theta=sin^(-1)(1/5sqrt(1+(8GM)/(5v_(0)^(2)R)))`