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?

Explanation

Let the speed of the instruement package is `v` when it grazes the surface of the planet.
Conserving angular momentum of the package about the centre of the planet
`mv_(0)xx5Rsin(pi-theta)=mvRsin90^(@)`
Conserving mechanical energy,
`-(GMm)/(5R)+(1)/(2)mv_(0)^(2)=-(GMm)/(R )+(1)/(2)mv^(2)`
`rArr (1)/(2)m(v^(2)-v_(0)^(2))=(4GMm)/(5R)`
`v^(2)-v_(0)^(2)=(8GM)/(5R)`
Subsituting the value of `v` from Eq. `(1)` in Eq. `(2)`
`25v_(0)^(2)sin^(2)theta-v_(0)^(2)=(8GM)/(5R)`
`rArr sintheta=(1)/(5)sqrt(1+(8GM)/(5v_(0)^(2)R))`
or `theta=sin^(-1)[(1)/(5)sqrt((1+(8GM)/(5v_(0)^(2)R)))]`