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A spaceship is sent to investigate a pla...

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?

Text Solution

Verified by Experts

Let the speed of the instrument 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)xx5R sin (pi - theta) = mvR sin 90^(@)`
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)`
substituting the value of v from equation (I) in equation (ii)
`25 v_(0)^(2) sin^(2)theta-v_(0)^(2)=(8GM)/(5R) rArr sin theta (1)/(5) sqrt(1+(8GM)/(5v_(0)^(2)R))`
or `theta = sin^(-1) [(1)/(5)sqrt((1+(8GM)/(5v_(0)^(2)R)))]`
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