With what speed `v_(0)` should a body be projected as shown in the figure, with respect to a planet of mass `M` so that it would just be able to graze the planet and escape ? The radius of the planet is `R`. (Assume that the planet is fixed ).

Explanation

By conservation of angular momentum we obtain
`mv_(0)d=mvr`
`impliesv=(mv_(0)d)/(mr)=v_(0)(d//r)`
By conservation of energy at `A` and `B`, we have
`(1)/(2)mv_(0)^(2)-(GMm)/(sqrt(D^(2)+d^(2)))=(1)/(2)mv^(2)-(GMm)/(r )`
When `r~~R` we obtain,
`(1)/(2)mv_(0)^(2)-(GMm)/(sqrt(D^(2)+d^(2)))-(1)/(2)m.(v_(0)^(2)d^(2))/(R^(2))-(GMm)/(R )`
`(1)/(2)mv_(0)^(2)((d^(2))/(R^(2))-1)=GMm[(1)/(R )-(1)/(sqrt(D^(2)+d^(2)))]`
`impliesv_(0)=sqrt((2GM((1)/(R )-(1)/(sqrt(D^(2)+d^(2)))))/(((d^(2))/(R^(2))-1)))`