A cosmic body `A` moves to the sun with velocity `v_(0)`(when far from the sun) and aiming parameter `l`, the arm of the vector `v_(0)`, relative to the centre of the sun. find the minimum distance by which this body will get to the sun. mass of the sum is `M`.

Options

A. `(GM)/(v_(0)^(2))[sqrt(1+((lv_(0)^(2))/(GM))^(2))-1]`
B. `(GM)/(v_(0)^(2))-1`
C. `(GM)/(v_(0)^(2))[sqrt(1+((lv_(0)^(2))/(GM))^(2))+1]`
D. `GMl v_(0)^(2) -1`

Answer

A

Explanation

Let `r` be the minimum distance and `m` the mass of cosmic body A. applying conservation of angular momentum (about sun) and conservation of mechanical energy, we have,
`mv_(0)l=mvr......(1)`
`1/2mv_(0)^(2)=1/2mv^(2)-(GMm)/r.....(2)`
Solving these two equation, we have
`r=(GM)/(v_(0)^(2))[sqrt(1+((lv_(0)^(2))/(GM))^(2))-1]`