A small particle of mass ` m` is projected at an angle `theta` with the ` x`- axis with an initial velocity ` v_(0)` in the ` x-y `plane as shown in the figure . At a time ` t lt ( v_(0) sin theta)/(g) `, the angular momentum of the particle is
where `hat (i) , hat (j) and hat(k)` are unit vectors along `x , y and z` - axis respectively.

Options

A. `(1)/(2)mgv_(0)t^(2) costhetahati`
B. `-mgv_(0)t^(2) costhetahatj`
C. `mgv_(0)t^(2) costhetahatk`
D. `-(1)/(2)mgv_(0)t^(2) costhetahatk`

Explanation

`t lt (v_(0)sintheta)/(g)`
`vecv=v_(0) cos theta hati+(v_(0) sin theta-"gt")hatj`
`vecr=x hati+y hatj=v_(0) cos theta t hati+(v_(0)sintheta t-(1)/(2)"gt"^(2))hatj`
`vecl_(0)=vecrxxmvecv=m(vecrxxvecv)`
`= -(1)/(2)mg v_(0) t^(2) cos theta hatk`