An object , initially at rest , explodes in three fragments. The momentum of two pieces are `- 3 p hat(i)` and `-4 p hat(j)` where `p` is a positive number. The momentum of the third piece
(i) will have magnitude `5p`
(ii) will make an angle `tan^(-1) (4//3)` with the `x`-axis
(iii) will make an angle `tan^(-1) (3//4)` with the `x`-axis
(iv) will have magnitude `7 p`

Options

A. `(i) ,(iii)`
B. `(i),(ii)`
C. `(ii) ,(iv)`
D. `(iii),(iv)`

Answer

B

Explanation

Let the momentum of the third piece be `p_(x) hat(i) + p_(y) hat(j)`. By the momentum conservation
` 0 = - 3p hat(i) - 4 p hat(j) + p_(x) hat(i) + p_(y) hat(j)`
`= ( - 3p + p_(x)) hat(i) + ( - 4p + p_(y)) hat(j)`
`- 3p + p_(x) = 0 rArr p_(x) = 3p`
`-4p + p_(y) = 0 rArr p_(y) = 4p`
The momentum of third piece
`|vec(p)| = sqrt((3p)^(2) + (4 p)^(2)) = 5p`
`tan theta = ( 4p)/(3p) rArr theta = tan^(-1) ((4)/(3))`