Two balls , having linear momenta `vec(p)_(1) = p hat(i)` and `vec(p)_(2) = - p hat(i)` , undergo a collision in free space. There is no external force acting on the balls. Let `vec(p)_(1)` and `vec(p)_(2)`, be their final momenta. The following option(s) is (are) NOT ALLOWED for any non -zero value of `p , a_(1) , a_(2) , b_(1) , b_(2) , c_(1)` and `c_(2)`
(i) `vec(p)_(1) = a_(1) hat(i) + b_(1) hat(j) + c_(1) hat(k) , vec(p)_(2) = a_(2) hat(i) + b_(2) hat(j)`
(ii) `vec(p)_(1) = c_(1) vec(k) , vec(p)_(2) = c_(2) hat(k)`
(iii) `vec(p)_(1) = a_(1) hat(i) + b_(1) hat(j) + c_(1) hat(k) ,vec(p)_(2) = a_(2) hat(i) + b_(2) hat(j) - c_(1) hat(k)`
(iv) `vec(p)_(1) = a_(1) hat(i) + b_(1) hat(j) , vec(p)_(2) = a_(2) hat(i) + b_(1) hat(j)`

Options

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

Answer

D

Explanation

Initial momentum `= 0`
Since there is no external force , the final momentum `= 0`
(i) `vec(p)_(1) + vec(p) != 0`
(ii) `vec(p)_(1) + vec(p)_(2) = 0` , if `c_(1) = -c_(2)`
(iii) `vec(p)_(1) + vec(p)_(2) = 0` , if `a_(1) = -a_(2) , b_(1) = - b_(2)`
(iv) `vec(p)_(1) + vec(p) != 0`