Two balls having linear momenta `vecp_(1)=phati` and `vecp_(2)=-phati, ` undergo a collision in fre space. There is no external force acting on the ball. Let `vecp_(1)^(')` and `vecp_(2)^(')` be their final moment. Which of the following option(s) is (are) NOT ALLOWED for an non zero value of `p,a_(1),a_(2),b_(1),b_(2), c_(1)` and `c_(2).`

Options

A. `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)`
B. `vec(p)_(1) = c_(1) hat(k)`
`vec(p)_(2) = c_(2)hat(k)`
C. `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_(2) hat(k)`
D. `vec(p)_(1) = a_(1)hat(i) + b_(1) hat(j)`
`vec(p)_(2) = a_(2) hat(i) + b_(1) hat(j)`

Answer

A, D

Explanation

As `vec(p)_(1) + vec(p)_(2) = vec(0)` so `vec(p)_(1') + vec(p)_(2') = 0`
For (A) `vec(p)_(1') + vec(p)_(2') = (a_(1) + a_(2)) hat(i) + (b_(1) + b_(2)) hat(j) + c_(1) hat(k)`
For (B) `vec(p)_(1') + vec(p)_(2') = (a_(1) + a_(2)) hat(i) + (b_(1) + b_(2)) hat(j)`
For (C ) `vec(p)_(1') + vec(p)_(2') = (c_(1) + c_(2)) hat(k)`
For (D) `vec(p)_(1') + vec(p)_(2') = (a_(1) + a_(2)) hat(i) + 2b_(1) hat(j)`
But `a_(1'), b_(1'), c_(1'), a_(2), b_(2), c_(2) != 0`
Therefore (A) & (D) is not possible to get
`vec(p)_(1') + vec(p)_(2') = vec(0)`