The balls, having linear momenta `vecp_1=vecpi and vecp_2_2=-vecpi`, undergo a collision in free space. There is no external force acting on the balls. Let `vecp'_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`.

Options

A. `vecp'_1=a_1hati+b_1hatj+c_1hatk`
`vecp'_2=a_2hati+b_2hatj `
B. `vecp'_1=c_1hatk`
`vecp'_2=c_2hatk`
C. `vecp'_1=a_1hati+b_1hatj+c_1hatk`
`vecp'_2=a_2hati+b_2hatj-c_1hatk`
D. `vecp'_1=a_1hati+b_1hatj`
`vecp'_2=a_2hati+b_1hatj`

Answer

A, D

Explanation

(a,d)
Use law of conservation of linear momentum.
The initial linear momentum of the system is `phati-phati=0`
Therefore the final linear momentum should also be
zero.
Option a:
`p'_1+p'_2=(a_1+a_2)hati+(b_1+b_2) hatj+c_1hatk=Final`
momentum.
It is given that `a_1,b_1,c_1,a_2,b_2 and c_2` have non-zero
values. If `a_1=x and a_2=-x. Also if b_1=y and b_2=`
`-y then the hati and hatj` components become zero. But
the third term having `hatk` components become is non-zero. This
gives a definite final momentum to the system which
violates conservation of linear momentum, so this is
an incorrect option.
Option d:
`p'_1+p'_2= (a_1+a_2)hati+2b hatj != 0 because b_1!=0`
Following the same reasoning as above this option is
also ruled out.