Three masses m , 2m and 3m are moving in X-Y plane with speed 3u , 2u and u respectively as shown in figure . The three masses collide at the same point at P and stick together . The velocity of resulting mass will be :

Options

A. `u/12 (hat(i) +sqrt(3)hat(j))`
B. `u/12 (hat(i) - sqrt(3)hat(j))`
C. `u/12 (-hat(i)+sqrt(3)hat(j)) `
D. `u/12 (-hat(i) - sqrt(3)hat(j))`

Answer

B

Explanation

Conservation of linear momentum
`m_(1)u_(1)+m_(2)u_(2) +m_(3)u_(3) = m_(1)v_(1)+m_(2)v(2)+m_(3)v_(3)`
Given , `m_(1)=m, m_(2)=2m and m_(3)=3m `
`u_(1)=3u,u_(2)=2u and u_(3)=u`
Let all the three masses moving with velocity = `vec(v)`
Taking component velocity of each mass on XY plane and solve as
`m xx 3u (hat(i)) + 2m xx 2u (-hat(i) cos 60^(@) - hat(j) sin 60^(@))`
`+3m xx u (-hat(i) cos 60^(@) +hat(j) sin 60^(@)) = (m+2m+3m)vec(v)`
`+3mxxu (-hat(i) cos 60^(@)+hat(j) sin 60^(@)) = (m+2m+3m) vec(v)`
`3"mu"hat(i) - 4"mu"(sqrt(3)/2hat(j)) - 3 "mu" (hat(i))/2 + 3"mu"(sqrt(3)/2 hat(j))`
` = 6m vec(v)`
`rArr "mu"hat(i) - 3/2 "mu"hat(i) - (sqrt(3))/2 "mu"hat(j) = 6m vec(v)`
`rArr -1/2"mu"hat(i) - (sqrt(3))/2 "mu"hat(j) = 6m vec(v)`
`rArr " " vec(v) = u/12 (-hat(i) -sqrt(3))hat(j))`