Two smooth balls `A` and `B`, each of mass `m` and radius `R`, have their centres as shown in fig. Ball `A`, moving along positive x-axis, collides with ball `B`. Just before the collision, speed of ball `'A'` is `4m//s` and ball `'B'` is stantionary. The collision

Velocity of the ball `'A'` just after the collision is

Options

A. `(i + sqrt(3)hat(j))m//s`
B. `(i - sqrt(3)hat(j))m//s`
C. `(2i + sqrt(3)hat(j))m//s`
D. `(2i + 2 hat(j))m//s`

Answer

A

Explanation

velocity of ball `'A'` after collision is
`= 4 sin 30 xx cos 60 hat(i) + 4 sin 30 xx sin 60 hat(j)`
`= (hat(i)+sqrt(3)hat(j))m//s`
Along common normal direction velocity of ball after collision is `'0'`
`e = (v_(2)-v_(1))/(u_(1)-u_(2)), 1 = (v_(2)-v_(1))/(4 cos 30-0)`
`v_(2)-v_(1) = 2sqrt(3)` ........(1)
`mxx4 cos 30 = mv_(1)+mv_(2)`
`v_(2)+v_(1) = 2sqrt(3)` ........(2)
From (1) & (2) `v_(1) = 0`
`v_(2) +v_(1) = 2sqrt(3) , v_(2)-v_(1) = 2sqrt(3)`
`rArr 2v_(2) = 4^(4)sqrt(3) , v_(2) = 2sqrt(3)`
Impluse of the force exerted by `A` and `B`
`= m[4 cos 30(cos 30 hat(i)-sin 30 hat(j))]`
`= (3m hat(i)-sqrt(3)mj)kg-m//s , (1)/(2) = (v_(2)-v_(1))/(4 cos 30)`
`v_(2)-v_(1) = sqrt(3)` ........(1)
`m xx 4 cos 30 = m v_(1)+m v_(2)`
`v_(2) + v_(1) = 2sqrt(3)`.........(2)
`v_(2) = (3sqrt(3))/(2)m//s , v_(2) = (3sqrt(3))/(2)[cos 30 hat(i)-sin 30 hat(j)]`
`v_(2) = ((9)/(4)hat(i)-(3sqrt(3))/(4) hat(j))`