On a frictionless horizontal surface , assumed to be the ` x-y` plane , a small trolley `A` is moving along a straight line parallel to the `y-axis `( see figure) with a constant velocity of `(sqrt(3)-1) m//s ` . At a particular instant , when the line `OA` makes an angle of `45(@)` with the `x - axis ` , a ball is thrown along the surface from the origin `O`. Its velocity makes an angle `phi` with the `x -axis and it hits the trolley .
(a) The motion of the ball is observed from the frame of the trolley . Calculate the angle `theta` made by the velocity vector of the ball with the ` x-axis in this frame .
(b) Find the speed of the ball with respect to the surface , if ` phi = (4 theta )//(4)`.

Answer

(a) `45^(@)`, (b) `2m//sec`

Explanation

(a) To hit trolley `vecV_(BA)` must be along `OA`, so `theta = 45^(@) & phi = (4 theta)/(3) rArr phi = 60^(@)`
`{:((b)vecV_(B),vecV_(B)-vecV_(A),,rArr,vecV_(B)=vecV_(BA)+vecV_(A),):}`
`{:(V_(B(x))=V_(BA(x)),,rArr,V_(B)cos 60^(@)=V_(BA)cos 45^(@),):}`
`(v_(B))/(2) = (v_(BA))/(sqrt(2)) rArr v_(B) = sqrt(2) v_(BA) ...(1)`

`v_(B(y)) = v_(BA(y)) +v_(A)`
`v_(B)sin 60^(@) = v_(BA) sin 45^(@) +v_(A)`
`v_(B) (sqrt(3))/(2) = (v_(BA))/(sqrt(2)) +(sqrt(3)-1) ..(2)`
form (1) & (2)
`v_(B) (sqrt(3))/(2) = (v_(B))/(2) +(sqrt(3)-1)`
`rArr v_(B) = ((sqrt(3-1)))/(2) -(sqrt(3)-1) rArr v_(B) =- 2m//s`