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 )//(3)`.