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)`.

Explanation

Let a body starting from rest travels a distance of .h. m from A to B during which it acquires a velocity V as shown in the figure .Its velocity becomes 2V at point C.

`v^(2)-u^(2)=2 as`
Apply the above formula for both the cases we get,
`V^(2)-0^(2)2gh" "(2V)^(2)V^(2)=2gh^(1)`
`impliesV^(2)=2gh` ......(1)
`implies3V^(2)=2gh^(1)`..........(2)
Eliminating the unconcerned terms by dividing equation (2) by equation (1) we get,
`(2)/(1)=(3V^(2))/(V^(2))=(2gh^(1))/(2gh)impliesh^(1)=3h`.