## 2002(5M)

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 φ 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 θ 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 φ = 4θ /3  .

Solution:  (a) If the trolley A is at rest, anything that comes to hit it must be along the line OA, so the angle should be $$\theta=45^{0}$$ with the x- axis.

(b) Let the ball is given a velocity v at an angle φ = 4θ /3, where $$\theta=45^{0}$$, the angle we get is $$\phi=60^{0}$$. The observer at trolley will see it coming at angle $$\theta=45^{0}$$ so the relative velocity . $$\vec{v}_{rel} = \vec{v}_{ball} - \vec{v}_{Trolley}$$ will have same x and y components.

Here $$\vec{v}_{ball} =v cos 60^{0}\vec{i} + v sin 60^{0}\vec{j}$$

and $$\vec{v}_{Trolley} = (\sqrt{3} -1)\vec{j}$$.

Now the x- component of $$v_{rel}$$ = $$\frac{v}{2}$$

and the y -component of $$v_{rel}$$ = $$\frac{\sqrt{3}}{2}v - \sqrt{3}+1$$. Since both the components are equal, we get v= 2m/s after a little bit of manipulation.