Question

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 $x-axis$ (see figure) with a constant velocity of $(\sqrt{3}-1)m/s.$. At a particular instant when the line OA makes an angle of $45^{\circ }$ with the $x$-axis, a ball is thrown along the surface from the origin $O$. Its velocity makes an angle $\varnothing$ 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 $\varnothing =4\theta /3.$

Answer 1

Let A stands for trolley and $B$ for ball.
Relative velocity of $B$ with respect to $A(v_{BA})$ should be along $OA$ for the ball to hit the trolley.
Hence, $v_{BA}$ will make an angle of $45^{\circ }$ with positive x-axis.
$\theta =45^{\circ }$
(b) Let $v$= absolute velocity of ball.
$\varnothing =\frac{4\theta }{3}=\frac{4}{3}(45^{\circ })=60^{\circ }\to$ with x-axis
$\therefore v_B=(v\cos \theta )\hat{i}+(v\sin \theta )\hat{j}=\frac{v}{2}\hat{i}+\frac{\sqrt{3}v}{2}\hat{j}$
$v_A=(\sqrt{3}-1)\hat{j}$
$\therefore v_{BA}=\frac{v}{2}\hat{i}+(\frac{\sqrt{3}v}{2}-\sqrt{3}+1)\hat{j}$
Since, $v_{BA}$ is at $45^{\circ }$
$\therefore \frac{v}{2}+\frac{\sqrt{3}v}{2}-\sqrt{3}+1$ or $v=2m/s$