Question

A truck starts from origin, accelerating with $am{s}^{-2}$ in positive x-axis direction. After $2s$, a man standing at the starting point of the truck projected a ball at an angle ${30}^o$ with velocity-$vm{s}^{-1}$. Find the relation between $a$ and $v$ such that the ball hits the truck. (Assume that truck is moving on horizontal plane and the man projected the ball from the same horizontal level of truck).

Answer 1

If the ball hits the truck, for the horizontal motion of truck and ball,
Displacement of ball in horizontal = Displacement of truck in horizontal direction
$vcos{30}^o\times (t-2)=\frac{1}{2}a{t}^2$
For the vertical motion of ball, the vertical displacement should be zero
$vsin{30}^o\times (t-2)-\frac{1}{2}g(t-2{)}^2=0$
$\Rightarrow (t-2)=\frac{2vsin30}{g}=\frac{v}{g}\Rightarrow t=2+\frac{v}{g}$
$\frac{\sqrt{3}}{2}\times \frac{v}{g}=\frac{1}{2}a{(2+\frac{v}{g})}^2$
$\frac{\sqrt{3}{v}^2}{g}=a{(2+\frac{v}{g})}^2\Rightarrow a=\frac{\sqrt{3}{v}^2}{g{(2+\frac{v}{g})}^2}$