A truck starts from origin, accelerating with `a m s^-2` in positive x - axis direction. After `2 s`, a man standing at the starting point of the truck projected a ball at an angle `30^@` with velocity `v m 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).

Explanation

If the ball hits truck, for the horizontal motion of truck and ball,
Displacement of ball in horizontal = Displacement of truck in horizontal direction
`v cos 30^@ cc (t - 2) = (1)/(2) at^2`
For the vertical motion of ball, the vertical displacement should be zero.
`v sin 30^@ xx (t - 2) - (1)/(2) g (t - 2)^2 = 0`
`rArr (t - 2) = (2 v sin 30)/(g) = (v)/(g) rArr = 2 + (v)/(g)`
`(sqrt(3 v)^2)/(2) xx (v)/(g) = (1)/(2) a (2 + (v)/(g))^2`
`(sqrt(3 v)^2)/(g) = a(2 + (v)/(g))^2 rArr a = (sqrt(3 v)^2)/(g(2 + (v)/(g))^2)`.