Question

An aircraft is flying horizontally with a constant velocity $200m{s}^{-1}$, at a height 1 km above the ground. At the moment shown, a bomb is released from the aircraft and the cannon gun below fires a shell with initial speed $200m{s}^{-1}$, at some angle $\theta$.
For what value of $\theta$ will the projectile shell destroy the bomb in mid-air?

Answer 1

a. Suppose the shell destroys the bomb at time t. Then for horizontal motion,
$t(200+200cos\theta )=\sqrt{3}\times 1000$
$\therefore t(1+cos\theta )=5\sqrt{3}$ ...(i)
For vertical motion,
$\frac{1}{2}g{t}^2+\left(200sin\theta \right)t-\frac{1}{2}g{t}^2=1000$
$sin\theta t=5$ ...(ii)
From (i) and (ii), $\frac{sin\theta }{1+cos\theta }=\frac{1}{\sqrt{3}}$
On solving we get $\theta ={60}^o$