Question

A bullet with muzzle velocity 100 m s ${}^{-1}$ is to be shot at a target 30 m away in the same horizontal line. How high above the target must the rifle be aimed so that the bullet will hit the target?

Answer 1

Horizontal range of bullet is $30$ m.
Using range formula, $R=\frac{u^2\sin 2\theta }{g}=30$
or $sin2\theta =\frac{30\times 10}{(100{)}^2}orsin2\theta =0.03$
For small $\theta ,sin\theta =\theta =tan\theta ,i.e.,2\theta =0.03$
Therefore, $\theta =0.015$ rad
In $\Delta OAP,tan\theta =\frac{AP}{OP}\Rightarrow AP=OPtan\theta$
The rifle must be aimed at an angle $\theta =0.015$ above horizontal.
Height to be aimed = $30tan\theta =30\left(\theta \right)=30\times 0.015=45cm$