Question

Ratio of the ranges of the bullets fired from a gun (of constant muzzle speed) at angle tetha, 2 tetha& 4tetha is
found in the ratio x: 2:2, then the value of x will be (Assume same muzzle speed of bullets is
​

Answer 1

Horizontal range of projectile is given by, $R=\frac{u^2sin2\theta}{g}$

if u remains constant then, horizontal range is directly proportional to $sin2\theta$.

here given, Ratio of the ranges of the bullets fired from a gun (of constant muzzle speed) at angle $\theta,2\theta$ and $4\theta$

found in the ratio x : 2 :2 respectively.

it means, $\frac{x}{sin\theta}=\frac{2}{sin2\theta}=\frac{2}{sin4\theta}$

taking $\frac{2}{sin2\theta}=\frac{2}{sin4\theta}$

or, $sin2\theta=sin4\theta$

or, $sin2\theta=2sin2\theta cos2\theta$

$sin2\theta=0,cos2\theta=\frac{1}{2}$

we can't take $sin2\theta$ = 0, because value of expression will be undefined.

or, $cos2\theta=\frac{1}{2}=cos\frac{\pi}{3}$

so, $\theta=\frac{\pi}{6}$

now $\frac{x}{sin\theta}=\frac{2}{sin2\theta}$

or, x/sin(π/6) = 2/sin(π/3)

or, x/(1/2) = 2/(√3/2)

or, x = 2/√3

hence, answer is x = 2/√3