Question

A shell is fired from a fixed artillery gun with an initial speed $u$ such that it hits the target on the ground at a distance $R$ from it. If $t_1\text{and}{t}_2$ are the values of the time taken by it to hit the target in two possible ways, the product $t_1{t}_2$ is :

• A. $\frac{2R}{g}$
• B. $\frac{R}{g}$
• C. $\frac{R}{2g}$
• D. $\frac{R}{4g}$

Answer 1

The correct option is A $\frac{2R}{g}$

Range of the projectile will be the same for complimentary angles of projection.

$\text{Let}{\theta }_1=\theta \text{and}{\theta }_2={90}^{\circ }-\theta$

Time of flight of the projectile is given by,

$t_1=\frac{2u\sin \theta }{g}$

$t_2=\frac{2u\sin ({90}^{\circ }-\theta )}{g}=\frac{2u\cos \theta }{g}$

$Now,t_1{t}_2=\left(\frac{2u\sin \theta }{g}\right)\left(\frac{2u\cos \theta }{g}\right)$

$\Rightarrow t_1{t}_2=\frac{2}{g}\left(\frac{u^2\sin 2\theta }{g}\right)=\frac{2R}{g}$

Hence, option $\left(D\right)$ is correct.