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 $t_1$ 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. $\frac{R}{4g}$
• B. $\frac{R}{g}$
• C. $\frac{R}{2g}$
• D. $\frac{2R}{g}$

Answer 1

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

$R$ will be same for $\theta$ and ${90}^{\circ }-\theta$.

Time of flights,

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

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

$\text{Now,}{t}_1{t}_2=\left(\frac{2u\sin \theta }{g}\right)\left(\frac{2u\cos \theta }{g}\right)$

$=\frac{2}{g}\left(\frac{u^2\sin 2\theta }{g}\right)=\frac{2R}{g}[\because R=\frac{u^2\sin 2\theta }{g}]$

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