question_answer

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: [JEE Main Held on 12-4-2019 Morning]

A) \[R/g\]

B) \[R/4g\]

C) \[2R/g\]

D) \[R/2g\]

Correct Answer: C

Solution :

[c] Range will be same for time \[{{t}_{1}}\And {{t}_{2}},\]so angles of projection will be \['\theta '\And '{{90}^{o}}-\theta '\]

\[{{t}_{1}}=\frac{2u\sin \theta }{g}{{t}_{2}}=\frac{2u\sin ({{90}^{o}}-\theta )}{g}\]

and\[R=\frac{{{u}^{2}}\sin 2\theta }{g}\]

\[{{t}_{1}}{{t}_{2}}=\frac{4{{u}^{2}}\sin \theta \cos \theta }{{{g}^{2}}}=\frac{2}{g}\left[ \frac{2{{u}^{2}}\sin \theta \cos \theta }{g} \right]\]\[=\frac{2R}{g}\]