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 \(t_1t_2\) is

\(\begin{array}{l} Range\,will\,be\,same\,for\,time\,{t_1}\,and\,{t_2},\,so\,\\ angles\,of\,projrection\,will\,be\,'\theta '\,\& '{90^ \circ } - \theta '\\ {t_1} = \frac{{2u\,\sin \,\theta }}{g}{t_2} = \frac{{2u\,\sin \,\left( {{{90}^ \circ } - \theta } \right)}}{g}\,and\\ \,\,\,\,\,\,\,R = \,\frac{{{u^2}\sin \,2\theta }}{g}\\ {t_1}{t_2} = \frac{{4{u^2}\sin \theta \cos \theta }}{{{g^2}}} = \frac{2}{g}\left[ {\frac{{2{u^2}\sin \theta \cos \theta }}{g}} \right]\\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = \frac{{2R}}{g} \end{array}\)