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If a stone is to hit at a point which is at a distance $d$ away and at a height $h$ above the point from where the stone starts, then what is the value of initial speed $u$ if the stone is launched at an angle $\theta $ ?
$\frac{g}{{\cos \,\theta }}\,\sqrt {\frac{d}{{2\left( {d\,\tan \,\theta \, - \,h} \right)}}} $
$\frac{d}{{\cos \,\theta }}\,\sqrt {\frac{g}{{2\left( {d\,\tan \,\theta \, - \,h} \right)}}} $
$\,\sqrt {\frac{{g{d^2}}}{{h\,{{\cos }^2}\,\theta }}} $
$\,\sqrt {\frac{{g{d^2}}}{{\left( {d\ -\ h} \right)}}} $
Solution
$h=(u \sin \theta) t-\frac{1}{2} g t^{2}$
$\quad d=(u \cos \theta) t$
or $\quad t=\frac{d}{u \cos \theta}$
$\therefore \quad h=u \sin \theta \cdot \frac{d}{u \cos \theta}-\frac{1}{2} g \cdot \frac{d^{2}}{u^{2} \cos ^{2} \theta}$
$\therefore \quad u=\frac{d}{\cos \theta} \sqrt{\frac{g}{2(d \tan \theta-h)}}$
