A stone projected with a velocity u at an angle with the horizont… - Vidyadip

Question

A stone projected with a velocity u at an angle $\theta$ with the horizontal reaches maximum height $H_1$. When it is projected with velocity u at an angle $\left( {\frac{\pi }{2} - \theta } \right)$ with the horizontal, it reaches maximum height $ H_2$. The relation between the horizontal range R of the projectile, $H_1$ and $H_2$ is

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Answer

(a)${H_1} = \frac{{{u^2}{{\sin }^2}\theta }}{{2g}}$ and ${H_2} = \frac{{{u^2}{{\sin }^2}(90 - \theta )}}{{2g}} = \frac{{{u^2}{{\cos }^2}\theta }}{{2g}}$

${H_1}{H_2} = \frac{{{u^2}{{\sin }^2}\theta }}{{2g}} \times \frac{{{u^2}{{\cos }^2}\theta }}{{2g}} = \frac{{{{({u^2}\sin 2\theta )}^2}}}{{16{g^2}}} = \frac{{{R^2}}}{{16}}$

$R = 4\sqrt {{H_1}{H_2}} $

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