Question

Prove that the maximum horizontal range is 4 times of the height attained by the particle.

Answer 1

$\because RangeR=\frac{u^{2}sin2\theta }{g}$
and maximum height $H=\frac{u^{2}si{n}^2\theta }{2g}$
$\because \frac{R}{H}=\frac{\frac{u^{2}sin2\theta }{g}}{\frac{u^{2}si{n}^2\theta }{2g}}=\frac{2sin2\theta }{si{n}^2\theta }$
or $\frac{R}{H}=\frac{2\times 2cos\theta .sin\theta }{si{n}^2\theta }=\frac{4cos\theta }{sin\theta }=\frac{4cos\theta }{sin\theta }$
or $R=H4\frac{cos\theta }{sin\theta }=4H.cot\theta$
$\because$ For maximum range $\theta ={45}^0$
$\therefore cot\theta =1$
$\therefore R=4H$