Question

If $h_1$ and $h_2$ be the greatest heights in the two paths of projectile for a given range, then the range is given by:

• A. $\sqrt{h_1{h}_2}$
• B. $\frac{h_1+h_2}{2}$
• C. $4\sqrt{h_1{h}_2}$
• D. $\frac{2\sqrt{h_1{h}_2}}{h_1+h_2}$

Answer 1

The correct option is C $4\sqrt{h_1{h}_2}$

For the same range of projectile fired with velocity u, there are two angle of projection, $\theta$ and $90-\theta$ with horizontal. Corresponding height
$h_1=\frac{u^{2}si{n}^2\theta }{2g}{h}_2=\frac{u^{4}si{n}^2(90-\theta )}{2g}=\frac{u^{2}co{s}^2\theta }{2g}{h}_1{h}_2=\frac{u^{2}si{n}^2\theta co{s}^2\theta }{4g^2}$
Multiply and divide RHS by 4
$h_1{h}_2=\frac{4u^{4}si{n}^2\theta co{s}^2\theta }{16g^2}16h_1{h}_2=\frac{(2u^{2}sin\theta cos\theta {)}^2}{g^2}16h_1{h}_2={\left[\frac{u^{2}si{n}^2\theta }{g}\right]}^{2}R=\frac{u^{2}sin2\theta }{g}16h_1{h}_2=R^{2}R=4\sqrt{h_1{h}_2}$