Question

The range R of projectile is same when its maximum heights are h₁ and h₂. What is the relation between R and h₁ and h₂

• A. $R = \sqrt{h_{1}h_{2}}$
• B. $R = \sqrt{2h_{1}h_{2}}$
• C. $R = 2\sqrt{h_{1}h_{2}}$
• D. $R = 4\sqrt{h_{1}h_{2}}$

Answer 1

Answer: (D)

$R = 4\sqrt{h_{1}h_{2}}$

Answer 2

For equal ranges body should be projected with angle $\theta$or $(90^{o} - \theta)$ from the horizontal.

And for these angles : $h_{1} = \frac{u^{2}\sin^{2}\theta}{2g}$ and $h_{2} = \frac{u^{2}\cos^{2}\theta}{2g}$

by multiplication of both height : $h_{1}h_{2} = \frac{u^{2}\sin^{2} ⥂ \theta\cos^{2} ⥂ \theta}{4g^{2}} = \frac{1}{16}\left( \frac{u^{2}\sin 2\theta}{g} \right)^{2}$

⇒ $16h_{1}h_{2} = R^{2}$ ⇒ $R = 4\sqrt{h_{1}h_{2}}$