A projectile has the same range R when the maximum height . . .

Question : A projectile has the same range R when the maximum height attained by it is either H₁ or H₂. Find the relation between R, H₁ and H₂. 

Doubt by Aakarsh

Solution : 

For the same range the angle of projections must be complimentary. 

Let θ₁=θ and θ₂=90-θ

R = u²sin2θ/g — (1) 

H1=u²sin²θ₁/2g 
H₁=u²sin²θ/2g — (2) 

H₂=u²sin²(90-θ)/2g 
H₂=u²cos²θ/2g — (3) 
[∵ sin(90-θ)=cosθ]

Multiple equation (1) and (2)

H₁×H₂=u²sin²θ/2g × u²cos²θ/2g

H₁×H₂=u²sin²θ/2g × u²cos²θ/2g H₁×H₂=[u²sinθcosθ/2g]² 
H₁×H₂=[u²(2sinθcosθ)/4g]² 
H₁×H₂=[u²sin2θ)/4g]²

H₁×H₂=[R/4]² 
[Using equation (1)] 
H₁×H₂=[R/4]² 
H₁×H₂=R²/16 
16H₁×H₂=R₂ 
OR 
R²=16H₁H₂  OR
R=4√[H₁H₂]
which is the required relation.