Question

A projecile can have the same range r for two angles of projection. If t1 and t2 be the time of flights in the two cases, then the product of the two times of flights is directly proportional to :

Answer 1

Let the angle of Projection be θ₁ and θ₂.

∴ R = u²Sin2θ₁/g

For second case, R = u²Sin2θ₂/g

Since, both the ranges are equal,

Therefore, u²Sin2θ₁/g = u²Sin2θ₂/g

⇒ u²Sin2θ₁ = u²Sin2θ₂

⇒ u²Sin2θ₁ = u²Sin2θ₂

θ₁ = 90 - θ₂

Now,

t₁ × t₂ = 2usinθ₁/g × 2usinθ₂/g

⇒ t₁ × t₂ = 4u²sinθ₁sinθ₂/g²

⇒t₁ × t₂ = 4u²sinθ₁cosθ₁/g²

∴ t₁ × t₂  = 2u²sin2θ₁/g²

∴ t₁ × t₂ = 2/g × R

∵ g is constant.

∴ t₁ × t₂ ∝ 2R

Hope it helps.