Three balls are kicked from the ground level at some angles above horizontal with different initial speeds. All three balls reached in their paths the samemaximum height. Ball A covered the total horizontal range D; ball B covered the total horizontal range 1.5D and ball C covered total horizontal range 2D when they hit the ground. Which of these three balls spent the greatest time in flight? (section 4.3) They all have the same time of flight ОА OB oc Band A and B

Time of flight A is greatest

Explanation:

Let u₁, u₂, u₃ be their initial velocity and θ₁, θ₂ and θ₃ be their angle of projection. They all achieve a common highest height of H.

So

H = u₁² sin²θ₁ / 2g

H = u₂² sin²θ₂ / 2g

H = u₃² sin²θ₃ / 2g

On the basis of these equation we can write

u₁ sinθ₁ = u₂ sinθ₂=u₃ sinθ₃

For maximum range we can write

D = u₁² sin2θ₁ / g

1.5 D = u₂² sin2θ₂ / g

2 D = u₃² sin2θ₃ / g

1.5 D / D = u₂² sin2θ₂ / u₁² sin2θ₁

1.5 = u₂ cosθ₂ / u₁ cosθ₁ (since, u₁ sinθ₁ = u₂ sinθ₂)

u₂ cosθ₂ >u₁ cosθ₁

u₂ sinθ₂ < u₁ sinθ₁

2u₂ sinθ₂ / g < 2u₁ sinθ₁ / g

Time of flight B < Time of flight A

Similarly we can prove

Time of flight C < Time of flight B

Hence Time of flight A is greatest.