A particle is projected from a point at height h above horizontal ground with fixed speed u. It must strike a point on the ground at horizontal distance R from the vertical through the point of projection. If two projection angles θ₁ and θ₂ are possible, then which relation is correct?
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Question

A particle is projected from a point at height h above horizontal ground with fixed speed u. It must strike a point on the ground at horizontal distance R from the vertical through the point of projection. If two projection angles θ₁ and θ₂ are possible, then which relation is correct?

Options

G1 Physics · Accepted Answer

The trajectory measured from the launch point is y = x tanθ - [gx²/(2u²)]sec²θ. At the target, x = R, y = -h. Let T = tanθ. Since sec²θ = 1+T², -h = RT - [gR²/(2u²)](1+T²). Rearranging, [gR²/(2u²)]T² - RT + [gR²/(2u²) - h] = 0. The two roots are T₁=tanθ₁ and T₂=tanθ₂. Therefore, T₁+T₂ = R / [gR²/(2u²)] = 2u²/(gR), and T₁T₂ = [gR²/(2u²)-h] / [gR²/(2u²)] = 1 - 2hu²/(gR²). Hence both A and B are correct.