A particle is projected horizontally with speed u from a point at height h above the foot O. It strikes normally a straight inclined plane passing through O and making angle α with the horizontal. The required relation between u, h and α is:
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Question

A particle is projected horizontally with speed u from a point at height h above the foot O. It strikes normally a straight inclined plane passing through O and making angle α with the horizontal. The required relation between u, h and α is:

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G1 Physics · Accepted Answer

The projectile trajectory is y = h - gx²/(2u²). At impact, vₓ = u, vᵧ = -gt = -gx/u. For normal impact on a plane of slope tanα, the velocity must have slope vᵧ/vₓ = -cotα. Hence, -gx/u² = -cotα, so x = u²cotα/g. The impact point also lies on the plane: h - gx²/(2u²) = x tanα. Substitute x: h - [g/(2u²)] [u⁴cot²α/g²] = u²/g. Thus, h = u²/g + u²cot²α/(2g) = [u²/(2g)](2+cot²α). Therefore, u² = 2gh/(2+cot²α). Using 2+cot²α = (1+sin²α)/sin²α, we get u² = 2gh sin²α/(1+sin²α).