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in Physics by (49.4k points)

A particle is projected with some speed at an angle a to the horizontal from the end B of the horizontal base BC of a triangle ABC. It rises to the vertex A and after just grazing it, falls down to reach point C of the base BC. If the base angles of the triangle are β and γ, show that 4 cot α = cot β + cot γ.

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Best answer

From the figure,

cot β = \(\frac{BD}{H_{max}}\)

and cot γ =\(\frac{DC}{H_{max}}\)

cot β + cot γ =\(\frac{DC+DC}{H_{max}}\)

\(\frac{Range}{H_{max}}\)

⇒ cot β + cot γ = \(\frac{(u^2sin\,2α)/g}{(u^2sin^2α)/2g}\)

⇒ cot β + cot γ = 2 × \(\frac{2sin\,α\,cos\,α}{sin^2α }\)

cot β + cot γ = 4 cot α.

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