A particle starts from rest from top of a smooth inclined plane whose base has fixed length b. What should be the angle θ (in degrees) of plane to horizontal so that time of fall is minimum.

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TestHub · Accepted Answer

Length of the incline: l = b sec θ ; acceleration down the smooth plane: a = g sin θ l = 2 1 ​ a t 2 ⇒ b sec θ = 2 1 ​ g sin θ ⋅ t 2 t = g sin θ cos θ 2 b ​ ​ = g sin 2 θ 4 b ​ ​ t is minimum when sin 2 θ is maximum = 1 , i.e. 2 θ = 9 0 ∘ θ = 4 5 ∘

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