A block is projected upward with speed u on a rough incline of angle θ. Coefficient of kinetic friction is μ and tanθ > μ. Let t₁ be the time to reach the highest point and t₂ the time to return from the highest point to the initial position. The ratio t₂/t₁ is:
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Question

A block is projected upward with speed u on a rough incline of angle θ. Coefficient of kinetic friction is μ and tanθ > μ. Let t₁ be the time to reach the highest point and t₂ the time to return from the highest point to the initial position. The ratio t₂/t₁ is:

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G1 Physics · Accepted Answer

Upward acceleration magnitude is a₁ = g(sinθ + μcosθ), so t₁ = u/a₁. Distance travelled upward is s = u²/(2a₁). During downward motion, acceleration magnitude is a₂ = g(sinθ - μcosθ). From s = ½a₂t₂², t₂ = u/√(a₁a₂). Hence t₂/t₁ = √(a₁/a₂) = √[(sinθ + μcosθ)/(sinθ - μcosθ)].