Here, we have two cases, one with friction and one without friction. The distance travelled by the block in both cases is same.
Let the distance travelled by the block be “s” the mass of the block be “m”, the coefficient of friction be “μ” and the acceleration due to gravity be “g”. The angle made by the incline with the ground is θ .
Case:1 No friction
The equation of motion is
Here, initial velocity, u = 0
Acceleration in the incline frame of reference is, a=g(sinθ)
Case:2 With Friction
Here, we first calculate the acceleration.
From the free body diagram,
Using this acceleration in the equation of motion. The time taken is given by t’.
We know t’=pt
Substituting the values of t’, and t
Squaring both sides and simplifying,
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