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A particle starts sliding down a friction less inclined plane. if Sn is distance travelled by it from time t=n-1 sec to n sec then find the ratio Sn/Sn+1.

The particle starts sliding, so initial velocity=0 first, find distance travelled in n seconds, s=ut+1/2gt^2 here, u=0, t=n. then find distance travelled in n-1 seconds. the difference between those two is Sn. Sn+1 denotes the distance that is travelled by particle from time n seconds to n+1 seconds. as you have found Sn, find Sn+1. then you get the ratio. if you are still facing problems in understanding, ask where you need the details. happy to help! :)

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Use the equations of motion: s = ut + 1/2 a t^2

let Ф be the angle of the inclined plane with the horizontal. Acceleration along the inclined plane = g sin Ф

Let initial velocity at t = 0 sec, be u = 0 m/s.

distance travelled up to n-1 sec: D(n-1) = 1/2 g SinФ * (n-1)² distance travelled up to n sec. : D(n) = 1/2 * g Sin Ф * n² Distance travelled upto n+1 sec: D(n+1) = 1/2 * g Sin Ф * (n+1)²

Sn = Distance travelled from t = n -1 sec. to n sec. = 1/2 g Sin Ф * [n² - (n-1)²] = 1/2 g Sin Ф * (2n - 1) Sn+1 = distance travelled from t = n sec to n+1 sec = 1/2 g Sin Ф * (2n +1)

Ratio : Sn / Sn+1 = (2n -1) / (2n +1) = 1 - 1 / (n +1/2)