Question

A particle starts sliding down a frictionless inclined plane. If S_{​​​​​n} is the distance travelled by it from time t=n-1 secto t=n sec, find the ratio of S_{​​​​​​n}/S_{​​​​​​n+1}.

Explain in detail.

Answer 1

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)