Question

A block is moving down a smooth inclined plane staring from rest at time t=0 let $S_n$ be the distance travel by the block in the interval t = n - 1 to t = n. The ratio $\frac{S_n}{S_{n+1}}is$

Answer 1

Given that,

S_n be the distance travel by the block in the interval t = n-1 and t = n.

Then,

Applying second equation of motion,

Then,

$s=ut+\frac{1}{2}a{t}^2....\left(I\right)$

Since u = 0

At t = n-1

Now, put the value of t in equation (I)

$s=0+\frac{1}{2}\times a\times {(n-1)}^2$

$s=\frac{a{(n-1)}^2}{2}$

Now, At t = n

Put the value of t in equation (I)

$s^{\prime }=\frac{1}{2}a{n}^2$

Now, the difference

$S_n=s^{\prime }-s$

$S_n=\frac{1}{2}a(n^2-{(n-1)}^2)$

$S_n=\frac{1}{2}a(n^2-n^2-1+2n)$

$S_n=\frac{1}{2}a(2n-1)......\left(II\right)$

Now, put n+1 in place of n in equation (II)

$S_{n+1}=\frac{1}{2}a(-1+2(n+1\left)\right)$

$S_{n+1}=\frac{1}{2}a(2n+1)$

Now, the ratio of S_n and S_n+1

$\frac{S_n}{S_{n+1}}=\frac{2n-1}{2n+1}$

Hence, the ratio of $\frac{S_n}{S_{n+1}}$ is $\frac{2n-1}{2n+1}$.