Question

Write the formula for distance traveled in n^th second.

Answer 1

Step 1: Given

$S=ut+\frac{1}{2}a{t}^2$

Here,

$S$ is the distance traveled.

$u$ is the initial velocity.

$t$ is the time.

$a$ is the acceleration

Step 2: For distance traveled in n and n-1 second

Let,

Initial velocity = $u$

Acceleration = $a$

Distance traveled in n second = $S_n$

By the second equation of motion,

Distance traveled in $t$ seconds,

$S=ut+\frac{1}{2}a{t}^2$

Distance traveled in n seconds,

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

Distance traveled in (n-1) seconds,

$S_{(n-1)}=u(n-1)+\frac{1}{2}a{(n-1)}^2$ ----- (ii)

Step 3: Formula for distance traveled in n and n-1 second

To calculate the distance traveled in n^th second,

$S_{n^{th}}=S_n-S_{(n-1)}$ ----- (iii)

Putting values of $S_n$ and $S_{(n-1)}$ from equations (i) and (ii)

$S_{n^{th}}=un+\frac{1}{2}a{n}^2-\left[u\right(n-1)+\frac{1}{2}a{(n-1)}^2]$

Simplifying the above equation,

$$\begin{gathered} S_{n^{th}}=u+\frac{1}{2}a(2n-1) \\ {S}_{n^{th}}=u+a(n-\frac{1}{2}) \end{gathered}$$

Therefore, the equation for distance traveled in n^th second = $S_{n^{th}}=u+a(n-\frac{1}{2})$.