Question

Derive an expression for the displacement travelled by a body in the $n^{th}$ second of its motion. The distance travelled by a body in the $n^{th}$ second is given by the expression ( $2 + 3 n$ ). Find the initial velocity and acceleration. Also, find its velocity at the end of $2 s$ .

Accepted Answer

Consider a body is moving with uniform acceleration $a$ with initial velocity $u$ travelled for time $t$ seconds and the displaced through displacement $2 + 3 n$ .

We know that, from the equations of motion, displacement covered in $t$ seconds is $s_{t} = u t + \frac{1}{2} a t^{2}$

Similarly, the displacement covered in $n$ seconds is $s_{n} = u n + \frac{1}{2} a n^{2}$ ....... ( $1$ )

Displacement covered in $(n - 1)$ seconds is $s_{n - 1} = u (n - 1) + \frac{1}{2} a {(n - 1)}^{2}$ .....( $2$ )

Therefore, the displacement covered in $n^{th}$ second we get by subtracting eq ( $2$ ) from eq ( $1$ )

$s_{n^{t h}} = u + \frac{a}{2} (n - 1)$ or $u + a (n - \frac{1}{2})$

As we know that the distance travelled by the body is given as $2 + 3 n$
Now as we know that if u is the initial velocity and a is the acceleration of the body then the distance travelled by it in n seconds $u + a (n - \frac{1}{2})$

So, $2 + 3 n = u + a (n - \frac{1}{2}) 2 + 3 n = u + a n - \frac{a}{2}$

Now let us compare the coefficient of n and constant terms on both sides of the above equation. We get,
$3 n = a n - - - - (1) A n d, 2 = u - \frac{a}{2} - - - - (2)$
On solving equation $(1)$ $we get a = 3 {ms}^{- 2}$ (acceleration of the body)
So, putting the value of an in equation $(2)$ We get,
$2 = u - \frac{3}{2} = u - 1.5 u = 3.5 {ms}^{- 1}$
Now as we know that according to the first law of motion.
The speed at the end of $t$ seconds is calculated as $v = u + a t$ where u is the initial velocity and a is the acceleration.
So, the velocity at the end $2 s$ will be
$v = 3.5 + 3 \times 2 = 9.5 {ms}^{- 1}$
Hence, the velocity at the end of two seconds will be $9.5 {ms}^{- 1}$ .

Derive an expression for the displacement travelled by a body in the $n^{th}$ second of its motion. The distance travelled by a body in the $n^{th}$ second is given by the expression ( $2 + 3 n$ ). Find the initial velocity and acceleration. Also, find its velocity at the end of $2 s$ .

Important Questions on Measurements and Motion

Learn and Prepare for any exam you want !

Derive an expression for the displacement travelled by a body in the nth second of its motion. The distance travelled by a body in the nthsecond is given by the expression (2+3n). Find the initial velocity and acceleration. Also, find its velocity at the end of 2 s.

Important Questions on Measurements and Motion

Learn and Prepare for any exam you want !

Derive an expression for the displacement travelled by a body in the $n^{th}$ second of its motion. The distance travelled by a body in the $n^{th}$ second is given by the expression ( $2 + 3 n$ ). Find the initial velocity and acceleration. Also, find its velocity at the end of $2 s$ .