Question

Write the formula for the distance travelled in nth second.

Answer 1

Here we will focus on the concept of distance, displacement and velocity; and how they are related to each other and then we will derive the formula for the distance travelled in nth second.

Complete step by step answer:
We know that the distance is the length of the path taken by a moving body between its initial point and its final point or location. We all recognise that the displacement of a moving body is the linear distance between its original and final positions.

We all know that a straight line drawn between two points has the shortest length of all the curves connecting them, so the linear distance between two points is the shortest path between them. As a result, we can assume that when distance is shortest, it becomes displacement. Otherwise, the displacement exceeds the gap.

By definition, velocity is a vector quantity that expresses the change in distance per unit of time and direction. Its units, including speed, are determined by length and time, but direction is also factored in. The SI unit of velocity is equal to$m{{s}^{-1}}$.

Let Initial velocity of an object be $u$. Acceleration be $a$. Let the distance travelled in n second from initial position be $s$. As per the second equation of motion,
Distance travelled in t second will be
$s=ut+\dfrac{1}{2}a{{t}^{2}}...\left( 1 \right)$
So distance travelled in $n$ second
${{s}_{n}}=u\left( n \right)+\dfrac{1}{2}a{{n}^{2}}...\left( 2 \right)$
Now distance travelled in $\left( n-1 \right)$ second
${{s}_{n-1}}=u\left( n-1 \right)+\dfrac{1}{2}a{{\left( n-1 \right)}^{2}}...\left( 3 \right)$
We can find distance travelled nth second as follows
${{s}_{nth}}={{s}_{n}}-{{s}_{\left( n-1 \right)}}$
$\Rightarrow {{s}_{nth}}=u\left( n \right)+\dfrac{1}{2}a{{n}^{2}}-u\left( n-1 \right)+\dfrac{1}{2}a{{\left( n-1 \right)}^{2}}$
On simplifying the equation we get
$\therefore {{s}_{nth}}=u+\dfrac{1}{2}a(2n-1)$

Note: There are in total three equations of motion among which here we have used equation (2) because it related distance with initial velocity. Acceleration and time which makes it easier to solve the problem.