Question

A body starts from rest with uniform acceleration. If its velocity after n seconds is v then assuming straight line motion, its displacement in last two seconds is
A. \[\dfrac{{v(n - 1)}}{n}\]
B. \[\dfrac{{2v(n - 1)}}{n}\]
C. \[\dfrac{{2v(n + 1)}}{n}\]
D. \[\dfrac{{v(n + 1)}}{n}\]

Answer 1

Hint: Given, the motion is a straight line motion, we can use the equations of motion for one dimensional movement. The motion of a body in a straight line is called one dimensional motion. These equations relate the displacement, velocity and acceleration with respect to time.

Formula used :
The equation of motion for velocity is,
\[v = u + at\]
Where, v – final velocity
u – initial velocity
a – acceleration
t – time
To find the displacement of the object,
\[s = ut + \dfrac{1}{2}a{t^2}\]
Where, s – displacement

Complete step by step solution:
The equations of motion can describe the behaviour of the body which is moving with respect to time. These equations will relate displacement or distance with velocity and acceleration with respect to time.

Here, we are asked to find the displacement of the object. The displacement can be described as the shortest distance an object can travel through the path. The velocity can be defined as the rate of change of displacement and acceleration is given as the rate of change of velocity.

Given, u = 0 m, t = n s
To find, s = ? at t = 2 s
Consider the acceleration of the body to be ‘a’.
So, the final velocity of the object is,
\[v = u + at\]
\[\Rightarrow v = 0 + an\]
The acceleration can be written as,
\[a = \dfrac{v}{n}\]
To find the displacement s, at t=2 s,
\[s = vt + \dfrac{1}{2}a{t^2} \\ \]
\[\Rightarrow s = (v{\rm{x2}}) + \dfrac{1}{2}\left( {\dfrac{v}{n}} \right){2^2} \\ \]
\[\Rightarrow s = 2v + \dfrac{v}{{2n}}{2^2} \\ \]
\[\therefore s = \dfrac{{2v}}{n}(n - 1)\]

So, the correct answer is option B.

Note : During the last step, we got the final answer using the arithmetic progression. Since the displacement is asked for the object after 2 seconds of its motion, it gives rise to an arithmetic series. The equations of motion are used because the motion is assumed to be in a straight line.