Question

A particle is moving in a straight line with constant acceleration $a$ and initial velocity ${v_0}$. What will be the average velocity during the first $t$ second?
A) ${v_0} + \dfrac{1}{2}at$
B) ${v_0} + at$
C) $\dfrac{{{v_0} + at}}{2}$
D) $\dfrac{{{v_0}}}{2}$

Answer 1

The average velocity of a particle in a time interval $t$ is the total displacement per unit time. So, first calculate the total displacement in the time interval. The total displacement can be calculated by using the formula $s = ut + \dfrac{1}{2}a{t^2}$ where $u$ is the initial velocity of the particle, $a$ is its constant acceleration and $t$ is the time.

Complete step by step answer:
As given in the question that the particle is moving in a straight line with constant acceleration $a$ and initial velocity ${v_0}$ and we have to calculate its average velocity during first $t$ second.
We know that the average velocity of a particle in a time interval $t$ is the total displacement per unit time. So, we have to calculate the total displacement in the time interval.
The total displacement can be calculated by using the formula $s = ut + \dfrac{1}{2}a{t^2}$ where $u$ is the initial velocity of the particle, $a$ is its constant acceleration and $t$ is the time.
So, according to the question, $u = {v_0}$
Therefore, the total displacement of the particle in first $t$ second is given by
$s = {v_0}t + \dfrac{1}{2}a{t^2}$
And we know that average velocity is given by ${v_{av}} = \dfrac{{{\text{Total Displacement}}}}{{{\text{Time taken}}}}$
Therefore, the average velocity of the particle in first $t$ second is given by
${v_{av}} = \dfrac{s}{t} = \dfrac{{{v_0}t + \dfrac{1}{2}a{t^2}}}{t} = {v_0} + \dfrac{1}{2}at$

Hence, option A is correct.

Note: We can apply the formula for displacement $s = ut + \dfrac{1}{2}a{t^2}$ only when the acceleration of the particle is constant throughout the time in consideration. This formula can be derived by both analytical and graphical methods. As velocity is a vector quantity it must have a direction. The direction of average velocity is in the direction of the total displacement.