Question

The displacement of a body is given by $s = \dfrac{1}{2}g{t^2}$, where g is acceleration due to gravity. The velocity of the body at any time t is
$\left( A \right)\dfrac{{g{t^3}}}{6}$
$\left( B \right)\dfrac{{g{t^2}}}{6}$
$\left( C \right)gt$
$\left( D \right)\dfrac{{gt}}{2}$

Answer 1

- Hint: In this question use the equation of motion and compare the given equation with these equations of motion and use the concept that displacement and distance are the two faces of a single coil i.e. they are not same so use these concepts to reach the solution of the question.

Formula used – $v = u + at$, $s = ut + \dfrac{1}{2}a{t^2}$

Complete step-by-step solution -
As we know that the velocity of any particle is the ratio of the distance covered by the particle in time interval t, and it is often measured in m/s.
Therefore v = (d/t), where v = velocity in m/s, d = distance covered by the particle in meters and t = time taken by the particle during the journey in seconds.
But in the given problem displacement is given not the distance covered by the particle as displacement is the distance between its initial point and the final point and the distance is the total distance covered by the particle in a particular time interval.
So we cannot calculate the velocity of the body directly; we have to use equations of law of motion.
Given equation:
$s = \dfrac{1}{2}g{t^2}$
Now as we know that first and second equation of motion which is given as
$v = u + at$............... (1)
And
$s = ut + \dfrac{1}{2}a{t^2}$................. (2)
Where, v = final velocity.
               u = initial velocity.
               a = acceleration due to gravity.
               s = distance
               t = time.
So compare the given equation with equation (2) we have,
$ \Rightarrow u = 0{\text{ and }}a = g$
Now substitute this value in equation (1) we have,
$ \Rightarrow v = 0 + gt$
$ \Rightarrow v = gt$
So this is the required velocity of the body at time t.
So this is the required answer.
Hence option (C) is the correct answer.

Note: Whenever we face such types of questions always recall the equations of motion which is stated above so when we compare the given equation with standard equation we got initial velocity of the body is zero and acceleration due to gravity (a) is same as earth gravity (g), so substitute this value in equation (1) we will get the required answer.