Question

If the velocity v increases linearly with time so that $v=kt$, where $k=4m/{{s}^{2}}$. Find the distance covered by it in the first three seconds.

Answer 1

This is an example of motion in a single dimension, or linear motion. In order to solve such problems, we can use the equations of motion with which we can easily generate our required solution. We can substitute the given values in the equation and solve it.

Formulas used:
Second equation of motion: If u is the initial velocity, t is the time and a is the acceleration of motion then the distance travelled by the body‘s’ can be defined as
$s=ut+\dfrac{1}{2}a{{t}^{2}}$

Complete step by step answer:
In the question, we are given that the velocity v and it increases linearly with time. This condition is expressed in the form $v=kt$.
We are given that $k=4m/{{s}^{2}}$. This is the acceleration of the body.
To find the distance travelled by the body, we can use the second equation of motion, that is
$s=ut+\dfrac{1}{2}a{{t}^{2}}$
The initial velocity u is zero.
We have to find the distance travelled in the first three seconds hence time=3s.
Upon substituting values we get,
$s=\dfrac{1}{2}\times 4\times {{3}^{2}}=18m$

Hence, in the first three seconds the body travels a distance of 18m.

Note: The change in velocity with respect to time creates acceleration and it can be either positive or negative. An increase in velocity causes positive acceleration and a decrease in velocity causes negative acceleration. The value of acceleration should be taken accordingly, during calculation. Students must be careful about this point.