Question

A time varying power $P=2t$ is applied on a particle of mass $m$. Find kinetic energy and velocity of the particle as a function of time.

Answer 1

1. Power is defined as the rate of doing work.
2. It is the amount of energy consumed per unit of time.
3. Power supplied to the particle is the rate of change of its kinetic energy.

Step1: Given data

Power as a function of time, $P=2t$

Mass of the particle = $m$

Step 2: Formula used and calculation

We know that, power supplied to the particle is the rate of change of its kinetic energy, where K is kinetic energy

$$\begin{gathered} P=\frac{dK}{dt} \\ \Rightarrow \frac{dK}{dt}=2t \\ \Rightarrow K=t^2-----\left(1\right) \end{gathered}$$

Hence, the kinetic energy of the particle as a function of time is $K=t^2$.

We know that, where v is the velocity

$K=\frac{1}{2}m{v}^2$

Putting the value from $eq\left(1\right)$

$$\begin{gathered} \frac{1}{2}m{v}^2=t^2 \\ \Rightarrow v=\sqrt{\frac{2}{m}}t \end{gathered}$$

Hence, the velocity of the particle as a function of time is $v=\sqrt{\frac{2}{m}}t$.