Power: Definition, Equations & Formula, Derivation, Applications, Strategies to Solve Problems

Often it is interesting to know not only the work done on an object, but also the rate at which this work is done. We say a person is physically fit if he not only climbs four floors of a building but climbs them fast. Power is defined as the time rate at which work is done or energy is transferred. The average power of a force is defined as the ratio of the work W, to the total time taken t.

The Mathematical Expression of Power is:

$P=\frac{dW}{dt}$

where $W$  = total work done

and $t$  = time taken.

If we want to calculate the average power for a particular time interval $\mathrm{\Delta }t$ (when work is done over a finite period, such as lifting an object or accelerating a vehicle), the formula used will be:

$P_{\mathrm{a}\mathrm{v}\mathrm{g}}=\frac{\mathrm{\Delta }W}{\mathrm{\Delta }t}$

For a constant force $\vec{F}$ acting on a body moving with velocity $\vec{v}$ , the power will be:

$P=\vec{F}\cdot \vec{v}=Fv\mathrm{c}\mathrm{o}\mathrm{s}\theta$

Where, $\mathit{\theta }$ is the angle between the force and velocity vectors. The SI unit of power, the watt, has dimensions ${\mathrm{M}}^1{\mathrm{L}}^2{\mathrm{T}}^{-3}$ . This formula highlights that power is maximized when force and velocity are aligned $\left(\theta =0^{\circ }\right)$  and zero when they are perpendicular $\left(\theta ={90}^{\circ }\right)$ .

 

Here is a step by step derivation using core physics fundamentals:

W=F⋅d (1)

P=tW (2)

When we substitute 1 into the equation 2, the result comes out to be:

P=t/F⋅d
But as you all know, v = d/t

So, P=F⋅v

Hence, proved!

Let’s dive into detailed aspects of the Work Energy theorem and check how is it related to Power!

Wnet​=ΔKE=21​mv₂−21​mu₂

Where,

W = Total Work Done

M = Mass of the object

V = final velocity

U = initial velocity

Now, the Formula for Power is

P=W/t​
Substitute work formula into power, and you get the answer as:

P=tΔKE

Depending on the type of energy being exerted on, power can be classified into the following categories:

1. Mechanical Power:

This is the power generated from exerting a physical force.

Formula: P = fv

where f = force and v = velocity.

Example: Car Engine, Turbine, etc.

2. Thermal Power:

This is the power generated through heat transfer.

Formula: P = Q/t

where Q = heat energy and t = time.

Example: Heater, Solar Panel, etc.

3. Electrical Power:

This is the power generated through electrical circuits.

Formula: P = vi

where v = voltage and I = electric current

Example: Bulb, motor, etc.

Power simply means how fast the energy is converted from one form to another or how quickly the work is done. Power is directly proportional to the magnitude of the force as well as the velocity of the object. In cases where the force is perpendicular to the velocity, such as the tension in a string during uniform circular motion, the power delivered will be zero because no work is done.

Given below are some of the core highlights of power which the JEE aspirants can refer to:

• Scalar Quantity: Power only has magnitude and doesn’t have any particular direction unlike force or velocity. This means that if the work is being completed faster, power generated will be higher.
• Dependence on Motion: Power is directly proportional on the motion of the object. If the object is not moving, it doesn't matter how much force is being applied because the power will always be zero.
• Energy Efficiency: A higher power cannot guarantee a better performance if the energy is being wasted. How much effectively the energy is being utilized will always be considered as an important factor while calculating the power generated.
• Proportionality: Power will always be directly proportional to the total work done and inversely proportional to the amount of time taken.

The power calculated can also vary depending on whether it is calculated based on a particular instance of time or for the average of the total time. Let’s summarize this difference for a clear understanding:

1. Instantaneous Power:

This is the Power calculated for a specific moment of time, which can be defined as:

Pinst​=V(t)⋅I(t)
where:

V(t) = voltage at time t

I(t) = current at time t

Application: AC Circuits

2. Average Power:

This is the Power calculated for the total amount of time, which can be defined as:

Pavg​=ttotal​Wtotal​​
Application: Electricity Bills