Power is defined as the measure of how quickly or slowly a work is being done. Work defines an amount of energy that we supplied, transferred or converted to be later retrieved or by means of natural physical systems. Work no doubt an important physical entity that allows us to understand underlying principles of the physical world, but we often need to know how quickly or slowly the work is being done.
The power is the main thing to determine how the work is being done. Power is the measure of how much work is done per unit time or the rate of change of work done. It is the total work done by a force divided by the time interval during the work is done.
Once you determine the total work done in a given interval of time, it's easier to find power. Let's consider a simplest case of a constant net force doing some work. Let \(s\) is the total displacement covered by a body under the action of a constant force of magnitude \(F\) in the direction of displacement in time interval \(\Delta t = {t_2} - {t_1}\). Then, the power is the total work done divided by the time interval that is,
\[P = \frac{{Fs}}{{{t_2} - {t_1}}} = \frac{W}{{\Delta t}}\]
In case of a variable force or the displacement is not along a straight line we can find the rate of change of work done at any particular instant. Let \(dW\) is a very small work done in a very small time interval \(dt\) in which \(\Delta t\) approaches zero. Note that in a very small time \(dt\) the force can be taken as constant and the displacement is also along a straight line, so the power at any instant is,
\[P = \frac{{dW}}{{dt}} \tag{4} \label{4}\]
The SI unit of power is watt denoted by \(\text{W}\) which is \(\text{J/s}\) or \(\text{kg} \cdot \text{m}^2/\text{s}^3\). Note that the unit of power and work done denoted by the same letter \(\text{W}\) are not the same things. The SI unit of power is named after the name of an inventor his Watt steam engine by improving Newcomen steam engine. A frequently used non-si unit of power is horsepower (hp) also developed by James Watt, and it is
\[1\text{hp} = 746 \text{W}\]
In the early times of steam engine, the steam engine replaced the work done by horses. They tried to figure out how much work is done by a steam engine in terms of work done by a horse for a long period. A trained athlete can easily overcome 1hp but not for longer periods of time. The horse power to watt relationship was overestimated for the work done by horses for a long period of time by an appreciable amount .
Only the work that has been done is not the main thing on understanding what's going on but it is important to see this considering the time it takes to do the given amount of work. For example, to bring your car into motion quickly from start to motion, you need to do some amount of work at some time interval. In that case the thing you apply to the car is the power.
