7.15:
Power Expended by a Constant Force
Power is the rate at which work is done. The formula for power can be rearranged to obtain the work done if the power output during a process is known.
If the output power is changing as a function of time, the work done is the integral of the power during that time period and hence the area under the curve gives the value of work done.
When a constant force is applied on an object, the power of the object can be expressed in terms of the force acting on it and the displacement covered by the object. Here, the derivative of displacement with time is velocity. Thus, power can also be represented as the product of force and velocity.
From the equation, power can be positive, negative, or zero depending on the angle between the force and the velocity vector.
7.15:
Power Expended by a Constant Force
The relationship between work done and the time taken to do it can be explained using the concept of power. For example, several sprinters in a race may have the same velocity when they reach the finish line, therefore doing the same amount of work, but the winner does it in the least amount of time. Thus, power is defined as the rate of doing work. Since work can vary as a function of time, the average power is defined as the work done during a time interval, divided by the time interval. Average power is called instantaneous power when the time interval is close to zero.
The power involved in moving a body can also be expressed in terms of the forces acting on it. If a force F acts on a body that is displaced ds in a time dt, the power expended by the force is
where v is the velocity of the body. The fact that the limits implied by the derivatives exist for the motion of a real body justifies the rearrangement of the infinitesimals.
This text is adapted from Openstax, University Physics Volume 1, Section 7.4: Power.