Work is physical quantity and is defined as the dot product
of force and displacement. This concept is valid when the applied force is
constant. If the applied force is variable, we shall calculate work done for
each part of the force by multiplying it with displacement and adding that
entire small works together. This can be done with a mathematical process
called integration.

If the applied force is variable, we shall integrate the
force and displacement equation to get the total work done in the process.

We can derive mathematical form of work done with respect to
variable force, using the definition of kinetic energy. We know the
mathematical form of kinetic energy. Let us differentiate the equation with
respect to time. Being mass of a physical quantity is constant, we can write it
outside the integration and we need to integrate the square of velocity with
respect to time.

By simplifying the equation further, we can get that change
in kinetic energy is the product of force and change in time. This is only for
a small magnitude of kinetic energy in a small interval of time.

By competing the required steps, we can get the equation for
change in kinetic energy. We also know that work done is nothing but change in
kinetic energy. It means that the change in kinetic energy is stored in the
form of work done.

To get the total work done, we shall integrate the equation.It is shown in detail in the following video lesson.

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