In physics, we often want to understand how an object's motion changes. We can use forces and acceleration, but there's often a more direct way: by looking at energy. The work-energy theorem provides a powerful bridge between the concepts of work and kinetic energy.

Essentially, the total work done by all forces acting on an object results in a change in its speed, and therefore a change in its kinetic energy. Let's see how this relationship is derived.

The derivation starts with familiar ground: Newton's second law. Consider a constant net force, $F_{net}$, acting on an object of mass $m$, causing it to accelerate over a distance $d$.

The work done by this net force is the force multiplied by the distance.

To connect this to velocity, we can use a kinematic equation that relates acceleration, distance, and velocity, and solve it for acceleration.

Now, we can substitute this expression for acceleration back into our equation for work.

The distance $d$ cancels out, leaving us with the final theorem. The work done is equal to the change in the quantity $\frac{1}{2}mv^2$, which we define as kinetic energy (KE).

The work-energy theorem gives us a clear way to understand how forces affect an object's speed. The sign of the net work tells you exactly what happens to the kinetic energy.

For example, when you push a box from rest, you do positive work, and its kinetic energy increases. When friction acts on a sliding puck, it does negative work, and the puck's kinetic energy decreases until it stops. If a satellite is in a stable circular orbit, the gravitational force is always perpendicular to its motion, so the net work done is zero, and its speed remains constant.

By focusing on the transfer of energy, the work-energy theorem offers a different and often simpler perspective for analysing motion.