Work-Energy theorem
Concrete
In the 1840's James Prescott Joule set out to verify that gravitational potential energy converts to heat in a consistent way.
Among the setups used by Joule was one of the type shown in the image. A weight is attached to a rope, the rope runs over a pulley. At the start the rope is coiled around a vertical axle that makes paddles turn in water; the water is inside a calorimeter.
The paddles are churning the water and the effect of this churning is that the temperature of the water rises. Joule confirmed that double the height gives double the temperature change. That is, Joule confirmed that the relation is linear: if the force is uniform then force multiplied by distance gives the amount of energy.
This exemplifies that it is very useful to have an equation that expresses force acting over distance.
In the case of the paddle wheel setup used by Joule the friction limits the rate at which the weight descends. Other experiments corroborate that the amount of potential energy release on descent does not depend on the velocity of descent. If an object falls without friction the velocity increases linear with time. That corroborates that for free fall also the potential energy release is linear with height.
The Work-Energy theorem has been recognized for centuries, but for a long time it was not yet stated in its current form. So the Work-Energy theorem is both old and new; a history of twists and turns. For the rest of this exposition that history will be ignored, making it seem as if the theorem was always clearly recognized.
Derivation
In the following exposition the Work-Energy theorem is derived twice. First in the way it could have been derived at the earliest opportunity. This derivation can also be seen as an exercise in economy; it uses the barest minimum of mathematical tools that will accomplish the job. The value of a minimal derivation is this: by removing everything that is *not* necessary it is demonstrated what the essence of the theorem is.
We have the following two expressions to describe motion of a point that is subject to uniform acceleration
$$ v = v_0 + at \qquad (1.1) $$
$$ s = s_0 + v_0t + \frac{1}{2}at^2 \qquad (1.2) $$
(1.1) is the derivative of (1.2), in that sense the two are not independent, but it's still possible to take advantage of the fact that there are two equations there.
If an object is subject to uniform acceleration $a$ over a distancs $(s - s_0)$ then how much change of velocity will that cause?
So, we want to start with acceleration and distance traveled as a given, and obtain velocity. That means we need to work towards an expression that contains position, acceleration and velocity, but not time.
We obtain an expression for '$t$' from (1.1), and substitute that into (1.2). After that substitution: while time is implicitly still present (velocity and acceleration are time derivatives), there is no explicit time.
$$ s = s_0 + v_0\frac{(v-v_0)}{a} + \tfrac{1}{2} a \frac{{(v-v_0)}^2}{a^2} \qquad (1.3