# How was the formula for kinetic energy found?

• Ganesh Ujwal

#### Ganesh Ujwal

How was the formula for kinetic energy found, and who found it?

My questions mostly concern the history of physics. Who found the formula for kinetic energy
##E_k =\frac{1}{2}mv^{2}##
and how was this formula actually discovered? I've recently watched Leonard Susskind's lecture where he proves that if you define kinetic and potential energy in this way, then you can show that the total energy is conserved. But that makes me wonder how anyone came to define kinetic energy in that way.

My guess is that someone thought along the following lines:

Energy is conserved, in the sense that when you lift something up you've done work,
but when you let it go back down you're basically back where you started.
So it seems that my work and the work of gravity just traded off.

But how do I make the concept mathematically rigorous? I suppose I need functions ##U## and ##V##, so that the total energy is their sum ##E=U+V##, and the time derivative is always zero, ##\frac{dE}{dt}=0##.

But where do I go from here? How do I leap to either

a) ##U=\frac{1}{2}mv^{2}##
b) ##F=-\frac{dV}{dt}?##

It seems to me that if you could get to either (a) or (b), then the rest is just algebra, but I do not see how to get to either of these without being told by a physics professor.

Hhhmmm...well, I don't know what came first either...how about the formula for Work=Force x Distance? Because that along with Newton's Second Law seems to yield the equation for Kinetic Energy.

The history has not been discussed here, as far as I know. Wikipedia is a good starting point.

http://en.wikipedia.org/wiki/Conservation_of_energy

However, it doesn't answer the question, who first derived the energy-work theorem. With this I mean that from
$$m \dot{\vec{v}}=\vec{F}(\vec{r})$$
one gets
$$\frac{m}{2} (\vec{v}_2^2-\vec{v}_1^2)=\int_{t_1}^{t_2} \mathrm{d} t \dot{\vec{r}} \cdot \vec{F}(\vec{r}).$$
The right-hand side is the line integral of the force along the trajectory of the particle under influence of this force.

The energy-conservation theorem in the stricter sense only holds for forces that have a potential (or cases like the Lorentz force of electromagnetism for time-independent electromagnetic fields, where the electric field has a potential and the magnetic force is always perpendicular to the velocity and thus doesn't contribute to work). In this case you have
$$\frac{m}{2} (\vec{v}_2^2-\vec{v}_1^2)=-\int_{t_1}^{t_2} \mathrm{d} t \vec{v} \cdot \vec{\nabla} U(\vec{r})=-U(\vec{r}_2)+U(\vec{r}_1).$$
Thus we have the conservation of total energy,
$$\frac{m}{2} \vec{v}_2^2 +U(\vec{r}_2) = \frac{m}{2} \vec{v}_1^2 + U(\vec{r}_1).$$