How was the formula for kinetic energy found?

[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.

 

[gsal]

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.

 

[DrClaude]

This has been discussed many times before on PF. See for instance https://www.physicsforums.com/threa...c-energy-1-2mv-2-derived.124750/#post-1021524

 

[vanhees71]

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,
$$\frac{m}{2} \vec{v}_2^2 +U(\vec{r}_2) = \frac{m}{2} \vec{v}_1^2 + U(\vec{r}_1).$$

 

[PJC01]

The formula for kinetic energy was first derived by the mathematician and physicist Gottfried Leibniz in the late 17th century. He developed the concept of vis viva, or living force, which he defined as the product of mass and velocity squared. This was later renamed kinetic energy by the physicist William Thomson (Lord Kelvin) in the 19th century.

Leibniz's formula for kinetic energy, ##E_k =\frac{1}{2}mv^{2}##, was based on his understanding of motion and his work on the principle of conservation of energy. He believed that energy could not be created or destroyed, only transformed from one form to another. In his work, he also considered the concept of force and its relationship to motion.

The idea that kinetic energy is proportional to mass and velocity squared can also be traced back to the work of Galileo and Newton. Galileo's experiments with inclined planes and rolling objects showed that the distance an object traveled was proportional to the square of its velocity. Newton's second law of motion, ##F=ma##, also showed that the force acting on an object is directly proportional to its mass and acceleration.

However, it was Leibniz who first combined these ideas and derived the formula for kinetic energy. Over the years, the formula has been refined and expanded upon by other scientists, including James Joule and Hermann von Helmholtz.

In summary, the formula for kinetic energy was found through a combination of theoretical reasoning and experimental evidence. It was first derived by Leibniz, but has been refined and developed by many other scientists since then.