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

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 December 20th, 2014, 02:02 AM #1 Senior Member   Joined: Aug 2014 From: India Posts: 355 Thanks: 1 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.
 December 20th, 2014, 02:45 AM #2 Math Team     Joined: Jul 2011 From: North America, 42nd parallel Posts: 3,372 Thanks: 233 Kinetic energy of an object depends on mass and velocity of that object , that's called momentum. Kinetic energy is the antiderivative of momentum of a constant mass with respect to it's velocity. I think Newton worked out the details but i'm not sure , maybe it was J. P. Joules? For constant mass $m$ it could look like this. $\int{m dv}= mv +c_0$ So the antiderivative of a constant mass with respect to velocity is momentum. Where $c_0$ is the initial momentum. If the initial velocity is 0 then initial momentum is 0 Integrate again , $\int{mv dv}= \frac{1}{2} mv^2 + c_1$ So the antiderivative of momentum of a constant mass with respect to velocity is kinetic energy. Where $c_1$ is the initial kinetic energy , but we just defined initial velocity to be zero so initial kinetic energy is 0 as well , $c_1= 0$ and we get the familiar $K_e= \frac{1}{2} mv^2$ Without the 'annoying' $c_1$
 December 21st, 2014, 04:46 AM #3 Senior Member   Joined: Nov 2013 From: Germany Posts: 179 Thanks: 1 Math Focus: Number Theory History was is written ..... I love this kind of history!
 January 5th, 2015, 01:58 AM #4 Senior Member   Joined: Apr 2014 From: Glasgow Posts: 2,155 Thanks: 731 Math Focus: Physics, mathematical modelling, numerical and computational solutions I watched a documentary on this a long time ago. I cannot remember the details, but from what I gather there were many people who figured out the bits and pieces required for Newton to put the pieces together and come up with the laws of motion. For example, I believe it was a Willem Gravesande and Emilie du Chatelet who discovered that the height for which an object was dropped was proportional to velocity squared multiplied by its mass, $\displaystyle height \propto mv^2$, by dropping ball bearings into clay from different heights and observing the depth of the impression of the ball bearing made into the clay. Some details are found on Wikipedia: Kinetic energy - New World Encyclopedia

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