Momentum and kinetic energy?

Question

i noticed that the integral of the momentum with respect to the velocity gives you the kinetic energy

my question is how does integrating momentum give you energy?

i know how to do the integral, but i don't understand how total momentum equals energy

Integral m v dv = one half m v squared

Answer 1

Momentum = Integral[m(dv)]

but

"mv(dv)" does not in any way, shape or form represent momentum nor does it resemble anything close.  

Answer 2

Interesting.

I think that's a fluke?

The actual relationship is KE = P²/2m

.

Answer 3

Well its not really that surprising. If you consider, in a mechanical system the infinitesimal energy dE = F.dx, where F is force and dx is the infinitesimal distance. Thus total energy is given by integrating both sides. Now F = m(dv/dt), so we have dE = m(dv/dt).dx, converting dx into dv (or alternately you can change dv -> dx) dx = (dx/dt).dt and dv = (dv/dt).dt, so dt = (dt/dv).dv (sensibly enough), subbing this into dx, we get dx = (dx/dt).(dt/dv).dv, now combine everything

dE = m.(dv/dt).(dt/dv).(dx/dt).dv -> dE = m(dx/dt).dv = m.v.dv = p.dv

Now physically this isnt surprising either, since momentum (at least in a classical sense) of a system is in effect its speed. However, momentum is a more fundamental property of a system, since it takes into account the mass of the system etc.

What we had done above is essentially mapped the changing momenta over velocity rather than space, and this is just as valid. The integral relation occurs from how energy is defined with respect to a force that causes it (or a field).

The mathematics I used is a very "physics" way of doing math, I hope you dont mind that.