Derivative of momentum with respect to position
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Momentum is the instantaneous product of mass and velocity, [math]\rho=mv[/math], rather than an integral of mass with respect to velocity.
Similarly kinetic energy is an instantaneousquantity equal to [math]\tfrac12mv^2[/math] or, if you prefer, [math]\tfrac12\rho v[/math], rather than an integral of mass with respect to velocity.
If mass were a function of velocity, which it is not, you might want to integrate to get a total momentum. In the same way that you can integrate velocity with respect to time to get a total distance. You might also want to get a total momentum by summing the individual momentums across a collection of particles: in the limit this would be an integration, but with respect to position rather than velocity.
The leading entity that comes up when you search for things that are dependent on velocity cubed is Wind Power generated by turbines which varies in proportion to the cube of the wind speed.
Similarly kinetic energy is an instantaneousquantity equal to [math]\tfrac12mv^2[/math] or, if you prefer, [math]\tfrac12\rho v[/math], rather than an integral of mass with respect to velocity.
If mass were a function of velocity, which it is not, you might want to integrate to get a total momentum. In the same way that you can integrate velocity with respect to time to get a total distance. You might also want to get a total momentum by summing the individual momentums across a collection of particles: in the limit this would be an integration, but with respect to position rather than velocity.
The leading entity that comes up when you search for things that are dependent on velocity cubed is Wind Power generated by turbines which varies in proportion to the cube of the wind speed.
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