Applications of Calculus

By now you will have learned to recognize the circumlocutions I use in the sections without calculus in order to introduce calculuslike concepts without using the notation, terminology, or techniques of calculus. It will therefore come as no surprise to you that the rate of change of momentum can be represented with a derivative,

And of course the business about the area under the F - t curve is really an integral, Δp_total = R F_totaldt , which can be made into an integral of a vector in the more general three-dimensional case:

Δp_total = ∫F_totaldt .

In the case of a material object that is neither losing nor picking up mass, these are just trivially rearranged versions of familiar equations, e.g., F = mdv/dt rewritten as F = d(mv)/dt. The following is a less trivial example, where F = ma alone would not have been very easy to work with.

Rain falling into a moving cart

Finally we note that there are cases where F = ma is not just less convenient than F = dp/dt but in fact F = ma is wrong and F = dp/dt is right. A good example is the formation of a comet's tail by sunlight. We cannot use F = ma to describe this process, since we are dealing with a collision of light with matter, whereas Newton's laws only apply to matter. The equation F = dp/dt, on the other hand, allows us to find the force experienced by an atom of gas in the comet's tail if we know the rate at which the momentum vectors of light rays are being turned around by reflection from the atom.