14.6 Applications of calculus by Benjamin Crowell, Light and Matter licensed under the Creative Commons Attribution-ShareAlike license.
By now you will have learned to recognize the circumlocutions I use in the sections without calculus in order to introduce calculus-like 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,
`F_"total"=(dp_"total")/dt`
And of course the business about the area under the `F-t` curve is really an integral, `Deltap_"total"=intF_"total"dt`, which can be made into an integral of a vector in the more general three-dimensional case:
`Deltap_"total"=intF_"total"dt`.
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.
`=>` If 1 kg/s of rain falls vertically into a 10-kg cart that is rolling without friction at an initial speed of 1.0 m/s, what is the effect on the speed of the cart when the rain first starts falling?
`=>` The rain and the cart make horizontal forces on each other, but there is no external horizontal force on the rain-plus-cart system, so the horizontal motion obeys
`F=(d(mv))/(dt)=0`
We use the product rule to find
`0=(dm)/(dt)v+m(dv)/(dt)`.
We are trying to find how `v` changes, so we solve for `dv"/"dt`,
`(dv)/(dt)=-v/m(dm)/(dt)`
`=-((1 m"/"s)/(10 kg))(1 kg"/"s)`
`=-0.1 m"/"s^2`.
(This is only at the moment when the rain starts to fall.)
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.
14.6 Applications of calculus by Benjamin Crowell, Light and Matter licensed under the Creative Commons Attribution-ShareAlike license.