6: General Planar Motion

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Although Newton’s laws of motion, the various force laws, and the three conservation laws we have derived, are all valid in three dimensions, we have so far restricted our study of motion almost exclusively to two special cases: linear motion in one dimension, and rotational motion in a plane, where the radius of the rotation is constant. Although for the second case we do need two directions to describe it, the motion itself is constricted to a circle, and thus essentially one-dimensional. In this section, we’ll look at general motion in a plane - which turns out to capture a large number of important nontrivial cases.

6.1: Projectile Motion
The simplest case of two-dimensional motion occurs when a particle experiences a force only in one direction. The prime example of this case is the motion of a projectile in Earth’s (or any other planet’s) gravitational field as locally described by Galilean gravity.
6.2: General Planar Motion in Polar Coordinates
Although in principle all planar motion can be described in Cartesian coordinates, they are not always the easiest choice. For example, a central force field (a force field whose magnitude only depends on the distance to the origin, and points in the radial direction). For such a force field polar coordinates are a more natural choice than Cartesians. However, polar coordinates do carry a few subtleties not present in the Cartesian system, because the direction of the axes depends on position.
6.3: Motion Under the Action of a Central Force
A central force is a force that points along the (positive or negative) radial direction \(\boldsymbol{\hat{r}}\), and whose magnitude depends only on the distance r to the origin - so \(\boldsymbol{F}(\boldsymbol{r})=F(r) \hat{\boldsymbol{r}}\).
6.4: Kepler's Laws
The fact that the planets move in elliptical orbits was first discovered by Kepler, based on observational data alone (he didn’t have the benefit, as we do, of living after Newton and thus knowing about Newton’s law of gravity). Kepler summarized his observational facts in three laws, which we can, with the benefit of hindsight, prove to be corollaries of Newton’s laws.
6.E: General Planar Motion (Exercises)

Thumbnail: Stock cars racing in the Grand National Divisional race at Iowa Speedway in May, 2015. Cars often reach speeds of 200 mph (320 km/h).