The next step is to mathematically describe the change in velocity, which results in the *acceleration*, ; this is defined as:

The form is the same as (8.1), except that has been replaced by . Approximations can similarly be made. For example, .

The acceleration itself can vary over time, resulting in . The following integral relates acceleration and velocity (compare to (8.4)):

Since acceleration may vary, you may wonder whether the naming process continues. It could, with the next derivative called *jerk*, followed by *snap*, *crackle*, and *pop*. In most cases, however, these higher-order derivatives are not necessary. One of the main reasons is that motions from classical physics are sufficiently characterized through forces and accelerations. For example, Newton's Second Law states that , in which is the force acting on a point, is its mass, and is the acceleration.

For a simple example that should be familiar, consider acceleration due to gravity, m/s. It is as if the ground were accelerating upward by ; hence, the point accelerates downward relative to the Earth. Using (8.6) to integrate the acceleration, the velocity over time is . Using (8.4) to integrate the velocity and supposing , we obtain

(8.7) |

Steven M LaValle 2020-11-11