After "jerk", the next three derivitives are affectionately known as "snap", "crackle" and "pop".
Ordinarily one does not attack these kinds of problems by looking at higher and higher order derivitives though. Instead one writes down a differential equation that relates, for instance, acceleration to velocity and position.
One can use differential calculus to attempt to solve such an equation, reducing it to a form that expresses position as a function of time.
Failing that, there are computational methods (such as Runge Kutta) that generate approximate solutions by running a kind of simulation and advancing stepwise. Such approaches often treat the second derivitive (acceleration) as a variable and work in part by estimating its average value over the duration of each small step.
This sounds similar to what you are talking about.
