Experiments to prove that the governing equation can not be taken as $F= m \frac{d^n x}{ dt^n}$ for $n$ except 2 in general

Newton's laws function less as a single claim than as a schema for analyzing motion. The second law supplies the form: causes enter as forces, and force equals mass times the second derivative of position. But the choice of second derivative is not neccesarily forced. Suppose we replaced it with $F= m \frac{d^nx}{dt^n}$ holding the other laws fixed and keeping force as the same independently measurable quantity — what a spring balance reads. For each $n \in \mathbb{N}$ with $n \neq 2$ , what experiment would expose that this schema is false?