I'm reading Harbovsky and Susskind's *The Theoretical Minimum* Vol I.

In *Lecture 3: Dynamics*, under *Aristotle's Law of Motion* is mentioned Aristotle's (fallacious) law $F(t) = m {dx\over dt}$. Later, time is reversed by "changing $t$ to $-t$" and the result is mentioned as $F(-t) = -m{dx\over dt}$. This made absolutely no sense to me. So I tried to formulate this in precise mathematical terms. The following is my result.

> Let $\tau: \mathbb{R\to R}$ be a function defined by $\tau (t):=-t$
> and let $\dot{f}$ denote the derivative of a differentiable function
> $f$. Let $F$ and $x$ be differentiable functions satisfying $F(t) = m\dot x(t)$ for some constant $m$. Then $F(-t) =-m\dot{(x\circ\tau)}(t)$.

**Question:** Am I correct?