Newton's Laws of Motion

Perhaps the most heavily used equation in physics is Newton's second law of motion:

$$\zbox {\mbox{\emph{Force = Mass $\times$\ Acceleration}}}$$

That is, when a force is applied to a mass, the mass experiences an acceleration proportional to the applied force. Denoting the mass by $m$, force at time $t$ by $f(t)$, and acceleration by

$$a(t)\isdef {\ddot x}(t) \isdef \frac{d^2 x(t)}{dt^2},$$

we have

$$\zbox {f(t) = m\,a(t) = m\,{\ddot x}(t).} \protect$$ (E.1)

In this formulation, the applied force $f(t)$ is considered positive in the direction of positive mass-position $x(t)$. The force $f(t)$ and acceleration $a(t)$ are, in general, vectors in three-dimensional space $x\in{\bf R}^3$. In other words, force and acceleration are generally vector-valued functions of time $t$. The mass $m$ is a scalar quantity, and can be considered a measure of the inertia of the physical system (see §E.1.2).