1: Velocity and Acceleration of a Point

Fix a right-handed (positively oriented) Cartesian coordinate system \(O x y z\) in physical space \(\mathbb{R}^3\).
The law of motion of a point \(M\) with position vector \(\boldsymbol{r}_M(t)=\boldsymbol{O} \boldsymbol{M}(t)\) is defined as the mapping

\[t \mapsto \boldsymbol{r}_M(t)=x(t) \boldsymbol{e}_x+y(t) \boldsymbol{e}_y+z(t) \boldsymbol{e}_z, \quad x(\cdot), y(\cdot), z(\cdot) \in C^2[0, T] .\notag \]

Here, \(\boldsymbol{e}_x, \boldsymbol{e}_y, \boldsymbol{e}_z\) denote the unit basis vectors of the corresponding coordinate axes.

The trajectory (path) of point \(M\) is the set of points

\[\left\{\boldsymbol{r}_M(t) \mid t \in[0, T]\right\} \subset \mathbb{R}^3,\notag\]

which is usually a piecewise smooth curve.

The velocity and acceleration of point \(M\) relative to \(O x y z\) are defined as the vectors

\[\boldsymbol{v}_M=\frac{d \boldsymbol{r}_M}{d t}=\dot{\boldsymbol{r}}_M=\dot{x}(t) \boldsymbol{e}_x+\dot{y}(t) \boldsymbol{e}_y+\dot{z}(t) \boldsymbol{e}_z, \quad \boldsymbol{a}_M=\dot{\boldsymbol{v}}_M .\notag\]

Unless specified otherwise, velocities and accelerations in what follows are considered relative to the frame \(O x y z\).

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