Forest UCM NLM GalileanTans

Assume that $S^{\prime}$ is a coordinate system moving at a CONSTANT speed $v$ with respect to a fixed coordinate system $S$.

Let $\vec{r}$ and $\vec{r}^{\prime}$ describe the position an object in motion using two different coordinate systems $S$ and $S^{\prime}$ respectively.

$\vec{R}$ represents a vector that locates the origin of the moving reference frame ($S^{\prime}$) with respect to the origin of reference from $S$.

Using the definition of vector addition

$\vec{r} = \vec{R} + \vec{r}^{\prime}$

Similarly

$\vec{v} = \frac{d \vec{r}}{dt} = \frac{d \vec{R}}{dt} + \frac{d \vec{r}^{\prime}}{dt}$

and

$\vec{a} = \frac{d^2 \vec{r}}{dt^2} = \frac{d^2 \vec{R}}{dt^2} + \frac{d^2 \vec{r}^{\prime}}{dt^2}$

Newton's law of motion may be written as

\vec{F} = m\vec{a} = m \left ( \frac{d^2 \vec{R}}{dt^2} + \frac{d^2 \vec{r}^{\prime}}{dt^2} \right )

Forest_UCM_NLM#Galilean_Transformations