Forest UCM NLM GalileanTans
Assume that [math]\displaystyle{ S^{\prime} }[/math] is a coordinate system moving at a CONSTANT speed [math]\displaystyle{ v }[/math] with respect to a fixed coordinate system [math]\displaystyle{ S }[/math].
Let [math]\displaystyle{ \vec{r} }[/math] and [math]\displaystyle{ \vec{r}^{\prime} }[/math] describe the position an object in motion using two different coordinate systems [math]\displaystyle{ S }[/math] and [math]\displaystyle{ S^{\prime} }[/math] respectively.
[math]\displaystyle{ \vec{R} }[/math] represents a vector that locates the origin of the moving reference frame ([math]\displaystyle{ S^{\prime} }[/math]) with respect to the origin of reference from [math]\displaystyle{ S }[/math].
Using the definition of vector addition
- [math]\displaystyle{ \vec{r} = \vec{R} + \vec{r}^{\prime} }[/math]
Similarly
- [math]\displaystyle{ \vec{v} = \frac{d \vec{r}}{dt} = \frac{d \vec{R}}{dt} + \frac{d \vec{r}^{\prime}}{dt} }[/math]
and
- [math]\displaystyle{ \vec{a} = \frac{d^2 \vec{r}}{dt^2} = \frac{d^2 \vec{R}}{dt^2} + \frac{d^2 \vec{r}^{\prime}}{dt^2} }[/math]
Newton's law of motion may be written as
- [math]\displaystyle{ \vec{F} = m\vec{a} = m \left ( \frac{d^2 \vec{R}}{dt^2} + \frac{d^2 \vec{r}^{\prime}}{dt^2} \right ) }[/math]
If
- [math]\displaystyle{ \frac{d^2 \vec{R}}{dt^2}=0 }[/math] [math]\displaystyle{ S^{\prime} }[/math] is moving at a constant velocity [math]\displaystyle{ \vec{V} }[/math]
Then
- [math]\displaystyle{ \vec{F} = m\vec{a} = m \left ( \frac{d^2 \vec{r}^{\prime}}{dt^2} \right ) }[/math]
Newton's law hold in coordinate system which move at a constant velocity (an inertial reference frame). Accelerating reference frames are know as non-inertial reference frames and will be discussed later. (a coordinate system fixed to the Earth is a non-inertial reference frame since the Earth is rotating about its axis and moving in orbit about the Sun)