I # In an intertial frame of reference (i.e. a frame of reference undergoing zero acceleration), an object at rest stays at rest (or, similarly, keeps its initial velocity) unless acted upon by an outside force.
\begin{equation} \sum \mathbf{F} = 0 \Leftrightarrow \frac{d\mathbf{v}}{dt} = 0 \end{equation}
II # In the simplest of terms:
\begin{equation} \mathbf{F} = m\mathbf{a} \end{equation}
which states that
The vector sum of the forces on an object is equal to that objects mass times the acceleration
...
Newtonian Spacetime differs from Relativistic spacetime in that time is considered an absolute (i.e. \(t^{\prime} = t\)). Transformations between reference frames can be achieved with Galilean transformations. A particle traveling along the x axis in \(S\) at a constant speed u, has a speed \(u^{\prime}\) found by:
\begin{equation} u^{\prime}_{x} = \frac{dx^{\prime}}{dt^{\prime}} = \frac{dx^{\prime}}{dt} = \frac{dx}{dt} - v = u_{x} - v\tag{1} \end{equation}
remembering that the first derivative of position is velocity, and the second derivative of motion is acceleration
...
In Newton’s formulation of spacetime, time is an absolute; meaning that every observer experiences the same flow of time. Assuming an inertial frame with only translation across the x axis, an event \(P\), can be translated between \(S\) and \(S^{\prime}\) by the following linear transformations:
\begin{equation} t^{\prime} = t\tag{1} \end{equation}
\begin{equation} x^{\prime} = x - vt\tag{2} \end{equation}
\begin{equation} y^{\prime} = y\tag{3} \end{equation}
\begin{equation} z^{\prime} = z\tag{4} \end{equation}
Switching between coordinates (\(S\) v.
...
Linear Motion # (under constant \(\hat{\mathbf{a}}\))
\begin{equation} \mathbf{v} = \mathbf{a}t + \mathbf{v_{0}}\tag{1} \end{equation}
\begin{equation} \mathbf{r} = \mathbf{r_{0}} + \mathbf{v_{0}}t + \frac{1}{2}\mathbf{a}t^{2}\tag{2} \end{equation}
\begin{equation} \mathbf{r} = \mathbf{r_{0}} + \frac{1}{2}(\mathbf{v} + \mathbf{v_0})t\tag{3} \end{equation}
\begin{equation} \mathbf{v^{2}} = \mathbf{v^{2}_0} + 2\mathbf{a}(\mathbf{r} - \mathbf{r_{0}})\tag{4} \end{equation}
\begin{equation} \mathbf{r} = \mathbf{r_{0}} + \mathbf{v}t - \frac{1}{2}\mathbf{a}t^{2}\tag{5} \end{equation}
where:
\(\mathbf{r_{0}}\) is the initial position \(\mathbf{r}\) is the final position \(\mathbf{v_{0}}\) is the initial velocity \(\mathbf{v}\) is the final velocity \(\mathbf{a}\) is the acceleration \(t\) is the time interval