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
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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
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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.
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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