Newton

## Newtons Laws of Motion

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. $$\sum \mathbf{F} = 0 \Leftrightarrow \frac{d\mathbf{v}}{dt} = 0$$ II # In the simplest of terms: $$\mathbf{F} = m\mathbf{a}$$ which states that The vector sum of the forces on an object is equal to that objects mass times the acceleration ...

## Newtonian Spacetime

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: $$u^{\prime}_{x} = \frac{dx^{\prime}}{dt^{\prime}} = \frac{dx^{\prime}}{dt} = \frac{dx}{dt} - v = u_{x} - v\tag{1}$$ remembering that the first derivative of position is velocity, and the second derivative of motion is acceleration ...

## Galilean Transformations

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: $$t^{\prime} = t\tag{1}$$ $$x^{\prime} = x - vt\tag{2}$$ $$y^{\prime} = y\tag{3}$$ $$z^{\prime} = z\tag{4}$$ Switching between coordinates ($$S$$ v. ...

## Equations of Motion

Linear Motion # (under constant $$\hat{\mathbf{a}}$$) $$\mathbf{v} = \mathbf{a}t + \mathbf{v_{0}}\tag{1}$$ $$\mathbf{r} = \mathbf{r_{0}} + \mathbf{v_{0}}t + \frac{1}{2}\mathbf{a}t^{2}\tag{2}$$ $$\mathbf{r} = \mathbf{r_{0}} + \frac{1}{2}(\mathbf{v} + \mathbf{v_0})t\tag{3}$$ $$\mathbf{v^{2}} = \mathbf{v^{2}_0} + 2\mathbf{a}(\mathbf{r} - \mathbf{r_{0}})\tag{4}$$ $$\mathbf{r} = \mathbf{r_{0}} + \mathbf{v}t - \frac{1}{2}\mathbf{a}t^{2}\tag{5}$$ 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