Equations of Motion

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