Mathematics, Physics

Kinematics

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*Note: for the purposes of this article, variables that represent a column vector may either be denoted using bolded letters (e.g., $\mathbf{a}$) or using over-head arrow notation (e.g., $\overrightarrow{a}$)

Linear Motion

 

Variables:

  • $\mathbf{x} =$  position vector (SI unit: $\textup{m}$)

$\Delta \mathbf{x} = \mathbf{x}_f – \mathbf{x}_i$

  • $\mathbf{v} =$ velocity vector (SI unit: $\textup{m/s}$)

$\mathbf{v} = \frac{\Delta \mathbf{x}}{\Delta t} = \mathbf{\dot{x}}$

  • $\mathbf{a} =$ acceleration vector (SI unit: $\textup{m} / \textup{s}^2 $)

$\mathbf{a} = \frac{\Delta \mathbf{v}}{\Delta t} =\mathbf{\dot{v}}= \mathbf{\ddot{x}}$

Kinematic formula (1D motion with constant acceleration):

$v(t)=v_0 + at$

$\Delta x (t) = (\frac{v(t)+v_0}{2})t$

$\Delta x (t) = v_0t + \frac{1}{2}at^2$

$v^2 (t) = v_0^2 + 2a \Delta x(t)$

Rotational Motion

Variables:

  • $\mathbf{\theta} =$  angle vector (SI unit: $\textup{rad}$)

$\Delta \mathbf{\theta} = \mathbf{\theta}_f – \mathbf{\theta}_i$

  • $\mathbf{\omega} =$  angular velocity vector (SI unit: $\textup{rad/s}$)

$\mathbf{\omega} = \frac{\Delta \mathbf{\theta}}{\Delta t} = \mathbf{\dot{\theta}}$

  • $\mathbf{\alpha} =$  angular acceleration vector (SI unit: $\textup{rad} / \textup{s}^2 $)

$\mathbf{\alpha} = \frac{\Delta \mathbf{\omega}}{\Delta t} =\mathbf{\dot{\omega}}= \mathbf{\ddot{\theta}}$

Kinematic formula (uniform circular motion)

$\omega (t)=\omega_0 + \alpha t$

$\Delta \theta (t)= (\frac{\omega (t)+\omega_0}{2})t$

$\Delta \theta (t) = \omega_0t + \frac{1}{2} \alpha t^2$

$\omega^2(t) = v_0^2 + 2\alpha \Delta \theta (t)$

Relations between linear and rotational variables

Let $\mathbf{x} = \mathbf{r} = x\mathbf{\hat{\imath}}+ y\mathbf{\hat{\jmath}}$, $r = \left|| \mathbf{r} \right||$, and $\omega = \left|| \overrightarrow{\omega} \right||$

$x = r \cos{\theta}$

$y = r \sin{\theta}$

$r^2 = x^2 + y^2$

$\theta = \tan^{-1}{(y/x)}$

$\mathbf{v} = \overrightarrow{\omega} \times \mathbf{r}$

$\mathbf{a} = \mathbf{\overrightarrow{\alpha}} \times \mathbf{r} – \omega^2 \mathbf{r} $

Sources

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