Exercise 8a

\[\vec{r}(t) = \left( \begin{matrix} \cos \omega t \\ \mathrm{e}^{\omega t} \end{matrix} \right)\]

\[\vec{v}(t) = \left( \begin{matrix} -\omega \sin \omega t \\ \omega\, \mathrm{e}^{\omega t}\end{matrix} \right)\] \[ \left| \vec{v}(t) \right| = \sqrt{ (-\omega \sin \omega t )^2 + (\omega\, \mathrm{e}^{\omega t})^2 } = \omega \sqrt{ \sin^2 \omega t + \mathrm{e}^{2 \omega t} }\] \[\vec{a}(t) = \left( \begin{matrix} -\omega^2 \cos \omega t \\ \omega^2 \mathrm{e}^{\omega t}\end{matrix} \right)\]

Animation

position vector: red
velocity vector: blue
acceleration vector: green

\(\omega = 0.628\), \(T = 10 \, \mathrm{s}\)

The animation shows one period, \(x\)-axis is scaled by factor 300, \(y\)-axis is scaled by factor 0.5. The \(x\) component of the position vector oscillates, the \(y\) component is growing very fast due to the exponential function.