Use geometry to find formulas for velocity and acceleration in polar coordinates.
On the figure below, draw \(\hat{s}\) and \(\hat{\phi}\) at \(P\).
- Find \(\frac{d}{dt}\hat{s}\) and \(\frac{d}{dt}\hat{\phi}\) in terms of \(\hat{s}\) and \(\hat{\phi}\).
- Find \(\vec{v}\) in terms of \(\hat{s}\) and \(\hat{\phi}\).
The activity begins by with a SWBQ asking the students to find \({\vec{v}} = \frac{d\vec{r}}{dt}\). Typically, students propose two alternatives, \begin{align*} \frac{d\vec{r}}{dt} &= \frac{ds}{dt} {\hat{s}}\\ \frac{d\vec{r}}{dt} &=\frac{ds}{dt} \hat{s} + \frac{d\phi}{dt} \hat{\phi}. \end{align*}
A discussion ensues about which is correct. The proper formula is derived using \begin{align*} \frac{d{\vec{r}}}{dt} &= \frac{d}{dt}(s{\hat{s}})\\ &=\frac{ds}{dt} {\hat{s}} + s \frac{d\hat{s}}{dt}. \end{align*}
This result is further justified by drawing the velocity vector and discussing the fact that \({\hat{r}}\) changes as \(\phi\) changes.
Students are then asked to find values for \({d \bf\hat{s}}\over{dt}\) and \({d{ {\bf\hat{\phi}}}\over{d{t}}}\). Students are given the attached handout and asked to carry out calculations of \(\vec{v}\) and \(\vec{a}\) in polar coordinates.
After students have completed the calculations, quickly review the answers and the procedure for calculating them. \begin{align*} {\vec{v}} &= {{\dot{\vec r}}}\\ &= \dot{r} {\hat s} + s \dot \phi {\hat \phi}\\ {\vec{a}} &= {{\dot{\vec v}}} \\ &= {{\ddot{\vec r}}}\\ &= \left(\ddot{s}- s \dot{\phi}^2\right) {\hat s} + \left( s \ddot{\phi} + 2 \dot{s}\dot{\phi}\right) {\hat \phi} \end{align*}
Notes on these derivations can be found in these lecture notes.
Also, ask students to look at the dimensions. Each term has two derivatives with respect to \(t\) and one factor of \(r\) or its derivatives.
This activity leads smoothly into a mini-lecture defining the kinetic energy and angular momentum in polar coordinates. If you have a group (or 2) that are ahead of the game, ask them to find \(\vec{L}\) in terms of polar coordinates and have them start the mini-lecture.