Polar Grapher with Velocity and Acceleration vectors

Fun little applet for graphing polar curves in Geogebra. You can enter any function you like for r, and then set the bounds near the slider. You can animate the graph by pressing the play button in the lower left corner of the screen (this is particularly nice for getting a feel for the speed of the particle at any point). Additionally, you may choose to display either the Velocity Vector, the Acceleration Vector, or both. You can control the point where the vectors are displayed using the slider near the vector checkboxes. NOTE: The Velocity and Acceleration vectors are not reproduced to scale. They are drawn on the graph as a percentage of the scale factor you choose on the "Vector" slider. This means that if the largest the acceleration vector ever gets is 45, and you set the scale factor to 4, the Acceleration vector will be drawn with a length of 4. This is done so the vectors can be seen on the graph (as their magnitude often takes them far off the region of the graph we are interested in). The actual magnitudes of each vector are displayed next to the checkbox for turning the vector on and off. The graph also plots the integral radian points (i.e. the position at pi, 2pi, 3pi, etc.) which you can turn on or off in the lower right corner of the screen
Pay special attention to the Velocity and Acceleration vectors and how they behave. What do you notice? In particular you may want to look at the functions for a Circle and an Ellipse. These have applications to orbital mechanics and the Velocity and Acceleration vectors have some important characteristics in those cases.