This Mathwright Web activity features a simulation of a falling diver. Students can observe the motion, and freeze the action at two times, in order to compute average speed over any interval. They are asked to use the simulation to answer the question: How fast is the diver going when she hits the water? In order to find an answer, students have to invent for themselves some notion of instantaneous velocity. There are three activity pages. The first screen is shown below.

Students set the two observation times using slider bars at the top of the screen. Clicking a button runs the simulation. That produces an animated display of the falling diver, and leaves two ghost images, one red and one green, at the positions of the specified times. The white text window shows the computation of average speed between to the two specified times.

The second screen shows the same simulation, but adds a graph for average velocity versus one specified time. See the screen image below.

One of the times is held fixed, and the students systematically vary the other time. For each value of this variable time, running the simulation results in a computed average velocity. These values are plotted as a function of the variable time. The results fall on a line. There is no point on this line when the variable and fixed times are equal. However, it is clear from the graph how to interpolate the position of the missing point. That interpolation corresponds to the limit definition of instantaneous velocity.

On the final screen, the calculation of average velocity is related to the slope of a chord on the standard graph of position versus time. See the screen image below.

This time there is no simulation shown. Instead, the computation of average velocity is made as before, and shown to be the same as computing a slope of a chord. By holding one time fixed, and varying the other, students can observe that their computations of velocity at one precise instant amounts to finding the slope of a tangent line.