Velocity vs Impulse to Stop
by Professor Rich Born
The intelino/Voyager Challenge
A Typical Solution to the PocketLab Voyager/intelino Challenge
Figure 1 - Setup for a possible solution to the Voyager / intelino challenge
Action Video of the Author's Solution to the Challenge
The short 12-second video below shows a typical run of this challenge. In particular notice the sudden spike in acceleration when intelino/Voyager encounter a "stop 2-sec" command. These spikes will be the basis for the data analysis phase of the experiment.
Figure 2 shows an Excel graph of force versus time. The force was obtained by multiplying Voyager's y-acceleration data by the mass of the engine plus Voyager (0.106 kg). Since impulse is the product of force by the time that the force is applied, the impulse can be obtained by finding the area under each of the three spikes. We notice that the time interval for each of the spikes are about the same length. The intelino engine has created a stronger force to stop the engine, as shown by the rise in the peaks as the initial speed increases from 30 to 50 to 70 cm/s.
Figure 2 - The three impulses to stop the intelino engine
Now let's take a look at how to determine the value of the area. Figure 3 zooms in on the region around 8 seconds, where the intelino engine stops after traveling at a rate of 50 cm/s or 0.02 second between points. Here is a great opportunity for your AP physics students to do a little numerical analysis in order to find the area of the graph during the impulse. They have likely learned how to integrate functions to find areas, but may not have encountered a situation where there is no obvious function y = f(x). A common method to find such an area is to use the trapezoidal rule for approximating an integral. Figure 3 shows the entire region divided into a series of trapezoids. The area of each of these trapezoids can be found by use of the formula shown in the upper right corner of the figure.
Figure 3 - Zoomed in graph of force versus time
The author used Excel to create a simple spreadsheet to compute this area, as shown in Figure 4. Values are shown on the left and formulas in the corresponding cells on the right. Column C makes use of the trapezoidal rule to compute the area of each thin trapezoid. The areas are summed in cell C25. The sum, which is equal to the impulse, is about 0.063 N-m. The other two impulses, at about 4 and 12 seconds are computed in a similar manner.
Figure 4 - Excel spreadsheet using numerical analysis to obtain the value of the impulse
The final step in the data analysis is to construct a graph of impulse versus speed. Figure 5 shows the result in Excel. The graph clearly shows that the impulse to stop the engine is proportional to the speed of the engine. We have accomplished the goal of this intelino/Voyager challenge.
Figure 5 - Graph of impulse versus speed