Impulse & Change in Momentum
by Professor Rich Born
This lesson features Voyager and the "intelino smart train" in a lab for AP physics students. Designed for all ages, intelino is intuitive with its app, has built-in sensors to provide an interactive experience for the user, and is easily programmed with color snaps that allow the user to control intelino's actions. Students are challenged to design and carry out an experiment to show that impulse is equal to change in momentum when Voyager is mounted to an intelino smart engine that suddenly reverses itself.
A Typical Solution to Challenge
Figure 1 - The author's setup for the AP physics impulse/change in momentum challenge
The 6-second video below shows the a typical run of the author's lab.
Figure 2 - Excel graphs of Voyager's acceleration and magnetic field data
Now let's take a look at how to determine the value of the impulse. Figure 3 zooms in on the region around 4 seconds where the intelino engine reverses itself. The data rate was 50 points/second, or 0.02 second between points. This graph is of force vs. time. The force was calculated by multiplying the acceleration at each data point by the mass of the engine plus Voyager (0.106 kg). 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 - 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. As shown in column C, we note that the force is obtained by multiplying each acceleration value by the mass of the engine plus Voyager. Column D makes use of the trapezoidal rule to compute the area of each thin trapezoid. The areas are summed in cell D30. The sum, which is equal to the impulse, is 0.130 N-m. This is in close agreement with the change in momentum of 0.128 kg-m/s, a difference of about 1.5%. We have strong evidence that impulse equals change in momentum.
Figure 4 - Excel spreadsheet using numerical analysis to obtain the value of the impulse
This lab employs several of the NGSS science and engineering practices:
Asking questions and defining problems
Planning and carrying out investigations
Analyzing and interpreting data
Using mathematics and computational thinking
Constructing explanations and designing solutions