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 shows the setup used by the author of this lesson to solve the proposed AP physics challenge. PocketLab has been attached to the top of an intelino smart engine with a 3M damage-free strip, and is then placed at the far right end of a series of straight tracks. Note that the orientation of Voyager has +Y pointing to the right and -Y to the left. Voyager is started at the "fast" speed while in autopilot mode moving in the direction of the blue arrow in Figure 1. With two magnets placed one meter apart, Voyager's magnetic field versus time graph will show a peak as Voyager passes each magnet. This will allow determining the initial velocity. When the intelino engine reaches the built-in reverse command at the far left end of the track, the engine will suddenly reverse itself and travel back toward the right. Once again the magnetic field peaks will provide information allowing determining the final velocity. Knowing the initial and final velocities, the change in momentum mΔv can be calculated.
The general formula for impulse if FΔt, where F is the force and Δt is the time interval during which the force is applied. Force is found by multiplying the mass of the engine plus Voyager by the acceleration. The acceleration during the applied impulse can be determined from Voyager's graph of y-acceleration versus time. Since the acceleration turns out to vary significantly during the impulse, the impulse must be computed from the area under the force vs. time graph. Our hope is that the impulse will equal the change in momentum.
The engine is stopped from the intelino app before running off the right end of the track after it has passed the red magnet while moving to the right. Alternatively, there is an "end route" color action snap command that could be used to stop the engine automatically before it runs off the track on the far right.
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 contains Excel charts of Voyager's Y acceleration and Z magnetic field data. The top graph shows that the acceleration averages around zero when the engine is traveling from the right to the left, and again when traveling from the left to the right on the track. These facts provide reason to believe that the speed is constant in those regions of the graph. The acceleration has a sudden peak around 4 seconds when the engine stops and reverses direction at the left end of the track. The area of the region during the engine's change in direction is proportional to the impulse. We'll see shortly how to determine the value of the impulse. Note that the impulse is a positive quantity. The engine is slowing down in the negative direction upon reaching the reverse command. This makes the acceleration positive. Then the engine speeds up in the positive direction, making the acceleration again positive.
The graph of Z magnetic field versus time at the bottom of Figure 2 shows how to calculate the change in momentum Δp. The value of m, the mass of the intelino engine plus Voyager, is 0.106 kg. It is important to consider the signs of the initial and final velocities. The initial velocity is negative since the engine is moving to the left (-Y direction). The final velocity, however, is positive since the engine is moving to the right (+Y direction). We see that the change in momentum is 0.128 kg-m/s.
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