Velocity vs. Impulse to Stop
by Professor Rich Born
While driving at 40 mph, you see a red stop light ahead. You press your brakes for several seconds, gradually coming to a stop. A little later on the same road at 40 mph, you approach another light, this time green. While approaching this light, it suddenly changes to yellow. You make a split-second decision to put on your brakes to avoid going through a red light. With the brakes applied quite hard, you quickly stop, waking up your sleeping friend in the front passenger seat.
What are the common features in both of these situations? In both cases you are traveling at 40 mph. In the first case, you apply a lesser force for a longer period of time. In the latter case, you apply a strong force for a short period of time. In both cases the car comes to a stop. In both cases you apply a force for a specific amount of time to bring the car to a stop. In physics terminology, you have provided an impulse to stop the car. Impulse is the product of the force F by the time Δt that the force is applied. In other words, impulse equals FΔt. This impulse produces a change in momentum Δp = mΔv, where m is the mass of the object and Δv is the change in velocity. In both cases the initial velocity is 40 mph and the final velocity is zero, resulting in a change in velocity of -40 mph.
In this lab, you will investigate the relationship between the initial speed and the impulse required to stop the object. The moving object will be an "intelino smart train" engine with PocketLab Voyager mounted to the top of the engine for acceleration data collection. 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. The intelino app allows setting the speed of the engine accurately anywhere between 30 cm/s and 70 cm/s, making it easy to find the relationship between velocity and the impulse required to stop the engine.
The intelino/Voyager Challenge
Students are challenged to design and carry out an experiment to show that impulse is proportional to the initial velocity when Voyager is mounted to an intelino smart engine that suddenly stops. Their design should make use of intelino color action snap commands and collect all necessary data in a single run of intelino/Voyager on the track. The intelino app can run on one device while the PocketLab app runs on another device.
A Typical Solution to the PocketLab Voyager/intelino Challenge
Figure 1 shows the setup that the author of this lesson used for this challenge. Voyager has been mounted to the top of the intelino engine using a 3M damage free strip. The -Y side of Voyager is facing the direction of motion. Voyager is set to record y-acceleration at a data rate of 50 points/second. As soon as the engine is started using the intelino app in "autopilot" mode, the engine encounters a custom color action command to set the speed to 30 cm/s. After traveling the length of about two tracks, the built-in color action command "stop 2 sec" is encountered. Voyager collects data on acceleration during this stop. The acceleration is positive since the engine is slowing down while moving in the -Y direction. After two seconds have elapsed, the engine starts up again and encounters another "stop 2 sec" command followed by a custom color action command to set the speed to 50 cm/s. After traveling another couple of track lengths, the engine stops once again for two seconds. Upon starting up again, a custom color action snap command is encountered that sets the speed to 70 cm/s. The engine stops for a final time after traveling a final couple lengths of track.
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
Extended Challenge: Application to the Real World
Take another look at the accompanying video. You will notice that the stopping distance of the engine increases as the initial speed of the engine increases. But how does in increase? Is stopping distance proportional to the speed, the square of the speed, or what? Do another run of the experiment, but this time determine the stopping distance for each of the speeds.
Physics tells us that doubling the speed results in quadrupling the stopping distance. In other words, the stopping distance is proportional to the square of the speed of the vehicle. Does your results agree with this? If not, how can you explain this disagreement?
Have you ever been told not to follow too close to the driver ahead of you? To keep a safe distance? To abide by the "3-second rule"? To keep a distance of at least one car length for every ten miles per hour of speed? These questions all deal with the issue of stopping distance to avoid crashes.