Projects in High-Speed Photography
Trapped! Analysis of a Real Collision
Note to teachers: This lab may be edited for use in classes.
Introduction
In one type of physics problem involving a collision, one is given initial and final velocities of an object, the mass of the object, and the duration of the collision. One can then calculate the average net force on the object using the relationship:
Fave = Δp/Δt,
where Δp is the change in momentum of the object. This method provides no information about how the force changes with time during the collision. Such information may be difficult to obtain; nevertheless, it can provide some interesting insights into the collision process.
The purpose of this lab to investigate a real collision that takes place over a time span of only about 5 milliseconds. Even with such a short duration, details of how the force changes as a function of time can be obtained from multiple-image high-speed photographs. Such a photograph is shown below.
The photo shows 8 successive images of an elastic strip being struck by a BB. (The BB's used in the experiment were 0.177 inch in diameter and had a mass of 0.34 g each.) The images were produced by 8 flash bursts, 0.000548 s apart in time. Different colored filters were placed over each flash unit, thus making it possible to distinguish overlapping images in the photo.
The strip was initially hanging vertically from a clip, and the BB approached at high speed from the left, striking the strip in the center. The strip hung freely, with nothing being attached to the lower end. The BB was able to stretch the strip, because the BB was moving much faster than the speed at which disturbances could travel along the length of the strip.
Interestingly, the BB did not penetrate the strip but was trapped instead. The BB is within the vertex of each wishbone-shaped image. (In the first, red image, the BB has barely begun to stretch the strip. The slight curvature of the strip on either side of the BB is due to the puff of air from the air gun.) Obviously, the strip is decelerating the BB at the same time that the BB is accelerating the strip. The purpose of the lab will be to determine how the force on the BB changes with time during the collision. This information can be obtained from position vs. time data.
An interesting paradox emerges from the results of the analysis. Trying to resolve this paradox should provide insight into the collision process.
Preparations
The photographs can be analyzed by printing out the images and then taking measurements with a ruler. If, however, one has access to a photo-editing program, one can read the coordinates of the BB's position directly from the computer screen. The following photo can be analyzed for practice.
This photograph shows 8 flash images (0.000298 s apart) of the BB before striking the strip. This is not the same BB that struck the strip in the first photograph. However, it was fired from the same gun under the same conditions. Previous tests with the gun show that the muzzle speed doesn't vary more than 5% for the same number of pumps.
This photograph and the superimposed distance scale will allow one to determine the initial velocity of the BB. The instructions below are written assuming the use of photo-editing or analysis software.
- Right click here and select the Save As option. Save the image to your working directory. Keep the given filename. This is a larger version of the image above.
- Open the photo in a photo-editing or analysis program. Find where the coordinates of the cursor are displayed. This will provide the data that you'll need.
- Prepare a data table with columns for time, horizontal position in pixels, and horizontal position in meters.
- Back in the photo-editing program, point the cursor to the brightest part of each image of the BB and read the x-coordinate in pixels. Record the values in your data table.
- Now read the x-coordinates of two positions on the meter scale. Pick two points that are distinct and as far apart as possible (to increase accuracy). For each point, record the x-coordinate in pixels and the scale reading in meters. Then show your calculation of the scale factor that you'll apply to the x-coordinates (in pixels) in order to obtain actual distances.
- Now use the scale factor to calculate the horizontal positions in meters.
- Plot a graph of Horizontal Position (m) vs. Time (s). Apply a linear fit and obtain the speed.
- Before proceeding with the lab, ask the instructor to look over your results.
Analysis of the Collision
Collecting data
Proceed with the analysis of the collision in a similar way to what you did above. That is, save the image file to your working directory and then open the file in your photo editor. Click here to download a version of the image with a portion of a meter scale overlaid. Prepare a data table. Note that for this photo, the time interval between flashes is 0.000548 s.
Locating the position of the BB is a bit tricky in this photo. You know the BB is at the vertex of each image. You can center the cursor on this vertex. However, locating the vertex is more difficult where two images overlap. The color that is produced by such an overlap can help you find the vertex. For example, where the 3rd image overlaps the 4th, the vertex appears white. Where the 5th, red image overlaps the green, you get yellow. Where green overlaps blue, you get cyan.
Once you've found the BB coordinates, determine the scale factor as before and calculate positions in meters.
Fitting the data
You will now need to enter your data into a data analysis program that will do polynomial fits. Vernier Software's Graphical Analysis is one such program.
Try a quadratic fit to the Position vs. Time data just to convince yourself that the acceleration is not uniform. But then, did you have any reason to expect it to be uniform?
Find the lowest-order polynomial fit that gives a good fit and also has at least some correspondence to the actual physical situation, namely, that the BB must come to a stop and turn around. Record the equation of fit.
Find the second derivative of the Position vs. Time fit. Then plot it. Are you surprised at the magnitudes of the accelerations? How can they be so large? Find out how large the corresponding forces are by calculating the product, ma = (mass of the BB)x(acceleration). Why can you use ma to represent the force of the elastic strip on the BB?
Now plot ma vs. Time. Keep that graph handy for the following.
Interpretations
Examine your graph of ma vs. Time and describe how the magnitude of the force changes as a function of time. What do you find interesting about the graph?
Now consider the following:
The Webster's Revised Unabridged Dictionary gives the following as a definition of paradox: "an assertion or sentiment seemingly contradictory, or opposed to common sense; that which in appearance or terms is absurd, but yet may be true in fact." The BB-elastic collision presents an opportunity to pose a paradox and then attempt to resolve it. Note that while a paradox may seem contradictory, it "yet may be true in fact." Therefore, your job will be to explain how the following paradox can be true.
An examination of your ma vs. Time graph should reveal that the magnitude of the net force on the BB is greatest at the beginning of the collision and then decreases. This force is due to the action of the elastic strip. Yet, you know from experience that the more you stretch an elastic band, the more it pulls back on you. The elastic strip in the photograph certainly seems to be stretched more at the end of the collision than at the beginning. How is it possible for the force of the strip on the BB to be greatest at the beginning of the collision? Give very careful thought to this question and also give the photograph a close examination. Then write a paragraph in explanation.
