L08.  The Ballistic Pendulum

About your lab report:  Write your report in the L08 wiki. Organize your report as shown below. For numbered items, number your responses the same as the questions. Phrase each response in such a way as to make it clear what question you're answering.  After the due date, the reports of all students will be made visible. Each student will then review the reports of two other students. Instructions for the reviews will be posted later.

Goal

Measure the speed of a bullet using a ballistic pendulum.

Introduction

The textbook gives the theory of the ballistic pendulum in Example 9-5.  This reading has already been assigned.  If you haven't read it, don't go any further without reading it now.  If you have read it, review it before answering these questions.

The textbook example solves the problem by applying conservation of momentum to the collision and conservation of energy to the upward swing.  You may be wondering why the author didn't pick the seemingly simpler approach of doing a single conservation of energy problem with the initial state being the bullet just before hitting the pendulum and the final state being the pendulum at its greatest height.  You can answer this question for yourself by considering the conditions under which these conservation laws can be used.  This, of course, is something you should always do before applying these methods.  To review, here are the conditions for applying these conservation laws.

  1. Condition for conservation of momentum:  The net, external force on the system is 0.
  2. Condition for conservation of mechanical energy, K + U:  The net work done by external forces on the system is 0.
  1. Why was the author's use of conservation of momentum in Example 9-5 valid?  In order to answer, first identify the forces acting on the system of object and bob during the collision. Which of these are external forces? Is the net, external force 0? Note that the collision refers to the very brief interval of time when the object is combining with the bob. The collision is essentially complete as the upward swing begins.
  2. Now consider just the upward swing. The initial state is when the collision is complete and the object-bob combination has its maximum velocity. The final state is when the combined object reaches its highest point. Should mechanical energy be conserved in the upward swing? Again, what are the external forces on the system? Take the system to be the object, bob, Earth, and gravity. Do any external forces do work on the system? If so, is the net work by these forces equal to 0?
  3. Consider once again the collision only. Is kinetic energy conserved in this collision? Tell how you know.

Suppose the initial state is the object just before hitting the pendulum and the final state is when the pendulum has reached its greatest height. Item 3 above should make it clear that mechanical energy is not conserved between these states. This is why the author didn't solve the problem as a single conservation of energy problem.

The Video Clip and Some Data

Designing a real ballistic pendulum to measure the speed of a projectile involves more than the theory in the textbook.  If the projectile is particularly fast, such as a bullet, then the pendulum bob needs to be massive and there has to be a way to prevent it from swaying from side-to-side after the collision.  The length of the pendulum generally needs to be long in order to provide a correspondingly large horizontal displacement.  The latter is used as a measure of the vertical displacement, because the vertical displacement is generally too small to measure accurately.

View the video clip Measuring the Speed of a Bullet with a Ballistic Pendulum before going on. Caution:  The gun shots are loud.  You may want to turn down the volume before the rifle is fired.

  1. As you watch the clip, some items of data are given.  Be prepared to record them as they're given.  In order to help with this, we've listed the items as they're presented in the clip.  One of the values won't be needed in the analysis.  (You needn't record the values given near the end of the video clip for the larger log.)

Weight of the log (pounds):
Length of the log (inches):
Weight of the bullet (grains):
Distance between pendulum supports (inches):
Vertical distance from support to hook (inches):

Experimental Design

Click on this link to open a photo from the first shot.  The photo is an overlay of two video frames.  One frame shows the log the instant before the bullet struck.  The other frame shows the log at the instant that it reaches its highest position.  The overlay makes it clear that the vertical displacement is much smaller than the horizontal displacement.  The percentage uncertainty in measuring the vertical displacement would be much greater than that in measuring the horizontal.  Here is a situation where you can make good use of the experimental maxim that it's usually better to measure a larger value when possible.

  1. The unknown in textbook example 9-5 is the height to which the pendulum rises.  For this lab, however, the unknown is the velocity of the bullet.  Solve the equation for vo.
  2. In order to use the equation you just found to calculate the speed of the bullet, you need to know the mass of the bullet, the mass of the log, and the vertical height through which the log rises.  See the diagram to the right, which shows the initial and final states superimposed.  The diagram exaggerates h.  You saw from the photo that h was much smaller than the horizontal displacement, d.  You'll be measuring d from the photograph or a video clip, and you'll use that value to determine h.  Do the geometry now to express h in terms of d and L.  Include a diagram in which you label all relevant distances.

Analysis

As you carry out the analysis, show your work in an organized way and label quantities clearly.  Show equations in symbols first before substituting values.

The plan of the analysis is to obtain the horizontal deflection from the video, scale it to the actual distance, and then use it together with other data from the video clip to obtain the speed of the bullet.

  1. Determine the horizontal deflection using Method A or Method B.

Method A

  1. Print one of the photo overlays according to this scheme:  Shot 1 (for students with last names starting with A to I)      Shot 2 (J-R)      Shot 3 (S-Z)
  2. On your photo, measure d and the distance that you need in order to determine a scale factor.
  3. Calculate the actual horizontal deflection using the measurement of d on the photo and your scale factor.  Give your result in units of meters.

Method B

  1. Download one of these QuickTime video clips :  Shot 1 (for students with last names starting with A to I)      Shot 2 (J-R)      Shot 3 (S-Z)
  2. Open the video clip in Tracker.  Create a point mass.  Identify the frame in which the pendulum has reached its highest point.  Use this frame as well as the first frame to determine the horizontal displacement of the log.
  3. Still in Tracker, add a scale in order to calculate the actual horizontal deflection.  Give your result in units of meters
  1. Having determined the horizontal deflection, calculate the corresponding vertical deflection using your equation from step 6.  Give your result in units of meters.  Make a visual comparison with the photograph or video to verify that your value is reasonable.
  2. Convert the masses to the same unit.  Whatever the unit is, it will divide out in the ratio (m + M)/m.  We recommend http://metric-conversions.org for a units converter.  
  3. You have all the information you need now to calculate the magnitude of the bullet's velocity.  Round to proper significant figures as always.  Check especially carefully that your units reduce as expected, since there are many ways to make mistakes in this calculation. 
  4. In order to gain some confidence in your final result, do an online search for the typical muzzle speed of a Remington .222-calibre cartridge. Give the value and the URL of the page on which you found the value. Calculate the percentage difference between your value and the value from the web page.

Discussion

Qualitatively discuss the sources of error that you judge to be the most significant in this experiment. Describe how the error would affect your results.

Conclusion

Write a conclusion as per the lab guide.