10. Ballistic Pendulum*

Transcription

1 10. Ballistic Pendulum* Use is made of a ballistic pendulum to determine projectile velocity. Learning Objectives: 1. Explore the ideas of energy and momentum conservation, particularly the conditions under which either of these two quantities, energy and momentum, is conserved. 2. Learn to solve a certain class of multi-concept problem in which momentum is conserved during part of the motion and energy is conserved during another part. 3. Make an interesting application involving a ballistics measurement in a forensic context. Up until fairly recently a device that might be found in a police ballistics laboratory was the ballistic pendulum. This can be used to measure muzzle velocity from of a pistol or rifle. Wooden block Inelastic collision h By measuring the maximum height h to which the block rises after the bullet strikes it, the initial muzzle velocity can be worked out. * William A Schwalm

2 Reading: Before coming to lab, read the sections in your textbook on work and mechanical energy, and on linear momentum. Especially you should read about the conditions under which energy is conserved and the conditions under which momentum is conserved. They are different. Knight, Jones & Field (161): 9.1 through 9.5, 10.3 and 10.4 Serway and Vuille (211): 5.5, 6.1, 6.2 and 6.3, Serway and Jewett (251): 8.1 & 8.2 PRE-LAB EXERCISES 1. When a bullet strikes a block of wood, mechanical energy (kinetic plus potential) is not conserved. Why is this? Under what conditions is mechanical energy conserved? 2. What quantity is always conserved during collisions, if you consider a short enough time interval from just before to just after the collision? 3. Under what conditions is the total linear momentum of a system conserved? 4. A bullet moving vertically strikes a wooden block, which was initially at rest. The mass of the bullet is m and the mass of the block is M. After the collision, the bullet is embedded in the block. The block does not rotate and it rises a maximum distance L before falling back down. What was the original speed of the bullet? 10-2

3 Equipment: You have a ballistic pendulum apparatus with a spring gun. It has a physical pendulum with a ball-catching device, and a ratchet to make it stick on a toothed track, so that the pendulum won t swing back before you measure the height. There is a brass ball with a hole in it to use for a projectile. The ball slides over the rod on the spring gun and then you can cock it and launch it by pressing on the trigger bar. In the lab there are also a triple beam balance for measuring mass and a meter stick and smaller rulers for measuring distances. There should also be a plumb bob, a paper target and carbon paper that can be used for artillery practice. Exploration: Your group should mess around with the apparatus. Everyone should have a turn at cocking it and shooting it. Cocking it takes a minimum amount of strength, but you should all be able to do it. Practice shooting the ball into the catcher device on the pendulum until you can get it to catch the ball, and get the pendulum to stick at its maximum height. These skills will be useful in the experimental work. Problem 1 The problem is to derive a formula for getting the initial velocity given to the projectile by the spring gun, then to perform measurements and apply your formula to work out the initial ball speed. Finally, as always, you need to estimate the accuracy of the measurements and assess the quality of your result. 1. Method problem: How should potential energy PE of the pendulum plus ball depend on the height h of the center of mass of the combination? Give an equation with m being the mass of the ball and M the mass of the pendulum. Explain this, commenting on the reference position for the potential energy. 2. Prediction question: (Swing of the pendulum) a. Should horizontal momentum of the ball plus pendulum be conserved while the pendulum swings up to the maximum height? b. If so, why? If not, why not? c. Should mechanical energy (PE+KE) be conserved during the swing? d. If so why? If not why not? e. Thus, denoting the pendulum mass as M, relate the speed of the pendulum plus the ball, just after the collision, to the maximum height where the pendulum comes to rest. 10-3

4 3. Prediction question: (During impact) a. Is the sum of horizontal momentum of the ball plus pendulum conserved during the impact? b. If so, why and if not why not? c. Is the total mechanical energy (KE+PE) of the ball plus pendulum conserved during impact? d. If so why and if not why not? e. Thus treating the pendulum as a point object of mass M, relate the speed V of the combination after impact to the initial speed v o of the ball before impact. 4. Prediction question: Thus putting together the two previous results, show that v o m M m 2gh 5. Method question: Recalling that the center of mass would be a balance point, explain how you could find the center of mass of the ball plus pendulum in the laboratory by unscrewing the assembly from the frame and balancing it on a sharp edge for example of a ruler. How will this measurement be important in your theory, based on the questions above? 10-4

5 Plan: Write out a measurement plan for your group. Explain all the measurements you will need to make and how the group members will make them. Then explain how the data will be used to verify the predicted formula. Implementation: Carry out the plan and record the necessary data. Analysis: Use the data you have to determine the initial speed of the ball, or projectile, when it leaves the spring gun. Conclusions: What have you learnt? 1. Comment on how you have estimated the precision. 2. In doing this set of calculations and measurements, you learned about when you can apply and when you can t apply two important conservation ideas. What two ideas where these and what can you say about when they can be applied? 3. What value did you get for the muzzle velocity of the spring gun and what do you estimate for the margin of error? In other word, plus or minus how much in meters per second? 10-5

6 Problem 2 Predict the range of a projectile fired horizontally from the spring gun out across the floor of the lab. In other words, if you were to place the target at a certain point on the floor, where would you center it so that the projectile would be most apt to hit it? 1. Prediction question: If an object drops, starting from rest at a height h, how long will it fall until it hits the floor? In other words, find t for the time when it hits. 2. Prediction question: Suppose v o is the muzzle velocity you found in problem 1. Then if the time you computed in the first question is t, how far out, measured from a point just below the end of the bar on the spring gun, will the ball get before it hits the ground? 3. Method question: What experimental means will you use to find a point on the floor right below where the ball leaves the spring gun? Plan: Write out a plan for finding where the target should be placed (including a range formula, which you have to figure out your selves, not copy from somewhere) and a plan for making measurements in order to place the target. Say what data you need and what you will do with them. Implementation: place the target according to your plan. When you are ready to fire, call your TA to witness the shot. If you hit the target on the first shot, each member of the group gets one extra point. Record the results of your predictions here. 10-6

7 Analysis: Did you hit the target? If not, where do you think the difficulty lies? Be as quantitative as you can. If you think you have located a systematic problem that you know how to correct, explain the problem here and ask to take a second shot. Conclusions: What did you learn? 1. Do you think, given enough patience, some practice and the best equipment that the ballistic pendulum method would be fairly successful in determining the muzzle velocity of a pistol? About how much error do you think you would have? You have to explain this estimate. 2. Suppose you could fire the gun several times? And average the results. How much do you think this would help? 3. From your experience with other modern measurement techniques, can you suggest a better way to measure the velocity of a bullet, one that would not have been as available a few decades ago? 4. About what two conservation laws did you learn from this lab unit? Comment on what these conservation laws are and what you learned about applying them. 10-7

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