Washington University in St. Louis Introductory Physics Lab Collisions Lab Fall 2015 Modeling Collisions Pre-­‐Lab: Modeling in Physics A Bit of History This is just a joke, but it’s an old joke, so perhaps you won’t be offended by the notion of referring to it as history. Milk production at a dairy farm was so low that the farmer wrote to the local university, asking help from academia. A multidisciplinary team of professors was assembled, headed by a theoretical physicist, and two weeks of intensive on-­‐site investigation took place. The scholars then returned to the university, notebooks crammed with data, where the task of writing the report was left to the team leader. Shortly thereafter the farmer received the write-­‐up, and opened it to read on the first line: “Consider a spherical cow…”1 Introduction to Modeling Modeling is something that physicists (and all scientists, really) do all the time. You might model the motion of a car starting from rest by treating it as a point object moving with a constant acceleration. One might model a boat bobbing up and down in the water as a point mass attached to a spring. A scientist trying to estimate the density of the earth might model the earth as a uniform sphere. The list goes on and on and on. Any source that you look at will give a slightly different definition of the word model as it relates to physics. Young & Freedman write “a model is a simplified version of a physical system that would be too complicated to analyze in full detail.” Moore states that “a good physical model captures a phenomenon’s essence while remaining small and simple enough to grasp.” These definitions are as good as any, as they capture the main point that comes up in any worthwhile definition of model: simplicity. In addition to terms related to simplicity, there are other words that commonly show up in discussions of modeling. Some of these are fun, happy words like creativity and imagination. Some of the simplest models are even referred to as “toy” models. Other discussions can be rather inspirational. For instance, in his introductory physics textbook, Eric Mazur claims that students can apply skills they learn creating models to just about any profession they pursue. Hopefully, this is starting to sound like this might be worth your while! Simplifying Objects and Interactions Now for the details. When making a model, you will simplify both objects and interactions.2 Ignoring certain parts of a problem makes other parts of the problem easier to evaluate. One of the most difficult 1 Washington University in St. Louis Introductory Physics Lab Collisions Lab Fall 2015 aspects of making a good model is deciding what to ignore. In his book on modeling environmental problems, John Harte refers to this important part of making a model as “the trick.” Part of the process of simplification is making assumptions. These assumptions can be about objects (like assuming a cow is spherical) or about interactions (like assuming friction is negligible). An assumption can even fall into a gray area between objects and interactions (like assuming that the speed of some object is small compared to some other speed). When making a model, you should always clearly state your assumptions. Failure to do so could lead to significant confusion. Models Have Limitations Anytime we create or use a model, we must realize that it has limitations. An alternate punch line of the cow joke goes something like “I have a solution, but it only works for a spherical cow in a vacuum.” We must understand that our simplifications only work under certain conditions where our assumptions are met. One must make sure that what is gained is greater than what is lost as the problem is pared down. There is always a tradeoff between simplicity and accuracy. How This Relates to Scale Models Before taking a look at a couple of examples of modeling in physics, we should address the relationship between scale models (like toy cars) and models in physics. Basically, scale models are one type of model that a scientist can use, though they are much less common than the models that show up on a piece of paper or a blackboard. In some situations scale models can be just what the doctor ordered. For instance, scale models of airplanes are often placed in wind tunnels to learn how air will move around the real plane. The scale model retains the relevant features of the plane (i.e. its shape) while ignoring things like passenger seating and electrical systems that have no effect on its aerodynamics. In this lab, you will use carts and/or boxes to represent various physical objects. Your use of these objects will resemble the use of scale models in a slightly abstract way. The two long examples below are not about scale models; they are about models in a more general sense. Example 1: Car Let’s consider a couple of examples of modeling in detail. First, consider a car accelerating from rest after a traffic light has turned green. How would we typically model such a situation? That is, what objects and interactions will we simplify? First, it is very likely that you would think of the car as a point mass. You would ignore the fact that the car has an engine and wheels and a sunroof. As the real car moves, it would interact with the air and with the road. Wind resistance is tough stuff, though, so we’d probably ignore that in an Intro Physics class. The car interacts with the road due to contact between the tires and the road. We’re definitely going to have to simplify something here since we have already ignored the tires! Probably the simplest thing we could do is just treat this interaction as a constant force acting on the car in the direction of its acceleration. In doing so, we ignore the details of the road surface and the tire treads, among other things. This simplification is equivalent to assuming that the acceleration of the car is constant. 2 Washington University in St. Louis Introductory Physics Lab Collisions Lab Fall 2015 In making these simplifications, we gain ease of computation. Our constant acceleration equations are pretty user-­‐friendly. But what do we lose? What are the limitations of the model? If you’ve ever stuck your hand out the window of a car driving down the interstate, you probably noticed that ignoring wind resistance isn’t an especially good approximation at such high speeds. However, if our car is starting from rest, ignoring air resistance is a decent approximation for a pretty long time. When the air becomes relevant, our model of the car as a point mass breaks down. In fact, whether or not a car has a sunroof might actually be important at that point! Shifting our focus to the road/tire interaction, treating the interaction between the tires and the road as a constant force would certainly break down if there are pot holes or oil slicks. So, it’s clear that there are situations where our model of the car is less than perfect. However, that is not the same thing as saying it’s a bad model. There are still many situations where it is a very good approximation (like low speeds on a smooth road). And even in other situations (like a lead-­‐foot driving on a washboard road), our model might be an easy way to get a first approximation of some quantity. There is definitely something to be said for so-­‐called “hand-­‐waving” approaches or “back-­‐of-­‐the-­‐
envelope” calculations. Example 2: Baseball Now let’s consider the flight of a baseball through the air. In many cases we would choose to simplify the baseball by treating it as a point mass. As a real baseball flies, it interacts with the air and with the earth (through gravity). Interactions with the air are usually a pain to deal with, so we often simply ignore the interaction between the ball and the air. (This is equivalent to acting as if the ball is in a vacuum.) We also typically simplify the gravitational interaction between the earth and the ball by treating the force as constant. (This is equivalent to saying the acceleration due to gravity is constant.) This is usually a good simplification of Newton’s Universal Law of Gravitation, which itself is a simplification of Einstein’s Theory of General Relativity. So what are the limitations of our model of a baseball? Ignoring the air is a good approximation for low speeds. And it really simplifies the math. But as the speed of the ball increases, the interaction between the real ball and the real air gets stronger and stronger, reducing the validity of our model. Further, without air we cannot model curveballs or knuckleballs or other interesting things like that. However, there are still a wide range of problems where our model would give good predictions using relatively simple math. Our simplification of the gravitational interaction doesn’t actually limit us much. We would basically just need to be careful that our ball does not get launched into orbit. As long as the ball stays on earth, our model of gravity is just fine. Once again, though our model of the baseball is not perfect for every situation, the fact that it is so simple still makes it a powerful tool for solving many problems. 3 Washington University in St. Louis Introductory Physics Lab Collisions Lab Fall 2015 Make Your Own Model In this lab, you will be modeling various collisions and taking measurements as objects collide. In doing so, you will hopefully become more comfortable with momentum, energy, and the conservation laws associated with these quantities. In this Pre-­‐Lab, rather than focusing on momentum and energy, we will focus on modeling. (Practice problems involving momentum and energy can be found throughout your text book.) Taking the expression “toy model” somewhat literally, you will discuss how you could model children on a playground. Read This: For each of the two situations given below, do the following: a) Describe how you could simplify the objects (i.e. the child and the playground equipment). b) Describe how you could simplify the interaction(s). c) Describe one limitation of your model. Read This: If you need some inspiration, you could re-­‐read the examples of the car and the baseball. PL1. A child goes down a slide. PL2. A child rides a merry-­‐go-­‐round, holding onto a bar near the outside edge. (See Figure 1.) Figure 1: Children on a merry-­‐go-­‐round.
End of Pre-­‐Lab 4 Washington University in St. Louis Introductory Physics Lab Collisions Lab Fall 2015 Part I: Modeling Collisions Here’s the Big Idea At the most superficial level, in today’s lab you will take measurements of three collisions and perform analysis to see whether momentum and kinetic energy are conserved in the collisions. At a deeper level, however, this lab gives you a chance to get some hands-­‐on experience with modeling. Each of your collisions will be a model of a collision that could happen in real life. You’ll get to show off your creative side as you decide what pieces of lab equipment you can use to model each real-­‐life object. You will also get to put your imagination on display as you decide how your equipment will collide such that the real-­‐
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Dynamics cart (with sides C and D) Dynamics cart with spring launcher (with sides A and B) Cardboard box Masses and various bits and pieces Vernier motion sensors Equipment Notes -­‐ Carts Your collisions will either be between two carts or between a cart and the box. Here’s what happens when the various labeled ends of the carts collide. Feel free to test these out yourself. •
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A-­‐C (plunger all the way in) – The carts will stick together using Velcro. A-­‐C (plunger all the way out) – That carts will bounce off of each other. A-­‐C (plunger cocked) – The plunger will fire when the carts collide. (This is a little unreliable.) A-­‐D (plunger all the way in) – The carts will bounce off of each other. A-­‐D (plunger all the way out) – That carts will bounce off of each other. A-­‐D (plunger cocked) – The plunger will fire when the carts collide. (This is a little unreliable.) B-­‐C – The carts will not make physical contact. Their magnets will repel each other. B-­‐D – The carts will not make physical contact. Their magnets will repel each other. Equipment Notes -­‐ Possible Modifications There are a few bits and pieces that are available to use in your collisions. You will notice that you are not provided tape of any kind. Please do not use any tape of any kind to modify the carts or tracks. Tape can leave behind residue that can adversely affect the performance of the carts and tracks. Wire and thread can be looped through the small holes in the black plastic pieces on the ends of the carts. •
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Masses Rubber bands Bungee cord Wire Thread 5 Washington University in St. Louis Introductory Physics Lab Collisions Lab Fall 2015 Be aware that adding too many modifications to the carts or box could make your model confusing and could make your data hard to understand. Modifications should only be made if they will improve your model. Clever modifications will probably make the lab more enjoyable to perform. In Section 1, you will be asked to restrict your modifications to adding masses. In Sections 2 and 3, you may use any of the available pieces to modify the equipment. But remember, no tape! The Questions In each of your three collisions, you will consider the following points. •
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How will you model each object? How will you model the interaction between the objects? Will momentum be conserved in the collision? Will kinetic energy be conserved in the collision? What are the limitations of your model? 1. Your First Model Here’s the list of collisions that you can choose to model in your first collision. (This is kind of a “choose your own adventure” lab.) a) Thomas the Tank Engine slams into and couples to Gordon, who was peacefully resting with his brakes off. (Gordon becomes very cross.) b) A large bumper car hits a small bumper car that is trying to get away. c) An alpha particle (two protons and two neutrons) runs head-­‐on into a resting lithium nucleus (three protons and four neutrons). They do not fuse. d) A bowling ball runs head-­‐on into a bowling pin. e) A linebacker breaks through the defensive line and tackles a helpless punter. f) Two billiard balls smack into one another. g) A reckless driver in a big hurry tries to pass a safe trucker on the interstate, but instead rear-­‐
ends the law-­‐abiding trucker. Let’s be very clear about how this works. Pick a scenario from the list above that you think you can model using the equipment on the lab bench. You will be asked to explain what piece of equipment models each real-­‐life object and why you think these choices are reasonable. You will also have to make an appropriate decision concerning how the equipment will interact (i.e. which parts of the equipment will collide). In short, you must convince your TA that the essence of each object and interaction is preserved in your simplified representation. Then, using your knowledge of collisions, you will predict what quantities should be conserved when your pieces of equipment collide. After that, you will test your prediction by performing and analyzing an experiment. Finally, you will discuss limitations. If that sounds like a lot to keep straight, don’t worry! Here are the step-­‐by-­‐step instructions. 6 Washington University in St. Louis Introductory Physics Lab Collisions Lab Fall 2015 STOP Do This: Make sure your track is level. You can adjust the length of the legs using the screws. Do This: Go to the In-­‐Lab Links page and download the Logger Pro template. Unzip the file (LoggerProCollisionsTemplate.zip) and open up LoggerProCollisionsTemplate.cmbl. This template corrects for the fact that the two motion sensors are pointing in opposite directions. Students who do not take the time to download and open this often get very confused. Read This: In your first collision, the only modification that you are allowed to make to the carts or box is to add mass. Please do not use any of the other bits and pieces in you first collision. You may do more interesting modifications in Sections 2 and 3. Checkpoint 1.1: What scenario are you going to model? Checkpoint 1.2: Which piece of equipment will you use to model each object? Don’t forget to include something about the mass of the objects and the analogous equipment. Justify your choices. Checkpoint 1.3: How will you model the interaction between the objects? That is, which sides of the equipment are going to collide? Justify your choices. Checkpoint 1.4: Consider the momentum of the system (i.e. the colliding equipment). Predict whether or not the momentum of the system will be conserved in the collision. Discuss your reasoning. Further, if you expect the momentum to decrease, where will this momentum end up? Or if you expect the momentum to increase, where does this momentum come from? Checkpoint 1.5: Consider the kinetic energy of the system. Predict whether or not the kinetic energy of the system will be conserved in the collision. Discuss your reasoning. Further, if you expect the kinetic energy to decrease, where will this energy end up? Or if you expect the kinetic energy to increase, where does this energy come from? Do This: Perform the collision, measuring the velocities of each piece of equipment before and after the collision using the motion detectors. See Appendix A for instructions on how to collect data and two different methods to determine the velocity of a cart from the plots produced in Logger Pro. Checkpoint 1.6: Draw a sketch of the carts before the collision. The sketch should include the mass of each cart and the velocity of each cart (magnitude and direction!). Checkpoint 1.7: Draw a sketch of the carts after the collision. The sketch should include the mass of each cart and the velocity of each cart (magnitude and direction!). Read This: What is the purpose of the sketches? Don’t forget that momentum is a vector, which means that the direction of the motion is important. If you sketch your experiment, you will be less likely to make the mistake of ignoring the direction of the motion. (Recall that kinetic energy is not a vector. It is calculated using the object’s speed, not its velocity.) 7 Washington University in St. Louis Introductory Physics Lab STOP Collisions Lab Fall 2015 Read This: In your responses to all three Synthesis Questions today, you will need to quote and discuss initial and final values for momentum and kinetic energy. However, you do NOT need to show HOW you calculate momentum and kinetic energy values. By now, we should all be !
comfortable with plugging numbers into 𝒑 = 𝑚𝒗 and 𝐾 = 𝑚𝑣 ! . As long as your measured !
masses and velocities are displayed clearly, these calculations are trivial and need not be shown. Your measured masses and velocities can be displayed with a sketch or in a data table. Read This: To reiterate, your TA wants to see your measured data, your initial and final momentum, and your initial and final kinetic energy. You do NOT need to show the calculations. Your measured data can be displayed with a sketch or in a data table. S1
Synthesis Response 1 (35 Points): First, explain what scenario you modeled and how you modeled it. Justify choices you made. Next, use your data to show whether or not kinetic energy and momentum were conserved in your model collision and explain why these quantities were or were not conserved. (When comparing final and initial values, please show a percent difference. See Appendix B for details.) Finally, comment on whether or not you would expect those same results to hold in the real-­‐life collision that you were modeling, being sure to mention at least one limitation of your model as part of your commentary. (Please make sure you understand the two most recent Read This paragraphs so that you don’t waste time.) 2. Your Second Model Here’s the list of collisions that you can choose to model in your second collision. This list is supposed to be a little bit tougher. a) A car carrying a suspicious amount of nitroglycerine in the trunk is rear-­‐ended. b) Ozzie Smith drops down a bunt that basically falls straight down. c) A jack-­‐in-­‐the-­‐box on its side is slid across a very smooth table and runs into a comparably sized box of Legos. d) A pickup truck carrying an unsecured payload rear-­‐ends another vehicle. The payload slides forward during the collision. e) Two children in shopping carts joust using pogo-­‐sticks. f) Two astronauts are on a space walk outside the ISS, connected to each other by a tether. One of the astronauts jumps away from the ISS, pulling the second astronaut off of the space station. g) A car with a well engineered crumple zone crashes into another vehicle. Read This: That list is the main difference between Section 1 and Section 2. The following steps should look pretty familiar. You may also feel free to use the available bits and pieces (rubber bands, etc., but not tape!) to modify your equipment. Checkpoint 2.1: What scenario are you going to model? 8 Washington University in St. Louis Introductory Physics Lab Collisions Lab Fall 2015 Checkpoint 2.2: Which piece of equipment will you use to model each object? Don’t forget to include something about the mass of the objects and the analogous equipment in your response. Explain any modifications that you make. Justify your choices. Checkpoint 2.3: How will you model the interaction between the objects? That is, tell you TA which sides of the equipment are going to collide and/or describe any modifications you made to the equipment that will affect the way they interact. Justify your choices. Checkpoint 2.4: Consider the momentum of the system (i.e. the colliding equipment). Predict whether or not the momentum of the system will be conserved in the collision. Discuss your reasoning. Further, if you expect the momentum to decrease, where will this momentum end up? Or if you expect the momentum to increase, where does this momentum come from? Checkpoint 2.5: Consider the kinetic energy of the system. Predict whether or not the kinetic energy of the system will be conserved in the collision. Discuss your reasoning. Further, if you expect the kinetic energy to decrease, where will this energy end up? Or if you expect the kinetic energy to increase, where does this energy come from? Do This: Perform the collision, measuring the velocities of each piece of equipment before and after the collision using the motion detectors. Checkpoint 2.6: Draw a sketch of the carts before the collision. The sketch should include the mass of each cart and the velocity of each cart (magnitude and direction!). Checkpoint 2.7: Draw a sketch of the carts after the collision. The sketch should include the mass of each cart and the velocity of each cart (magnitude and direction!). S2
Synthesis Response 2 (35 Points): First, explain what scenario you modeled and how you modeled it. Justify choices you made. Next, use your data to show whether or not kinetic energy and momentum were conserved in your model collision and explain why these quantities were or were not conserved. Finally, comment on whether or not you would expect those same results to hold in the real-­‐life collision that you were modeling, being sure to mention at least one limitation of your model as part of your commentary. 3. A Novel Scenario For the final section, create your own scenario where objects collide and model it as you did in Section 1 and Section 2. You will perform the same kind of analysis as well. Here’s a chance to be really creative and wow your TA. Checkpoint 3.1: What scenario are you going to model? Checkpoint 3.2: Which piece of equipment will you use to model each object? Don’t forget to include something about the mass of the objects and the analogous equipment in your response. Explain any modifications that you make. Justify your choices. 9 Washington University in St. Louis Introductory Physics Lab Collisions Lab Fall 2015 Checkpoint 3.3: How will you model the interaction between the objects? That is, tell you TA which sides of the equipment are going to collide and/or describe any modifications you made to the equipment that will affect the way they interact. Justify your choices. Checkpoint 3.4: Consider the momentum of the system (i.e. the colliding equipment). Predict whether or not the momentum of the system will be conserved in the collision. Discuss your reasoning. Further, if you expect the momentum to decrease, where will this momentum end up? Or if you expect the momentum to increase, where does this momentum come from? Checkpoint 3.5: Consider the kinetic energy of the system. Predict whether or not the kinetic energy of the system will be conserved in the collision. Discuss your reasoning. Further, if you expect the kinetic energy to decrease, where will this energy end up? Or if you expect the kinetic energy to increase, where does this energy come from? Do This: Perform the collision, measuring the velocities of each piece of equipment before and after the collision using the motion detectors. Checkpoint 3.6: Draw a sketch of the carts before the collision. The sketch should include the mass of each cart and the velocity of each cart (magnitude and direction!). Checkpoint 3.7: Draw a sketch of the carts after the collision. The sketch should include the mass of each cart and the velocity of each cart (magnitude and direction!). S3
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Synthesis Response 3 (30 Points): First, explain what scenario you modeled and how you modeled it. Justify choices you made. Next, use your data to show whether or not kinetic energy and momentum were conserved in your model collision and explain why these quantities were or were not conserved. Finally, comment on whether or not you would expect those same results to hold in the real-­‐life collision that you were modeling, being sure to mention at least one limitation of your model as part of your commentary. Read This: One part of many of these collisions that you likely chose to ignore was human brains. While it’s very difficult to delve into the details of how brains react to a collision using intro lab equipment, Professor Bayly of the Wash U Mechanical Engineering Department uses sophisticated instruments to study how the brain reacts to rapid accelerations. Check out the lab website for links to more information. Time to Clean Up! Please clean up your station according to the Cleanup! Slideshow found on the lab website. 10 Washington University in St. Louis Introductory Physics Lab Collisions Lab Fall 2015 References [1] Harte, John. (1985). Consider a Spherical Cow. William Kaufmann, Inc. Los Altos, CA. [2] Etkina et al. “The Role of Models in Physics Instruction,” The Physics Teacher, Volume 43, 2005. 11 Washington University in St. Louis Introductory Physics Lab Collisions Lab Fall 2015 Appendix A: Producing and Analyzing Velocity Plots Collecting Data Start collecting data by clicking . Data will be collected for 10 seconds, after which time data collection will automatically stop. If you are unsure about how the sensors work and what the plots mean, we would recommend that you play around with things until you get it figured out. Once you have collected data, here are two ways you might want to analyze it. Finding an Average Velocity Click and drag the mouse over a portion of the plot that you would like to average. A blue rectangle indicates the portion of the plot that you have selected. Then click the average velocity over the time that you have selected. . This will calculate and display Finding an Instantaneous Velocity When you turn the “Examine” tool on, a vertical line will appear on the plot. As you move your mouse, the vertical line will move. Logger Pro will display the y-­‐value at which this vertical line intersects your data. That is, it gives an instantaneous velocity value. To turn on the “Examine” tool, click 12 . Washington University in St. Louis Introductory Physics Lab Collisions Lab Fall 2015 Appendix B: Comparing Values Quantitatively When comparing two values, do so quantitatively. Terms like “very close” are not sufficient. Ideally, two values would be compared using their uncertainties. We did this in the Measurement Lab. However, in this course we don’t always have rigorous uncertainty values. In cases where we don’t have a good estimate of the uncertainty, our best option is usually a percent difference. The percent difference between two values 𝑎 and 𝑏 is given by 𝑎−𝑏
𝑎−𝑏
precent difference = ×100% or ×100% 𝑎
(𝑎 + 𝑏)/2
The first formula can be used when there is one value (𝑎) that you trust more. 13