Impulse Brief

8/13/2019 Impulse Brief

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Teaching About

ImpulseandMomentum

an AAPT/PTRA Manualby

Bill FranklinPTRA and retired physics teacher

Technical ConsultantRobert Beck Clark

Texas A&M University, College Station, TX

Editorial Review BoardLarry Bader

Case Western Reserve University, Cleveland, OHRobert Beck Clark

Texas A&M University, College Station, TXJim Nelson

Seminole County Public Schools, Sanford, FL

Supported in part by

The National Science Foundation

Opinions expressed are those of theauthors and not necessarily thoseof the Foundation.


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2005 AAPT Teaching About Impulse and Momentum i

Table of Contents

Preface and Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .i i i

1 Introduction

What This Manual Is All About . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1.22 Background Material

The Impulse-Momentum Equation Is Just Newtons Second Law. . . . . . . . . . . .2.2When Are Momentum and Energy Conserved? . . . . . . . . . . . . . . . . . . . . . . . .2.2Newtons Cradle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3Bumpers, Fender Crumpling, Seat Belts, and Air Bags . . . . . . . . . . . . . . . . . . . 2.4Analysis of a Ticker Tape Collision Lab . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4Low-Friction Environments: Hovercraft and Winter Sports . . . . . . . . . . . . . . . .2.6Sample Data and Error Discussion for the Air Impulse Rocket Lab . . . . . . . . . . 2.7Slow Collisions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.8Hoses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2.10Rockets, Propellers, and Sails . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.12Airplane Lift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.12Curve Balls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2.14An Interesting ApproachWould You Use It? . . . . . . . . . . . . . . . . . . . . . . . .2.15Articles Reprinted by Permission From The Physics Teacher

Newtons Cradle and Scientific Explanation . . . . . . . . . . . . . . . . . . . . .2.16Figuring Physics (ice sailcraft) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2.21The Answer Is Obvious. Isnt It? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2.22

3 Laboratory ActivitiesEgg Pitching Activity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.2Labs With One-Dimensional Collisions and Explosions . . . . . . . . . . . . . . . . . .3.2

1. Head-on collisions of rolling carts or air track gliders . . . . . . . . . . . . . .3.22. Head-on collisions of bifilar pendula . . . . . . . . . . . . . . . . . . . . . . . . . .3.33. Air impulse rocket . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.3

4. Collision of a cart or air track glider with a force sensor . . . . . . . . . . . .3.35. Spring powered explosions of rolling carts or air track gliders . . . . . . . .3.46. Tennis ball or bottle stopper cannons . . . . . . . . . . . . . . . . . . . . . . . . . .3.47. Firecracker between cans . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.48. Ballistics pendulum (or box) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.4

Labs With Two-Dimensional Collisions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.51. Colliding air table pucks, air pucks, and billiard balls . . . . . . . . . . . . . .3.52. Colliding pendula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.5

Air Impulse Rocket and Launcher . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.6Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.6Parts List . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.8An Impulse Lab Using the Air Rocket . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.9Data Collection Sheet for the Air Impulse Rocket . . . . . . . . . . . . . . . . . . 3.11

The Reaction Force on a Hose . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.12Bungee Cart (CBL) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.13Student Hovercraft Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.15

4 DemonstrationsThe Tablecloth Trick . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4.2Multi-Purpose Cart Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4.3Cart Accessories: Fan Cart With Sails . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4.7Cart Accessories: Elastic and Inelastic Collisions . . . . . . . . . . . . . . . . . . . . . . .4.10The Fan-Driven Sailboat . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4.11

Building a Fan Cart to Demonstrate Momentum Exchanges with Air . . . .4.12Parts List for the Fan Cart . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4.14


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ii Teaching About Impulse and Momentum 2005 AAPT

Newtons Cradle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4.15Clackers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4.15Toppling a Block With a Pendulum Blow . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.16Slime Balls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.16Sad and Happy Balls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.16Cheapskate Elastic/Inelastic Collision Demonstrator . . . . . . . . . . . . . . . . . . . 4.17Stacked Balls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.18Coefficient of Restitution Apparatus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.18Rotating Sail Magnus Effect Demonstrator . . . . . . . . . . . . . . . . . . . . . . . . . . .4.19

Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4.19The Forces on a Rotating Object in Flowing Air . . . . . . . . . . . . . . . . . . . 4.20Parts for the Rotating Sail Magnus Effect Demonstrator . . . . . . . . . . . . .4.20

Homemade Fan Unit, Robert Morse Design . . . . . . . . . . . . . . . . . . . . . . . . . .4.21

5 Assignments and Homework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5.2

6 Computer ApplicationsOverview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2

1. Laboratory interfacing programs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.22. Data analysis programs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.23. Simulation programs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2

4. Problem generating programs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.25. Websites with useful information . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2Simulation Software . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3

1. Interactive Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.32. OnScreen Particle Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.33. Websites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.34. Physlets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .6.3

A Graphing Calculator Solution for Elastic Collisions . . . . . . . . . . . . . . . . . . . 6.4

7 Media ResourcesCinema Classics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .7.2Songs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .7.2

M Times V . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .7.2Pool Table Physics Rap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2

An Impulse to Sing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.38 Physics Olympics or Contests

Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2Inertia Ball . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2Egg Drop, No Soup Please . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3Bernoulli Ball . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3

9 Modern Physics ApplicationsOverview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2Classical (Except for Varying Mass) Applications of Momentum . . . . . . . . . . . . 9.2Nonclassical Applications of Momentum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3Symmetries and Conservation Laws in Quantum Mechanics . . . . . . . . . . . . . . 9.3Laboratory Activities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4

10 Assessment QuestionsIntroduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2Sample Test Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .10.2Sample Test Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .10.3

11 Appendix APeriodical Subject Categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .11.1Periodical Resource List . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .11.2

12 Appendix BApproximate Timing for Some Activities Appropriate for Teacher Workshops . .12.1


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2005 AAPT Teaching About Impulse and Momentum iii

PrefaceAs a veteran of more than 30 years of teaching high school physics students and 20 years of pro-

viding in-service training for other physics and physical science teachers, I wrote this manual as aguide for teaching about a topic that I consider to be very important in developing a big-pictureunderstanding of the physical world. I hope that it will prove to be useful for classroom teachers, mytarget audience, and also for those providing pre-service or in-service training for teachers.

The manual includes a number of unique demonstrations and laboratories that I have developedover the years, along with instructions for building the apparatus for them quite inexpensively.Several of these designs have won low-cost apparatus awards in AAPT apparatus contests. It also con-tains references to many standard laboratories, which are not duplicated in the manual, but which arewidely available elsewhere.

By design, I do not provide a rigid sequence of activities. Rather, I present an array of possibilitiesthat the teacher can select to suit a given audience. Some of the included activities are suitable forintroductory level students; others are challenging for the advanced placement or college level. Myaim is to build teachers conceptual understanding so that they will feel comfortable making theirown curriculum decisions, rather than being dependent upon the decisions of others.

I do encourage the reader to include an emphasis on automobile safety for every audience.Everyone needs to be persuaded to use seat belts to reduce collision forces and to keep passengers

within the vehicle structure. Everyone should also be aware of how the survival of occupants dependsupon the relative velocities and the crumple zones of colliding vehicles. In my opinion, physics stu-dents should leave our classrooms with a heightened awareness of the importance of driving safely,and as advocates for laws requiring crumple zones in all vehicles. Some guidance for preparingteacher workshops is provided in Appendix B.

AcknowledgmentsI would like to thank the following people for their contributions to this manual:

Robert Beck Clark, Texas A&M University, for specific suggestions and permission to reprint hisarticle about driving a boat with a fan as well as for much support and encouragement.

Judy Franklin, my wife, for her active involvement at every stage, from encouraging me to talk through

explanations, to proofreading, to song writing, to shopping for materials and bagging them into kits. Albert Bartlett, University of Colorado, Boulder, for many patient exchanges about hoses and nozzles. David Gavenda, University of Texas, Austin, for permission to reprint his article about Newtons cra-

dle and his help in improving my discussion of it. Mike Saathoff, for help with computer problems, for his help over the years in improving the test writ-

ing program that we shared and the bank of questions we wrote for it, and for many hours talkingphysics.

Jim Nelson, for his encouragement and his tireless efforts on behalf of physics teaching in generaland the PTRA program in particular.

Clarence Bakken, for offering his bungee lab and providing access to it through his website. Paul Hewitt, Judith Edgington, and James Earle, for permission to reprint their materials. Cathy Ezrailson, Jane Nelson, Rob Adams, Nina Daye, and Steven Henning, for suggesting available

materials that should be included in the manual. Scot Gill, Warren Hein, and Robert Morse, for reporting problems and offering suggestions regarding thefan cart.

Peggy Schweiger, for allowing me to include a graphing calculator solution for one-dimensional elas-tic collision problems, as well as suggesting some Internet resources.

Robert Morse, for allowing me to reprint his fan unit construction plans. Jane Chambers, for reformatting the entire book, checking it carefully, and incorporating many

additions and corrections patiently.Any errors in the manual are the responsibility of the author, not of these generous people whohelped. Bill Franklin


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Section

1

Introduction


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What This Manual Is All About

The title of this manual contains its central message: that impulse should be uttered in the samebreath with momentum. There are many situations in which momentum is conserved (or nearly so),and impulse need not be explicitly invoked to get a useful solution using momentum. But there aremany more situations in which momentum changes, and solutions can be found only by considering

impulse. Even when momentum is conserved, you can only know that by convincing yourself thatthe net impulse on the system in question is zero. The proper approach, then, is always to examineimpulse first.

Even when introducing momentum to students, it is useful to provide experiences involvingimpulse. Most of us develop a definition of momentum as the product of mass and velocity by askingstudents to consider the difficulty of stopping objects of various masses and speeds. Why not letthem feel it? Roll balls of different masses across the table at them at about the same speed. Then rollthe same ball at different speeds. Which is harder to stop? This works best when one of the balls isquite massive, such as a bowling ball, but be careful to avoid injuries! Depending upon the age andexperiences of your students, you may find it adequate to do this with a single student, or you maywant to have the whole class pair up and roll balls to one another.

A good activity at this point is having students ride on a hovercraft. Instructions for making andusing one appear in Section 3 (Laboratory Activities). A brief discussion, combined with winter sports,is in Section 2 (Background Materials).

Another excellent activity is to have an inertia ball contest. Details are given in Section 8(Physics Olympics), but the basic idea is to have students use a broom to push a bowling ball around acourse with turns, so that at various points in the path they have to speed it up, slow it down, or turnit. In the process, they learn that objects require forward forces to speed them up, backward forces toslow them down, and sideways forces to make them turn. This will help them understand the vectornature of momentum and, again, they feel this by providing the impulse necessary to cause a changein momentum.

Students who have learned to appreciate momentum by applying impulse will find it very natural

to define net impulse as the product of net force and the time it acts, and to connect this to themomentum change that it produces, arriving at net impulse = change in momentum ornet F t= (mv). There are many good examples of this, such as seat belts and air bags, or bendingyour knees as you come down from a jump.

This equation can be shown to be equivalent to Newtons laws, and should be for students inphysics, but perhaps not for those in lower level courses. However, that should come after the rela-tionship is developed experientially. A derivation is included in Section 2 (Background Material).

After the impulse-momentum connection is made plausible through demonstration (Section 4)and discussion, it should be tested experimentally. Several options for doing this are provided inSection 3 (Laboratory Activities). Once the relationship has a firm experimental validity, then it istime to explore less obvious applications, such as airplane lift or boosting spacecraft during planetary

encounters. Some of these appear in Section 2.Eventually, after students have studied work and energy, the parallels between momentum and

energy should be emphasized. In the same way that the impulse-momentum equation tells us not justwhen the momentum of a system is conserved, but also how much it is changed by a net impulse;the first law of thermodynamics tells us not just when the energy of a system is conserved, but alsohow much it is changed by work or heating. In both cases, it is crucial to identify the system thatyou wish to study, and to find all of the external forces (and thermal inputs, in the case of energy).

Section

1


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Section

2

Background Material


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Section

2

Rockets, Propellers, and Sails

Rocket propulsion is just like the hose reaction discussed above. The rocket expels hot gases athigh speed, and the gases exert an equal reaction force on the rocket. The same is true of a jet engineon an airplane, or a balloon with air escaping from its neck. Even more like the hose is the water jetof a jet ski or a squid. (The air impulse rocket discussed above is less a rocket than a projectile, since

it gets only an initial pulse, rather than continual propulsion.)Not very different is the deflection of a fluid by a propeller or a sail, both of which are involved in

the Fan Cart demonstration (Section 4). In either case, the fluid and the surface interacting with itexert impulses on one another.

A boat or airplane propeller propels the vehicle forward by pushing fluid backward. Of course,the vehicle then has to push fluid out of the way, encountering drag. In the case of a boat, drag canbe reduced by raising the hull most or all of the way out of the water. At high speeds this is accom-plished with a relatively flat, sloping bottom that pushes downward on the water, and is pushed upby it. At more moderate speeds, underwater planes, essentially wings beneath the waves, can lift thehull above the waves, making travel more comfortable, as well as more efficient. Hovercraft stayabove the waves by pushing air downward, and propel themselves forward by pushing air backward.This nearly eliminates any water drag, making them capable of relatively high speeds.

Boats generally meet much less resistance moving forward than moving sideways. In the case ofsailboats, an extended keel is usually used to exaggerate this. Wind deflected by the sail exerts animpulse on it. The momentum change of the deflected air varies with the airs speed, its directionchange, and the area of the sail. A skilled sailor adjusts the area and angle of the sails to get a largeforward force without tipping the boat over. This is easier if the wind is from the side or behind, butit is possible to sail surprisingly close to directly upwind. The keel makes this possible by preventingsideways motion. If the force exerted on the sail has even a small forward component, forwardprogress is possible, while the keel resists a large sideways component. Paul Hewitt describes how touse a sail-equipped cart to show this in his article Sailboat Demonstration in the February 1968issue of The Physics Teacher.

Propelling a cart (or boat) by aiming a fan into a sail, both of which are mounted on it, is the

subject of a demonstration treated in some detail in Section 4. Related articles from The PhysicsTeacher appear at the end of this section.

Airplane Lift

In 1970, Norman Smith, a NASA engineer disgusted by text-book treatments of airplane lift, submitted an article on the subjectto The Physics Teacher. To his astonishment, it was rejected. Heasked a friend, the science coordinator in my district, to set up ameeting with physics teachers so that he could find out whatphysics teachers didnt like about his article.

He showed us a textbook illustration similar to the top sketch at

the right, which pictured a wing with a flat horizontal bottom anda curved top with air flowing around it. The text claimed that theair flowing over the top had to travel farther, and because it had tomeet back up with the air it was separated from at the leading edge of the wing, had to travel faster.

Smith quite correctly denied that that was the case, pointing out that some wing shapes weresymmetrical. But he went on to say that because this explanation of why air travels above the wingfaster than it does below it was incorrect, the Bernoulli principle argument used to explain lift wasincorrect. He argued that the correctway to explain lift was in terms of Newtons third law. The wing


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2005 AAPT Teaching About Impulse and Momentum 2.13

is tilted upward at the leading edge, as in the bottom sketch to deflect air downward, forcing the airto push up on the wing.

This certainly is a correct way to explain lift, but I was surethat the Bernoulli effect was valid as well, and that it was hisdenial of this that had triggered the rejection of his article. I drewa sketch like that at the right, and pointed out that tilting thewing required it to push forward, as well as downward, on the airbelow it. This slows the air as well as deflecting it downward. Onthe other hand, the air above the wing is pushed backward as wellas upward by the wing, speeding it up. This means that air ismoving faster over the top of the wing, and gives a valid way toexplain the speed and pressure differences needed for a Bernoulli explanation. (In the sketch only theforce is labeled on the top, and only the components on the bottom, to reduce the welter of arrows.)

I urged him to rewrite the article to advocate the third law argument as a betterway to deal withlift, but to avoid saying that a Bernoulli explanation was wrong. His revised article was printed in theNovember 1972 issue. It helped to extinguish many textbook errors, and it remains a good reference.

One advantage of the momentum approach is that it can deal more naturally with the fact that

wings need to be tilted more steeply (or have flaps lowered) to maintain lift at low speeds. A lowerspeed means that the mass of air, m, flowing by in a given time interval, t, is smaller. The impulsemust remain equal to the weight of the plane times t, however, or the plane would fall. Thereduced m must be compensated by a larger v (in the vertical direction), which requires a greatertilt of the wing (or flaps).

Recently, I found a wonderful source for explaining lift (and many other aspects of flying). It is asoon-to-be-published book See How It Flies(ISBN 7016405) byJohn Denker. It will be available from McGraw Hill for about $35.As of March 2004, it is available on the Internet. The main menuis at: http://www.av8n.com/how. Chapters can be accessed fromthere.

Denker pointed out another way to explain the pressure differ-

ences that provide lift. The sketch at the right approximates theflow around a wing producing lift by deflecting air downward.Note that this causes the air to follow a curving path. Like any-thing in a curving path, it requires a centripetal force and can onlyget it from the pressure of the air around it. That means the pressure must decrease downward fromthe wing so that each layer (shaded band) of the flow has higher pressure above than below it. Sincethe pressure is atmospheric far away from the wing, it must be higher than atmospheric near it.

A similar argument applies to the air above the wing. The pressure must increase upward above itso that each layer has more pressure above it than below it. Again, the pressure is atmospheric faraway, therefore must be lower than atmospheric near the wing. Thus the pressure below the wing ishigher than that above, based only on the fact that air is deflected downward by the wing, with no

specific reference to either the Bernoulli effect or Newtons third law.In addition, lift can be ascribed to circulation around the wing or to the production of vortices.These also are treated by Denker in an unusually accessible manner, but I think that these moreabstract approaches are more difficult for high school students.

The nice thing is that all of these explanations are compatible. You could begin early in the yearwith a Newtons laws explanation. Then you could return to look at it in terms of impulse andmomentum, which is very nearly the same approach. When centripetal force is studied, lift becomesanother interesting application for that. And when you get to fluids, you can add the Bernoullisprinciple, and perhaps even circulation and vortices. What a lot of approaches there are, and whatconnectedness lift provides among them!

Background

Material


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Section

3

Laboratory Activities


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An Impulse Lab Using the Air Rocket

Overview and apparatus:

The rocket is accelerated by a pulse of air pressure. We will calculate the net impulse during theacceleration from pressure measurements and compare it to the momentum change found by mea-suring the launch speed directly, using a light sensor and a flag attached to one fin of the rocket.

Only if these agree within our experimental uncertainty, will we find support for the statement thatnet impulse equals momentum change.

To collect the necessary data, drill a 3/8 hole in the elbow and insert a 1/4 I.D., 3/8 O.D.polyethylene tube in it, long enough to extend to the exit end of the launch tube, as shown below.The 1/4 O.D. tubing of a pressure sensor fits tightly into the lower end of the tube, and the upperend samples the pressure experienced by the rocket. Also tape a 10 cm long posterboard flag to onefin of the rocket. We will find the launch speed by measuring the time that this flag interrupts thelight reaching a light sensor.

The pressure sensor must be connected to a computer or calculator capable of collecting data veryquickly; the launch will take only milliseconds! This can be done with almost any interface system,since data can be collectedin the lab by firing therocket into a box. Butthese instructions willrefer to the TexasInstruments CBL system,on the grounds that it isjust more fun to go out-side where the rocket can go places.

Setting up the program:

Verniers PHYSICS program makes data collection pretty easy. If you dont have it, you can copy itfrom the calculator of someone who does, or you can download it from TI (http://www.ti.com) or Ver-

nier (http://www.vernier.com). To get it from your computer to your calculator, you will need TI Graph-Link software and cables. There are different versions for the TI-82, 83, 86, etc. The software is free.Once the program is in your calculator, connect the calculator to the CBL with a link cable, plug

the pressure sensor into channel 1 and the light sensor into channel 2 of the CBL, and connect thepressure sensor tube to the tube emerging from the elbow of the launch tube.

Turn on both the CBL and the calculator. Select the PHYSICS program and follow the menus.Specify 2 probes, pressure in channel 1 and light in channel 2. Accept the stored calibrations for bothsensors. For pressure use kPa for units. (1 kPa = 1000 N/m2.) The range for light isnt critical, sincewe only look for a dip in the level as the flag goes by.

Next go to the options menu and set up triggering on channel 1. For a threshold, use 110. (At-mospheric pressure is about 100 kPa.) For prestore, use 20. This means that the calculator will keepsome data prior to the trigger point, in this case, 20% of the total. This is needed to ensure that you

dont miss the beginning of the event. It will also give you a reference level for atmospheric pressure.From the collect data menu, pick time graph. For time between samples, use 0.0003 seconds. For

number of samples, use 300. (Except for the TI-82, which can only collect 99 samples. For it, use0.0006 seconds and 99 samples. You will get less detailed, but usable, data.) Then you agree to usethe time setup, and alert the CBL. The calculator tells you that it is waiting for the trigger.

Collecting data:

Inflate the bottle if it is flat, place the rocket over the launch tube, adjust the launch angle, and besure that the flag will interrupt light to the light sensor very shortly after the rocket has cleared the

LaboratoryA

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Section

3


end of the launch tube. Make sure that the rocket wont hit anyone, then stomp on the center of thebottle. In a few seconds, the calculator will tell you that time data is in list 1, pressure data is in list 2,and light data is in list 3. When you press enter, the pressure vs. time graph is displayed. Press enteragain for the light intensity vs. time graph.

The calculator is in trace mode during the display of these graphs, so you can read points of inter-est by moving the cursor with the right and left arrows andreading the x (that is, t) and y (pressure or light intensi-ty) values at the bottom of the screen. If you are satisfiedwith the graphs or if you want to change the settings, thendecline the offer to repeat by selecting no. If the settingsare OK, but the data is not, then select yes.

If you selected no, then you are back at the main menu.You may change the triggering or data collection values andtry again, or if you were happy with the data, you mightwant to consider the options menu, where you can select aregion or integrate to find the area of the pressure vs. timegraph during the acceleration. Or you can quit, then look at

the data columns bygoing to STAT andEDIT. If you haveGraphLink, you cantransfer the data to acomputer to get amore detailed graph,to print it, or tohave more sophisti-cated ways to ana-lyze the data.

Analyzing the data:We want to find

the net impulse (theproduct of net F andtime) given to therocket and, fromthat, the launchspeed. Since p =F/A, then F = p*A,where A is the areaof the rocket cross

section. Since wewant net F, andatmospheric pressureacts on the front of the rocket, we need to find (p p atm)*A. The net impulse is the net F*t, summedup over the time of the impulse. This is just the shaded area of the graph. Find it in any convenientway. (One way is using integrate from the options menu, but be sure to subtract atmospheric pres-sure.)

We can estimate the launch speed by setting the net impulse equal to the momentum change,m*v. Check this against the speed from the light sensor data. Do they agree?

Do you see any other interesting features of the graph?

The same graph as above, drawn on a computer.

A graph from the screen of a TI-83.


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Data Collection Sheet for the Air Impulse Rocket

Name:________________________________________________Date:_____________________

Partners:________________________________________________________________________

Be very careful with units. Record uncertainties for all measurements. Check all results forreasonableness.

1. The mass of the rocket is ____________________ g = __________________kg.

2. The length of the flag is __________________ cm = _________________m.

3. The flag interrupted the light from __________ s to ___________________s.

4. This time interval is __________________s.

5. The launch speed is _______________________ m/s.

6. The rockets momentum change during launch is ____________________kg m/s.7. The launch tube diameter is ________________ cm = ________________m.

8. Its cross-sectional area is ____________________ m2

9. Atmospheric pressure is ______________ kPa = _______________N/m2. (1 kPa = 1000 N/m2)

10. The pressure rises above atmospheric at _________s and drops sharply at ________s.

11. The duration of the pressure pulse (difference between times in step above) t = _______s.

12. The area of the pressure vs. time graph during the impulse is pavgt = _____________N s/m2.

13. Atmospheric pressure times the time of the impulse is patm t = __________________N s/m2.

14. The difference between the last two items (1011) is (pavg patm)t =______________N s/m2.

15. The net impulse, net F t, = __________________________________N s.

16. We predict that net impulse equals momentum change. To test your results, find the percent dif-ference between the net impulse acting on the rocket and the rockets momentum change, as follows:

100 x (net impulse momentum change) = 100 x ( ) = ___________%

average of the two ( )

17. What we ask of this (or any) experimental test is that the results be within experimental uncer-

tainty. If possible, estimate the percent uncertainty in each factor in the net impulse and in themomentum change and add all of these together to find the experimental uncertainty. How does itcompare to the percent difference? What results did other groups get? What does this mean?

Other appropriate topics for discussion are: the relationship of the results of this lab to other labs and to the laws of nature that we

have studied,

any observations you made during the lab that are not dealt with in the above calculations, and

variations, extensions, or improvements for this lab.

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Section

4

Demonstrations


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The Fan-Driven Sailboat

The idea of propelling a boat by directing a fan at its sail has been around longer than I have. It isone of those cartoon physics ideas thats funny because it contradicts your experience. Robert BeckClark tells about a test question based on it that he encountered in the early 1960s, and the confu-sion generated when one student respondedwith a good argumentthat it was possible. (See The

Physics TeacherJanuary 1986, pages 3839, reproduced for your convenience in Section 2. )The student argued that air bouncing off the sail would undergo a larger momentum change thanit gained from the fan. If its velocity could be reversed by the sail without loss of speed relative to theboat, then the momentum change would be double that supplied by the fan to speed the air up fromrest. Even if you account for air initially moving relative to the fan and for speed loss upon reversalby the sail, you might still have an imbalance capable of propelling the boat. The student in the storyreceived credit, although he contradicted the expected answer.

In his article, Clark refers to the thrust reversers on some jet aircraft that work much like the fanand sail in the problem. Air is sped up relative to the plane by the jet engines, then turned around byclam-shell thrust reversers. The momentum change at the reverser is greater than that in the engine,giving a net momentum change for the air opposite that normally provided by the engine. The resultis that the plane is slowed, rather than being driven forward. The air is also deflected downward,

which makes it push up on the plane, as required by Newtons third law. This provides additional lift,which is useful at low landing speeds. The sketch on the left below shows the normal operation ofthe engine; that on the right shows the reverse thrust mode.

An older example is the Pelton water wheel. In that device, a jet of water was directed into curvedblades to reverse the waters direction and nearly double the force exerted on the blade, compared toletting a flat blade stop the water. More exquisitely designed blades are used today to squeeze asmuch thrust as possible from the steam or exhaust gases in steam or gas turbines.

The fan-sail effects can be effectively demonstrated with a fan cart. Make your own using low-friction wheels or an air track glider. Or buy one from any of several vendors. They can be used toshow how the air stream from the fan can be deflected by a sail to produce forward or backwardmotion or, with a flat sail perpendicular to the air stream, no motion at all. Actually, the greatesteffect is obtained by removing the sail entirely! In that case, the impulse acting to speed the air isaccompanied by an equal impulse acting on the fan (Newtons third law again), without being par-tially canceled by an impulse on the sail.

All four cases are sketched below, where the arrows are a rough indication of the air flow. The cartsare omitted for clarity, but in each sketch, the fan and the sail must be attached to the same cart.

Demonstratio

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Complete construction plans for a device to demonstrate the behavior of fan-driven sails follow. Itconsists of an inexpensive DC motor and fan blade mounted on the same base as a sail made of aplastic plate. The base is made to fit a PASCO dynamics cart. If you do not have one, low-frictionwheels, such as PASCOs ME-9492 (four wheels on two axles for $19) or ME-6957 (eight wheels onfour axles for $30). can be attached to the base with one screw per axle. This allows the wheels to beeasily removed to be used elsewhere. The other parts cost less than $5. Adding the sails to an existing

fan cart reduces the cost to only about $1.

Building a Fan Cart to Demonstrate Momentum Exchanges with Air

Construction:

Check off each step as you complete it.

1. Make the smaller plastic plate into a shallowdish by cutting off the outer rim, leaving no lip.

Also make the larger plastic plate into a flat disk bycutting just inside the first bend. The resulting sailswill both be about 7-3/4 in diameter. See thesketches in Step 1.

2. Cut two of the craft sticks in half, making four 3sticks. Use hot melt glue to attach two of the 3sticks and one 6 stick to one of the pipes, as shownin the Step 2 drawing. Repeat with the other pipeand the remaining craft sticks. This makes two mastsfor the sails.

3. Lay the flat disk on the table. Run a bead of hotglue on the craft sticks of one mast, and quicklypress the hot glue against the disk, centering the craftstick cross on the disk. In the same way, attach theother mast to the convex side of the shallow dish.See the sketch in Step 3.

Section

4

Step 1.

Step 2.

Step 3.

5 or 4


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4. Drill the shaft hole of the propeller out to 5/64and press the propeller onto the motor shaft. Attachthe motor to the mount with the cable tie, asshown in the Step 4 drawing. Run a bead of hot gluealong each side of the motor to keep it from sliding.

5. Hot glue the motor mount into the 1/2 hole in thebase, as shown in the drawing for Steps 5-7. Use hot

melt glue to seal the CPVC coupling into the 13/16hole.

6. Cut the connector off of the battery holder wires, ifthere is a connector. Cut off half of the red wire. Stripthe ends of both wires. If there is double-sided foamtape on the battery holder, use it to attach the holder tothe base. If not, use hot glue.

7. Solder the black lead from the battery holder to thenegative terminal (-) of the motor. Solder the alligatorclip lead to the positive terminal (+) of the motor.

Connecting the alligator clip to the stripped end of thered wire turns the fan on. (Or you can attach a switch tothe mount and replace the alligator clip lead with ashort wire. )

9. If you dont have a PASCO cart and track, you can keep

costs to a minimum by cutting shallow grooves in thebottom of the base to hold the axles of low-frictionwheels, such as the PASCO ME-9492 (four wheels ontwo axles for $19) or ME 6957 (eight wheels on fouraxles for $30). The axles can be attached with oneshort screw per axle. This makes them easily removableto use elsewhere. See the sketch for Step 9, which is abottom view of the base.

8. Now insert one of the masts into the coupling. Place the base into the tray of a PASCOdynamics cart, and place the cart near the center of a carefully leveled track. Then add AAcells, turn on the fan, and see what happens. Small lengths of Christmas tree tinsel or othersmall streamers can be taped to the edges of the sails to make the direction of the air streamvisible. The three sail configurations are shown below (without the carts). The fourthpossibility is to have no sail at all, as in steps 57.

Demonstratio

ns

Step 4.

Steps 57.

Step 8.

Step 9.


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Parts List for the Fan Cart(Prices effective April 2004, use 10 or more prices; tax and shipping not included)

Section

4

Item Source Quantity Cost

lumber, 1x4, 8 building supply 5.25 $0.16

wood dowel, 36 building supply 5 $0.13

CPVC pipe, 10 building supply two 8 $0.28

CPVC coupling building supply 1 $0.16

6d coated sinker nails, 1 lb. building supply 1 $0.01

Solo plastic plates, 10.25 grocery 1 $0.10

Solo plastic plates, 9 grocery 1 $0.10

craft sticks, 3/4x6, pkg. 300 craft store 4 $0.04

propeller, 5, #850632, pkg. 25 kelvin.com 1 $0.35

DC motor, 1.5-6V electronics outlet* 1 $0.50

jumper leads, pkg. 10 electronics outlet* 0.5 $0.13

battery holder, 4 AA cells electronics outlet* 1 $0.75

total per kit $2.78

* Try allelectronics.com, jameco.com, and kelvin.com. Radio Shack is generally higher.

Also needed: screwdriver, hot melt glue gun, soldering iron, wire cutters, drill, 5/64, 1/2 and 13/16 bits


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Newtons Cradle

A popular toy these days is a row of bifilar pendula that justtouch when hanging at rest. It goes by various names, one ofwhich is Newtons Cradle. It is actually quite an old device thatwas instrumental in the development of the concepts of momen-

tum and kinetic energy. See Section 2 for a fuller discussion ofthis.

The device can be used to demonstrate highly elastic collisionsin which both momentum and kinetic energy are very nearly con-served. Pull one pendulum back, as shown in the diagram at theright, and release it. When it strikes the remaining pendula, thelast in the row, the one nearest us, flies off almost as fast as thefirst one hit the row, while all of the others, including the first, remain nearly at rest. Remarkableenough, but it gets better. Release two, and two fly off the other end. Release three, and when theystrike the remaining two, three fly off, and two remain at rest. Release four, and four fly off, leavingonly the first at rest. All pretty spooky.

Even if you look at the total momentum and conclude that it is conserved, you have to admit

that many other outcomes could also conserve momentum. If one ball is released, for example,momentum would also be conserved if the first ball stopped and two balls left the other end of therow half as fast. It would even be conserved if the first ball bounced back just as fast as it came in,while a ball left the other end twice as fast. Yet only the first possibility ever occurs. What other ruleis being followed that excludes other possibilities?

Christian Huygens proposed a second rule: that the product of mass and the square of speed isalso conserved. Today, we use the term kinetic energy for 1/2 m v2, half of the quantity that Huygensused. But if some quantity is conserved, then half of it is also conserved. This rule fits the facts. It isconsistent with one ball moving at the end, but not with two. As long as you use highly elastic balls,both momentum and kinetic energy are conserved, to a good first approximation. Actually, two rulesare adequate only as long as only two balls interact, with only two unknown velocities after the colli-sion. This is the case if the balls are slightly separated, so that there is a series of two ball collisions.

Commercial versions usually have the balls touching, which greatly complicates the analysis. Formore details, see Section 2.

Of course many collisions are only partially elastic, that is, the objects dont stick together, butsome kinetic energy is still lost. The next few items deal with that.

Clackers

A simpler device that displays fairly elastic collisions is sometimes calledClackers. The device has two fairly elastic plastic balls connected to ahandle by plastic supports in such a way that they can rotate freely. Whenthe handle is held horizontally and jiggled, two patterns of collisionsbetween the balls are possible.

By jiggling the handle up and down, you can get both balls moving sothat they make a collision at the top, then at the bottom, etc., somewhatlike flapping bird wings. A circular motion of the handle can make oneball make one revolution and collide with the other. The first ball stops,and the second makes one revolution before hitting the first, and so on.The impression this motion gives is that one ball is moving, but changescolor with each collision. Because the collisions are not perfectly elastic,either of these motions damps out rather quickly if you stop pumpingenergy into the system by jiggling the handle.

Demonstratio

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Rotating Sail Magnus Effect Demonstrator

Construction

Check off each step as you complete it.

1. Drill a 3/16 hole about 1 from each end of the short wood block

(cart socket). Place it near the center of the bottom (unmarked) side ofthe bottom block. Attach it with two 1-1/4 dry-wall screws throughthe holes in the cart socket, as shown in the Step 1 drawing.

2. Drill 3/16 holes in the center of each 1/2 PVC end cap. Use the nailand a hammer to make starter holes at the marks near each end of thetop and bottom boards. Attach an end cap at each starter hole usingfour 1/2 #8 screws, as shown in the Step 2 drawing.

3. Press the two PVC pipes gently into the end caps to tentatively connectthe top and bottom boards into one unit, as shown in Step 3.

4. Cut the alligator clip lead into two equal parts and strip 3/8 of insula-tion from each of the cut ends. Slip one stripped end through the holein one motor terminal and wrap it around the terminal, making a good

mechanical connection. Repeat with the other wire and motor termi-nal. Solder both connections, heating the terminal from one side andtouching the solder to the other side. Do not touch the solder directlyto the soldering iron.

5. Run the wires along the side of the motor to the shaft end. Being care-ful to avoid shorting the wires to the metal motor parts, press themotor (terminal end first) into the 1 hole, as shown in Step 5.

6. Drill a 3/16 hole in the center of the bottom of the 2-liter bottle and a1/16 hole in the center of the cap. Screw the cap firmly onto the bottle.Slip the cap onto the motor shaft. Push the nail through the hole in thetop board and into the hole in the bottom of the bottle. Press the top

and bottom boards together firmly, but not so tightly that the bottlerubs on the top board. The bottle should spin freely.

7.Attach the battery holder to the bottom block and add a AA cell (onefresh cell is enough), with the flat end of the cell in contact with thespring of the holder. Place the apparatus on a low-friction cart. Thecart socket is made to fit a PASCO dynamics cart. Connect the wires tothe battery holder. The motor should spin the bottle. Aim a fan at theapparatus from the side. The cart should move either forward orbackward. Reverse the battery connection to reverse the spin direction.This should also reverse the direction that the cart moves.

Demonstratio

ns

Step 1.

Step 2.

Step 3.

Step 4.

Step 5.

Steps 6 & 7.

cart


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The Forces on a Rotating Object in Flowing Air

The sketch at the right approximates the air flow around therotating sail. The surface drag slows the air on one side (the bottomside in the sketch) and speeds it on the other (top) side. It also dragsthe air flowing past the top beyond the midpoint, while retardingthe air flowing past the bottom. The result is that the air flowing

near the rotating sail is deflected somewhat downward. Recognizingthis, you can explain the forces in any of three ways.

The easiest, especially if fluids have not been studied yet, is toappeal to Newtons laws. Since air is deflected downward, the sailmust be exerting a downward force on it (Newtons second law).Then by Newtons third law, air must be exerting an upward force on the sail. Or note that the airgains a downward momentum, which requires a downward impulse. Then it must exert an upwardimpulse on the sail.

A third explanation is in terms of the Bernoulli principle, which is basically a conservation ofenergy statement. It applies only approximately here, since it assumes a non-compressible, non-vis-cous, streamline flow, but it still gives results that are consistent with observation. It says that pressureand speed are inversely related in a flowing fluid. In this case, the flow is slower at the bottom, so thepressure should be higher there. The pressure difference between bottom and top then provides theforce that pushes the sail upward in the sketch.

I refer to this device as a rotating sail not only because you candrive the cart with it, but also because I was inspired to build thedevice by a 1920s attempt to sail a ship with such rotating cylin-ders. The idea didnt catch on, but it did work. Anton Flettnerbuilt two ships with rotating sails and crossed the Atlantic withthem. The first, in 1922, was an old sailing ship. The second, in1927, was a cargo ship with a 1000 hp diesel engine that drovethe ship at 9 knots with an underwater screw propeller. Usingrotors alone, it achieved 6 knots. Using both propeller and rotorsraised the speed to 13 knots in a good wind.

Of course, the device also models the behavior of a spinningball moving through air. The effect is important in such sports asbaseball, volleyball, tennis, golf, and ping-pong. Since viscous drag is important, as well as differencesin air speed, the effect is not fully described by the Bernoulli equation. It was explained by HeinrichMagnus in 1851 and is called the Magnus effect.

Flettners first ship. Its rotors were

about 17 m high and 3 m in diameter.

Item Source Quantity Cost

lumber, 1x4, 8 building supply 13.75 $0.44

thinwall PVC pipe, 10 building supply two 12.5 $0.23

PVC cap building supply 4 $0.72

#8 pan head screws building supply 4 $0.101.25 drywall screws, 1 lb, about 288 building supply 2 $0.03

heavy duty double sided foam tape, x75 building supply 1.5 $0.06

DC motor, 1.5V-6V electronics outlet* 1 $0.50

jumper leads, pkg. 10 electronics outlet* 1 $0.25

battery holder, 1 AA cell electronics outlet* 1 $0.45

2-liter plastic bottle with cap trash can 1 --

total per kit $3.54

*Try allelectronics.com, jameco.com, and kelvin.com. Radio Shack is generally higher.

Also needed: soldering iron, wire cutters, Phillips screwdriver, hammer, hot melt glue, drill, and 5/64, 5/32,

3/16, & 1 bits

Section

4

Parts for the Rotating Sail Magnus Effect Demonstrator(prices effective April 2004, use 10 or more prices; tax and shipping not included)


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Section

7

Media Resources


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3. An Impulse to Sing

Written by Judy and Bill Franklin for the Impulse and Momentum unitfor the 1999 PTRA summer workshop in San Antonio, Texas.

Sung to the tune of The Yellow Rose of Texas

Momentum is a concept

As real as it can be;

The product of an objects mass

And its velocity.

Its direction and its size are fixed.

They never change a bit,

Unless a force acts for a time

(An impulse acts on it).

It takes a mile to stop a train

Thats loaded down with freight.

Although velocity is slow,

Its mass is very great.

A bullet from a rifle

Is rather short on mass.

But deadly speed it has to spare,

So duck and let it pass.

To set a thing in motion,To get it off the dime,

We must apply an impulse;

We push it for a time.

Its just as hard to stop it.

Here, too, impulse we need.

Both time and force will slow it

And make it give up speed.

Momentum is a vector,

So objects will not swerve,Unless some sideways impulse

Is used to make them curve.

Momentum is a concept

As real as it can be;

The product of an objects mass

And its velocity.

MediaResou

rces


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Section

8

Physics Olympicsor Contests


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Physics Olympics or Contests

Section 8: Table of ContentsOverview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .8.2Inertia Ball . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .8.2Egg Drop, No Soup Please . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .8.3

Bernoulli Ball . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .8.3

OverviewThis section has a few ideas that can be used for a Physics Olympics or just a classroom competi-

tion. In general, the more involved you get students in using concepts, the better they learn them.Contests often help.

Inertia BallI have used Inertia Ball early in the year as an introduction to centripetal force. Students discover

that it takes a sideways force to coax a moving object into a circular path, as well as forces to speed itup and slow it down. If saved for momentum, the focus is on changes in either the magnitude or the

direction and the impulses required to produce them.The shape and size of the course can be adjusted to fit your area. Chalk marks on a sidewalk work

well, or you can use masking tape on floor tile or carpet. The pylons can be 2-liter bottles. Thestart/finish box should be about 1 meter square. A relatively stiff broom makes the direction of theforce more obvious.

For Olympics, the score is the team average. For classroom use, you could use either individual orgroup scores. I like to let only one group at a time be in the area. They can learn from watchingthose going before them, and they can coach one another.

Inertia Ball Instructions (modified by Bill Franklin from an idea by Tom Gordon)

The object is to push a bowling ball around the course as quickly as you can by touching it onlywith a broom. The following penalties apply:

hitting a pylon: 1 s for the 1st one hit, 2 s for the 2nd, 3 s for the 3rd, etc. touching the ball with your foot or hand: 5 s per occurrence. overshooting the finish square: 10 s. going out of bounds: 2 s. The ball is replaced where it went out; the clock keeps running.

Section

8


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Section

9

Modern PhysicsApplications


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Nonclassical Applications of Momentum

In the previous section, the particles traveled in rather definite paths. The Heisenberg uncertaintyprinciple, discussed below, limits our knowledge of when a particle is at a particular point in thepath, but that wasnt something that we needed to know. Things get much stranger for particles withsmall momenta, such as the outer electrons of an atom or those wandering about in the crystalline

lattice of a metal or semiconductor. Their positions can only be described in terms of probability dis-tributions that can extend over many atomic diameters. These spatial distributions can be calculated,at least in principle, from the Schrodinger wave equation. Schrodingers equation does not explicitlyinvolve momentum, but its development did.

In 1905, Einsteins paper on the photoelectric effect assigned a definite energy to each frequency,just Plancks constant, h, times the frequency, f. In 1923, de Broglie applied Einsteins celebratedequation, E = mc2, to photons also. Since photons travel at c (in a vacuum), he took the momentum,p, for a photon to be mc and wrote Einsteins equation as E = p c. Equating this to h f, he obtainedpc = h f, or p f = hf, since v = f for any wave. Dividing by p f, he got = h/p. Having identifiedthe wavelength of a photon with a momentum, he asserted that his equation provided a wavelengthfor any particle. He had some success applying the idea to the hydrogen electron, finding agreementwith Bohrs scheme by putting the electron in standing wave patterns circling around the nucleus.

He had difficulty selling this wild scheme to the faculty at the University of Paris, but eventuallysucceeded with Einsteins help. Experimental confirmation soon came from the experiments ofDavisson and Germer, who found diffraction and interference effects when they directed an electronbeam at a crystal lattice. Although de Broglies equation gave good results for free particles, it wassoon superseded by the work of Schrodinger and Heisenberg for more complicated situations, such asthe electrons in atoms and crystals.

One of Heisenbergs results, part of his uncertainty principle, is pxx h / 2, where px isthe uncertainty in the x component of momentum, and x is the uncertainty in the x position. Itspecifies a minimum amount of uncertainty in the product of our measurements of position andmomentum. If a way is found to measure the one very accurately, the other will become very uncer-tain. Einstein had trouble with this idea. In fact, he proposed several thought experiments in an

attempt to defeat it, but others found flaws in each attempt. A second part of the uncertainty princi-ple is E t h/2, which specifies a similar limit on the uncertainty of measurements of a parti-cles energy and the time required for its measurement.

Symmetries and Conservation Laws in Quantum Mechanics

There are a number of symmetries in nature that are tied to conservation laws at the quantummechanical level. That is to say that if nature allows us to make some shift in say position, or velocity,or time, without changing the way that things happen, then some conservation law is associated withthat invariance. In particular, if we translate our apparatus to a new position in space without rotat-ing it or otherwise changing it, and if the conditions in the new location are identical to those in theold one, then the apparatus will work just the same. Surprisingly, this is equivalent to saying that

momentum is conserved. There are a number of other ties between conservation laws and such shifts,some of which correspond to things that are notconserved in our universe. An accessible treatment ofthis is chapter 52 of Richard Feynmans Lectures in Physics, Vol. 1, which was reproduced in ThePhysics Teacherin April 1966.

ModernPhys

ics

Impulse Brief

Documents

Transcript of Impulse Brief