Chapter 2The Momentum Principle2.1 The Momentum Principle 522.1.1 Force 532.1.2 Impulse 552.1.3 Predictions using the Momentum Principle 562.2 The superposition principle 572.2.1 Net force 582.3 System and surroundings 582.4 Applying the Momentum Principle to a system 602.4.1 Example: Position and momentum of a ball 602.4.2 Example: A fan cart (1D, constant net force) 612.4.3 Example: A thrown ball (2D, constant net force) 632.4.4 Graphical prediction of motion 682.4.5 Example: Block on spring (1D, nonconstant net force) 692.4.6 Example: Fast proton (1D, constant net force, relativistic) 722.5 Problems of greater complexity 732.5.1 Example: Strike a hockey puck 732.5.2 Example: Colliding students 762.5.3 Physical models 792.6 Fundamental forces 802.7 The gravitational force law 802.7.1 Understanding the gravitational force law 812.7.2 Calculating the gravitational force on a planet 832.7.3 Using the gravitational force to predict motion 862.7.4 Telling a computer what to do 892.7.5 Approximate gravitational force near the Earth’s surface 912.8 The electric force law: Coulomb’s law 922.8.1 Interatomic forces 922.9 Reciprocity 932.10 The Newtonian synthesis 942.11 *Derivation of special average velocity result 952.12 *Points and spheres 962.13 *Measuring the universal gravitational constant G 972.14 *The Momentum Principle is valid only in inertial frames 972.15 *Updating position at high speed 982.16 *Definitions, measurements, and units 982.17 Summary 1012.18 Review questions 1032.19 Problems 1042.20 Answers to exercises 112Copyright 2005 John Wiley & Sons. Adopters of Matter & Interactions by RuthChabay and Bruce Sherwood may provide this revised chapter to their students.52 Chapter 2: The Momentum PrincipleChapter 2The Momentum PrincipleIn this chapter we introduce the Momentum Principle, the first of three fun-damental principles of mechanics which together make it possible to pre-dict and explain a very broad range of real-world phenomena (the other twoare the Energy Principle and the Angular Momentum Principle). In Chap-ter 1 you learned how to describe positions and motions in 3D, and we dis-cussed the notion of “interaction” where change is an indicator that aninteraction has occurred. We introduced the concept of momentum as aquantity whose change is related to the amount of interaction occurring.The Momentum Principle makes a quantitative connection betweenamount of interaction and change of momentum.The major topics in this chapter are:• The Momentum Principle, relating momentum change to interaction• Force as a quantitative measure of interaction• The concept of a “system” to which to apply the Momentum Principle2.1 The Momentum PrincipleNewton’s first law of motion, “the stronger the interaction, the bigger thechange in the momentum,” states a qualitative relationship between mo-mentum and interaction. The Momentum Principle restates this relation ina powerful quantitative form that can be used to predict the behavior of ob-jects. The validity of the Momentum Principle has been verified through avery wide variety of observations and experiments. It is a summary of the wayinteractions affect motion in the real world.As usual, the capital Greek letter delta (∆) means “change of” (some-thing), or “final minus initial.” The “net” force is the vector sum of allthe forces acting on an object. We will study forces in detail in this chapter.Examples of forces include • the repulsive electric force a proton exerts on another proton• the attractive gravitational force the Earth exerts on you• the force that a compressed spring exerts on your hand• the force on a spacecraft of expanding gases in a rocket engine• the force of the air on the propeller of an airplane or swamp boatTime interval short enoughWe require a “short enough time interval” for the Momentum Principle tobe valid, in the sense that the net force shouldn’t change very much duringthe time interval. If the net force hardly changes during the motion, we canuse a very large time interval. If the net force changes rapidly, we need toTHE MOMENTUM PRINCIPLE (for a short enough time interval ∆t)In words: change of momentum (the effect) is equal to the net forceacting on an object times the duration of the interaction (the cause).∆pFnet∆t=Fnet2.1: The Momentum Principle 53use a series of small time intervals for accuracy. This is the same issue we metwith the position update relation, , where we need to use ashort enough time interval that the velocity isn’t changing very much, orelse we need to know the average velocity during the time interval.Since (“final minus initial”), we can rearrange theMomentum Principle like this:UPDATE FORM OF THE MOMENTUM PRINCIPLE (for a short enough time interval ∆t)or, written out:This update version of the Momentum Principle emphasizes the fact that ifyou know the initial momentum, and you know the net force acting duringa “short enough” time interval, you can predict the final momentum. It’s aninteresting fact of nature that the x component of a force doesn’t affect they or z components of momentum, as you can see from these equations. The Momentum Principle written in terms of vectors can be interpretedas three ordinary scalar equations, for components of the motion along thex, y, and z axes:Note how much information is expressed compactly in the vector form ofthis equation, . In some simple situations, for example, ifwe know that the y and z components of an object’s momentum are notchanging, we may choose to work only with the x component of the momen-tum update equation.The Momentum Principle has been experimentally verified in a very widerange of phenomena. We will see later that it can be restated in a very gen-eral form: the change in momentum of an object plus the change in mo-mentum of its surroundings is zero (Conservation of Momentum). In thisform the principle can be applied to all objects, from the very small (atomsand nuclei) to the very large (galaxies and black holes), though understand-ing these systems in detail requires quantum mechanics or general relativi-ty.Historically, the Momentum Principle is often called “Newton’s secondlaw of motion.” We will refer to it as the Momentum Principle to emphasizethe key role played by momentum in physical processes.You are already familiar with change of momentum and with time in-terval . The new element is the concept of