Giancoli Chapter 2 Guide & Recommended Problems

From Eckerd Academic Wiki

Guide for Giancoli Chapter 2 (2008)

An object occupies a position in space (x,y,z) at each time (t). As t changes, so does x, y, and z: that is, the object moves through space as time flows. We begin our model-building by developing a mathematical language to describe this motion. In this chapter, we limit the object's motion to one dimension (that is, along a straight line). If the "given" line is horizontal, then the object's motion can be to the right or left, it can reverse direction, it can move with a constant ("steady") speed or it can accelerate (speed up) or decelerate (slow down). We want to be able to describe all this mathematically. The chapter is 25 pages long, divided into 9 sections.

Take a look at the "Summary" on page 43. There are exactly eight equations in the summary. These are the tools we will use to work the problems! (We will, of course, also use your tool-box from high-school algebra, things such as the quadratic formula - as in Example 2-19 on p. 38). All the other material on the 25 pages consists of explanations of these basic eight equations: what the symbols mean and how they relate to each other, when the equations are applicable, and how to use them to model the physical situations described in the examples and problems.

Section 1 underlies everything else. You must understand what displacement is and how it differs from distance traveled. My favorite example to illustrate the difference is running laps around a track: for each complete lap, the distance traveled is (for a circular track) 2 $\pi\$ R, whereas the displacement is zero (since x_final = x_initial which means that $\Delta\$ x = 0). NOTE WELL that $\Delta\$ x = x_final - x_initial always, and not the other way (i.e., not $\Delta\$ x = x_initial - x_final).

Section 2 defines average speed and average velocity. Equation 2-2 on p. 21 is the first of the eight important equations mentioned above. The two worked-out examples are worth a look. Section 3 defines instantaneous velocity, (i.e., velocity at some instant) which is a derivative (Eqn. 2-4, p. 22: second of "the eight"). Giancoli spends considerable time reviewing what a derivative is and how to "read" it off a graph: dx/dt is the "slope" of the graph of the function x(t). So if x(t) is the function representing an object's position x (meters) at time t (seconds), then dx/dt is the speed (meters/second) at time t.

Section 4 defines average acceleration (Eqn. 2-5, p. 25) and instantaneous acceleration (Eqn. 2-6, p. 27). These are the third and fourth of "the eight," and look a lot like the first two, but with v replacing x in the numerators. The idea is that (instantaneous) velocity is the rate-of-change of position (v = dx/dt) and (instantaneous) acceleration is the rate-of-change of velocity (a = dv/dt). Thus acceleration is the second derivative of position w.r.t. time: a = d²x/dt². Velocity and acceleration are related but different concepts.

Section 5 is devoted to a "special case" that occurs when the acceleration is constant (i.e., the velocity changes at a steady rate). In this case (and only in this case) we find the x, v, a, and t are related by Equations 2-12 (p. 29), which are the remaining four of "the eight" that we need to solve the problems in this chapter! I want you to understand that only two of these four are "independent:" you can start with any two, and obtain the others by algebraic manipulation. Remember that these four work only for constant acceleration.

Section 6 is simply advice on how to solve problems. I recommend it!

Section 7 uses the eight equations we now have to solve problems related to objects falling (near the earth's surface) under the influence of gravity (and neglecting air friction). As Galileo discovered around the year 1620 CE, such objects drop with a constant acceleration of approximately 9.8 meters/sec2. That is, in such cases, a = 9.80 m/s2 (denoted by g) downward, and this value is constant (i.e., it does not change) as the object drops. (If we get too far away from the earth's surface, the acceleration does change from this number, as we will see in a later chapter on gravity, but in Chapter 2 we limit our models to "near- earth's-surface.") The worked-out-examples comprise most of this section. There are no new equations; we use the eight we have, and apply them to various cases.

Section 8 takes up the more difficult case of an object moving so that its acceleration is nota constant. That is, a is a function of time: a = a(t). In this case, we cannot use Equations 2-12, and if we need to find v(t) and x(t) we must integrate a(t). Since integration is usually more difficult than differentiation, we will not do much with this section. However, Giancoli shows us (and I definitely want you to understand) how the set of Equation 2-12 came about in the first place: we actually DID integrate a(t) in the special case where a(t) = constant. Integration resulted in

v(t) = v(0) + at. Then a second integration yielded x(t) = x(0) + v(0)t + at²/2. I will go over this in class.

Section 9 will be omitted.

Giancoli Chapter 2 Guide & Recommended Problems

From Eckerd Academic Wiki

Guide for Giancoli Chapter 2 (2008)

Recommended Questions and Problems for Chapter 2

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