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Motion Along a Straight Line

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On September 26, 1993, Dave Munday, a diesel mechanic by trade, went over the Canadian edge of Niagara Falls for the second time, freely falling 48 m to the water (and rocks) below. On this attempt, he rode in a steel chamber with an airhole. Munday, keen on surviving this plunge that had killed other stuntmen, had done considerable research on the physics and engineering aspects of the plunge


If he fell straight down, how could we predict the speed at which he would hit the water?


The world, and everything in it, moves. Even a seemingly stationary thing, such as a roadway, moves because the Earth is moving. Not only is the Earth rotating and orbiting the Sun, but the Sun is also moving through space. The motion of objects can take many different forms. For example, a moving object’s path might be a straight line, a curve, a circle, or something more complicated. The entity in motion might be something simple, like a ball, or something complex, like a human being or galaxy. In physics, when we want to understand a phenomenon such as motion, we begin by exploring relatively simple motions. For this reason, in the study of motion we start with kinematics, which focuses on describing motion, rather than on dynamics, which deals with the causes of motion. Further, we begin our study of kinematics by developing the concepts required to measure motion and mathematical tools needed to describe them in one dimension (or in 1D). Only then do we extend our study to include a consideration of the causes of motion and motions in two and three dimensions. Further simplifications are helpful.

Thus, in this chapter, our description of the motion of objects is restricted in two ways:

  1. The motion of the object is along a straight line. The motion may be purely vertical (that of a falling stone), purely horizontal (that of a car on a level highway), or slanted (that of an airplane rising at an angle from a runway), but it must be a straight line.
  2. The object is effectively a particle because its size and shape are not important to its motion. By “particle” we mean either:

(a) a point-like object with dimensions that are small compared to the distance over which it moves (such as the size of the Earth relative to its orbit around the Sun),

(b) an extended object in which all its parts move together (such as a falling basketball that is not spinning), or

(c) that we are only interested in the path of a special point associated with the object (such as the belt buckle on a walking person). We will start by introducing very precise definitions of words commonly used to describe motion like speed, velocity, and acceleration. These definitions may conflict with the way these terms are used in everyday speech. However, by using precise defi- nitions rather than our casual definitions, we will be able to describe and predict the characteristics of common motions in graphical and mathematical terms. These mathematical descriptions of phenomena form the basic vocabulary of physics and engineering. Although our treatment may seem ridiculously formal, we need to provide a foundation for the analysis of more complex and interesting motions.

READING EXERCISE 2-1: Which of the following motions are along a straight line:

(a) a string of carts traveling up and down along a roller coaster,

(b) a cannonball shot straight up,

(c) a car traveling along a straight city street,

(d) a ball rolling along a straight ramp tilted at a 45° angle.

? READING EXERCISE 2-2: In reality there are no point particles. Rank the following everyday items from most particle-like to least particle-like: (a) a 2-m-tall long jumper relative to a 25 m distance covered in a jump, (b) a piece of lead shot from a shotgun shell relative to its range of 5 m, (c) the Earth of diameter 13 106 m relative to the approximate diameter of its orbit about the Sun of 3 1011 m.

? 26 CHAPTER 2 Motion Along a Straight Line Position and Displacement Along a Line 27 FIGURE 2-1

? Position is determined on an axis that is marked in units of meters and that extends indefinitely in opposite directions. 2-2 Position and Displacement Along a Line Defining a Coordinate System In order to study motion along a straight line, we must be able to specify the location of an object and how it changes over time. A convenient way to locate a point of interest on an object is to define a coordinate system. Houses in Costa Rican towns are commonly located with addresses such as “200 meters east of the Post Office.” In order to locate a house, a distance scale must be agreed upon (meters are used in the example), a reference point or origin must be specified (in this case the Post Office), and a direction (in this case east).

Thus, in locating an object that can move along a straight line, it is convenient to specify its position by choosing a one-dimensional coordinate system. The system consists of a point of reference known as the origin (or zero point), a line that passes through the chosen origin called a coordinate axis, one direction along the coordinate axis, chosen as positive and the other direction as negative, and the units we use to measure a quantity. We have labeled the coordinate axis as the x axis, in and placed an origin on it. The direction of increasing numbers (coordinates) is called the positive direction, which is toward the right in.

The opposite direction is the negative direction. drawn in the traditional fashion, with negative coordinates to the left of the origin and positive coordinates to the right. It is also traditional in physics to use meters as the standard scale for distance. However, we have freedom to choose other units and to decide which side of the origin is labeled with negative coordinates and which is labeled with positive coordinates. Furthermore, we can choose to define an x axis that is vertical rather than horizontal, or inclined at some angle. In short, we are free to make choices about how we define our coordinate system. Good choices make describing a situation much easier.

For example, in our consideration of motion along a straight line, we would want to align the axis of our onedimensional coordinate system along the line of motion. In Chapters 5 and 6, when we consider motions in two dimensions, we will be using more complex coordinate systems with a set of mutually perpendicular coordinate axes. Choosing a coordinate system that is appropriate to the physical situation being described can simplify your mathematical description of the situation. To describe a particle moving in a circle, you would probably choose a two-dimensional coordinate system in the plane of the circle with the origin placed at its center.

Defining Position as a Vector Quantity The reason for choosing our standard one-dimensional coordinate axis and orienting it along the direction of motion is to be able to define the position of an object relative to our chosen origin, and then be able to keep track of how its position changes as the object moves. It turns out that the position of an object relative to a coordinate system can be described by a mathematical entity known as a vector. This is because, in order to find the position of an object, we must specify both how far and in which direction the object is from the origin of a coordinate system.

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