Position, displacement, and average velocity
1. What is Physics?
One purpose of physics is to study motion of objects – how fast they move, for example, and how far they move in a given amount of time. NASCAR engineers are fanatical about this aspect of physics as they determine the performance of their cars before and durinng a race. Geologists use this physics to measure tectonic-plate motion as they attempt to predict earthquakes. Medical researchers need this physics to map the blood flow through a patient when diagnosing a partially closed artery, and motorists use it to determine how they might slow sufficiently when their radar detector sounds a waarning. There are countless other examples. In this chapter, we study the basic physics of motion where the object (race car, teconic plate, blood cell, or any other object) moves along a single axis. Such motion is called one-dimensional motion.
2. Motion
The world, and everything in it, moves. Even seemingly stationary things, such as a roadway, move with Earth’s ratation, Earth’s orbit around the Sun, the Sun’s orbit around the center of the Milky Way galaxy, and that galaxy’s migration relative to other galaxies. The classification and comparison of motion (called kinematics) is often challenging. What exactly do you measure, and how do you compare?
Before we attempt an answer, we shall examine some general properties of motion that is restricted in three ways.
+ The motion is along a straight line only. The line may be vertical, horizontal, or slanted, but it must be straight.
+ Forces (pushes and pulls) cause motion but will not be discussed until Chapter 5. In this chapter we discuss only the motion itself and changes in the motion. Does the moving object speed up, slow down, stop, or reverse direction? If the motion does change, how is time involved in the change?
+ The moving object is either a particle (by which we mean a point-like object such as an electron) or an object that moves like a particle (such that every portion moves in the same direction and at the same rate). A stiff pig slipping down a straight playground slide might be considered to be moving like a particle; however, a tumbling tumbleweed would not.
3. Postion and Displacement
To locate an object means to find its position relative to some reference point, often the origin (or zero point) of an axis such as the x axis in Fig.2-1. The positive direction of the axis is in the drection of increasing numbers (coordinates), which is to the right in Fig.2-1. The opposite is the negative direction.
For example, a particle might be located at x = 5 m, which means it is 5 m in the positive direction form the origin. If it were at x = -5 m, it would be just as far from the origin but in the opposite direction. On the axis, a coordinate of -5 m is less than a coordinate of -1 m, and both coordinates are less than a coordinate of +5 m. A plus sign for a coordinate need not be shown, but a minus sign must always be shown.
A change from position x1 to position x2 is called a displacement \( \Delta x \), where
\( \Delta x={{x}_{2}}-{{x}_{1}} \) (2-1)
(The symbol \( \Delta \), the Greek uppercase delta, represent a change in a quantily, and it means the final value of that quantity minus the initial value.) When numbers are inserted for the position values x1 and x2 in Eq.2-1 a displacement in the positive direction (to the right in Fig.2-1) always comes out positive, and a displacement in the opposite direction (left in the figure) always comes out negative. For example, if the particle moves from x1 = 5 m to x2 = 12 m, then the displacement is \( \Delta x=(12m)-(5m)=7m \). The positive result indicates that the motion is in the positive direction. If, instead, the particle moves from x1 = 5 m to x2 = 1 m, then . The negative result indicates that the motion is the negative direction.
The actual number of meters covered for a trip is irrelevant; displacement involves only the original and final positions. For example, if particle moves from x = 5 m out to x = 200 m and then back to x = 5 m, the displacement from start to finish is \( \Delta x=(5\text{ }m)-(5\text{ }m)=0 \).
Signs. A plus sign for a displacement need not be shown , but a minus sign must always be shown. If we ignore the sign (and thus the direction) of a displacement, we are left with the magnitude (or absolute value) of displacement. For example, a displacement of \( \Delta x=-4\text{ }m \) has a magnitude of 4 m.
Displacement is an example of a vector quantity, which is a quantity that has both a direction and a magnitude. We explore vectors more fully in Chapter 3, but here all we need is the idea that displacement has two features: (1) Its magnitude is the distance (such as the number of meters) between the original and final positions. (2) Its direction, from an original position to a final position, can be represented by a plus sign or a minus sign if the motion is along a single axis.
4. Average Velocity and Average Speed
A compact way to describe position is with a graph of position x plotted as a function of time t – a graph of x(t). (The notation x(t) represents a function x of t, not the product x time t.) As a simple example,, Fig.2-2 shows the position function x(t) for a stationary armadillo (which we treat as a particle) over a 7 s time interval. The animal’s position stays at \( x=-2\text{ }m \).
Figure 2-3 is more interesting, because it involves motion. The armadillo is apparently first noticed at t = 0 when it is at the position \(x=-5\text{ }m\). It moves toward x = 0, passes through that piont at t = 3 s, and then moves on to increasingly larger positive values of x. Figure 2-3 also depicts the straight-line motion of the armadillo (at three times) and is something like what you would see. The graph in Fig.2-3 is more abstract, but it reveals how fast the armadillo moves.
Actually, several quantities are associated with the phrase “how fast”. One of them is the average velocity \( {{\nu }_{avg}}=\frac{\Delta x}{\Delta t}=\frac{{{x}_{2}}-{{x}_{1}}}{{{t}_{2}}-{{t}_{1}}} \) (2-2)
The natation means that the position is x1 at time t1 and then x2 at time t2. A common unit for \( {{\nu }_{avg}} \) is the meter per sencond (m/s). You may see other units in the problems, but they are always in the form of length/time.
Graphs. On a graph of x versus t, \( {{\nu }_{avg}} \) is the slope of the straight line that connects two particular points on the x(t)curve: one is the point that corresponds to x2 and t2, and the other is the point that corresponds to x1 and t1. Like displacement, \( {{\nu }_{avg}} \) has both magnitude and direction (it is another vector quantily). Its magnitude is the magnitude of the line’s slope. A positive \( {{\nu }_{avg}} \) (and slope) tells us that the line slants upward to the right; a negative \( {{\nu }_{avg}} \) (and slope) tells us that the line slants downward to the right. The average velocity \( {{\nu }_{avg}} \) always has the same sign as the displacement \( \Delta x \) because \( \Delta t \) in Eq.2-2 is always positive.
Figure 2-4 shows how to find \( \Delta t \) in Fig.2-3 for the time interval t = 1 s to t = 4 s. We draw the straight line that connects the point on the position curve at the beginning of the interval and the point on the curve at the end of the interval. Then we find the slope \( \frac{\Delta x}{\Delta t} \) of the straight line. For the given time interval, the average velocity is
\( {{\nu }_{avg}}=\frac{6\text{ }m}{3\text{ }s}=2\text{ }m/s \)
Average speed \( {{s}_{avg}} \) is a different way of describing “how fast” a particle moves. Whereas the average velocity involves the particle’s displacement \( \Delta x \), the average speed involves the total distance covered (for example, the number of meters moved), independent of direction; that is,
\( {{s}_{avg}}=\frac{\text{total distance}}{\Delta t} \) (2-3)
Because average speed does not include direction, it lacks any algebraic aign. Sometimes \( {{s}_{avg}} \) is the same (except for the absence of a sign) as \( {{\nu }_{avg}} \). However, the can be quite different.