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Physics For Scientists And Engineers 6E - part 7

S E C T I O N 2 . 1 • Position, Velocity, and Speed

–60

–50

–40

–30

–20

–10

LIMIT

30 km

x(m)

–60

–50

–40

–30

–20

–10

LIMIT

30 km

x(m)

(a)

–40

–60

–20

∆t

∆x

x(m)

t(s)

(b)

Active Figure 2.1 (a) A car moves back and

forth along a straight line taken to be the x

axis. Because we are interested only in the

car’s translational motion, we can model it as

a particle. (b) Position–time graph for the

motion of the “particle.”

Position

t(s)

x(m)

Table 2.1

Position of the Car at
Various Times

defined as the positive direction) during the first 10 s of motion, from position ! to
position ". After ", the position values begin to decrease, suggesting that the car is
backing up from position " through position &. In fact, at $, 30 s after we start mea-
suring, the car is alongside the road sign (see Figure 2.1a) that we are using to mark
our origin of coordinates. It continues moving to the left and is more than 50 m to the
left of the sign when we stop recording information after our sixth data point. A graph-
ical representation of this information is presented in Figure 2.1b. Such a plot is called
a position–time graph.

Given the data in Table 2.1, we can easily determine the change in position of the

car for various time intervals. The

displacement of a particle is defined as its change

in position in some time interval. As it moves from an initial position x

to a final posi-

tion x

, the displacement of the particle is given by x

. We use the Greek letter

delta ($) to denote the change in a quantity. Therefore, we write the displacement, or
change in position, of the particle as

(2.1)

! x

Displacement

At the Active Figures link at

http://www.pse6.com, you can move

each of the six points ! through & and

observe the motion of the car pictorially

and graphically as it follows a smooth

path through the six points.

C H A P T E R 2 • Motion in One Dimension

From this definition we see that $x is positive if x

is greater than x

and negative if x

less than x

It is very important to recognize the difference between displacement and distance

traveled.

Distance is the length of a path followed by a particle. Consider, for example,

the basketball players in Figure 2.2. If a player runs from his own basket down the
court to the other team’s basket and then returns to his own basket, the displacement of
the player during this time interval is zero, because he ended up at the same point as
he started. During this time interval, however, he covered a distance of twice the length
of the basketball court.

Displacement is an example of a vector quantity. Many other physical quantities, in-

cluding position, velocity, and acceleration, also are vectors. In general,

a vector quan-

tity  requires  the  specification  of  both  direction  and  magnitude. By  contrast,  a
scalar quantity has a numerical value and no direction. In this chapter, we use pos-
itive (%) and negative (#) signs to indicate vector direction. We can do this because
the  chapter  deals  with  one-dimensional  motion  only;  this  means  that  any  object  we
study can be moving only along a straight line. For example, for horizontal motion let
us  arbitrarily  specify  to  the  right  as  being  the  positive  direction.  It  follows  that  any
object  always  moving  to  the  right  undergoes  a  positive  displacement  $x & 0,  and
any object moving to the left undergoes a negative displacement, so that $x ' 0. We
shall treat vector quantities in greater detail in Chapter 3.

For our basketball player in Figure 2.2, if the trip from his own basket to the oppos-

ing basket is described by a displacement of % 28 m, the trip in the reverse direction
represents a displacement of # 28 m. Each trip, however, represents a distance of
28 m, because distance is a scalar quantity. The total distance for the trip down the
court and back is 56 m. Distance, therefore, is always represented as a positive number,
while displacement can be either positive or negative.

There is one very important point that has not yet been mentioned. Note that the

data in Table 2.1 results only in the six data points in the graph in Figure 2.1b. The
smooth curve drawn through the six points in the graph is only a possibility of the actual
motion of the car. We only have information about six instants of time—we have no
idea what happened in between the data points. The smooth curve is a guess as to what
happened, but keep in mind that it is only a guess.

If the smooth curve does represent the actual motion of the car, the graph con-

tains information about the entire 50-s interval during which we watch the car move.
It is much easier to see changes in position from the graph than from a verbal de-
scription or even a table of numbers. For example, it is clear that the car was cover-
ing more ground during the middle of the 50-s interval than at the end. Between po-
sitions  # and  $,  the  car  traveled  almost  40 m,  but  during  the  last  10 s,  between
positions  % and  &,  it  moved  less  than  half  that  far.  A  common  way  of  comparing
these  different  motions  is  to  divide  the  displacement  $x that  occurs  between  two
clock readings by the length of that particular time interval $t. This turns out to be a
very useful ratio, one that we shall use many times. This ratio has been given a special
name—average  velocity.

The average velocity v–

of a particle is defined as the

Figure 2.2 On this basketball court,

players run back and forth for the entire

game. The distance that the players run

over the duration of the game is non-

zero. The displacement of the players

over the duration of the game is

approximately zero because they keep

returning to the same point over and

over again.

Ken White/Allsport/Getty Images

Average speed

S E C T I O N 2 . 1 • Position, Velocity, and Speed

particle’s displacement

∆x divided by the time interval ∆t during which that

displacement occurs:

(2.2)

where the subscript x indicates motion along the x axis. From this definition we see
that average velocity has dimensions of length divided by time (L/T)—meters per sec-
ond in SI units.

The average velocity of a particle moving in one dimension can be positive or nega-

tive, depending on the sign of the displacement. (The time interval $t is always posi-
tive.) If the coordinate of the particle increases in time (that is, if x

), then $x is

positive and

is positive. This case corresponds to a particle moving in the

positive x direction, that is, toward larger values of x. If the coordinate decreases in
time (that is, if x

) then $x is negative and hence

is negative. This case corre-

sponds to a particle moving in the negative x direction.

We can interpret average velocity geometrically by drawing a straight line between

any two points on the position–time graph in Figure 2.1b. This line forms the hy-
potenuse of a right triangle of height $x and base $t. The slope of this line is the ratio
$

x/$t, which is what we have defined as average velocity in Equation 2.2. For example,

the line between positions ! and " in Figure 2.1b has a slope equal to the average ve-
locity of the car between those two times, (52 m # 30 m)/(10 s # 0) ! 2.2 m/s.

In everyday usage, the terms speed and velocity are interchangeable. In physics, how-

ever, there is a clear distinction between these two quantities. Consider a marathon
runner who runs more than 40 km, yet ends up at his starting point. His total displace-
ment is zero, so his average velocity is zero! Nonetheless, we need to be able to quantify
how fast he was running. A slightly different ratio accomplishes this for us. The

aver-

age speed of a particle, a scalar quantity, is defined as the total distance traveled di-
vided by the total time interval required to travel that distance:

(2.3)

The SI unit of average speed is the same as the unit of average velocity: meters per sec-
ond. However, unlike average velocity, average speed has no direction and hence car-
ries no algebraic sign. Notice the distinction between average velocity and average
speed—average velocity (Eq. 2.2) is the displacement divided by the time interval, while
average speed (Eq. 2.3) is the distance divided by the time interval.

Knowledge of the average velocity or average speed of a particle does not provide in-

formation about the details of the trip. For example, suppose it takes you 45.0 s to travel
100 m down a long straight hallway toward your departure gate at an airport. At the 100-m
mark,  you  realize  you  missed  the  rest  room,  and  you  return  back  25.0 m  along  the
same  hallway,  taking  10.0 s  to  make  the  return  trip.  The  magnitude  of  the  average
velocity for  your  trip  is % 75.0 m/55.0 s ! % 1.36 m/s.  The  average  speed for  your  trip  is
125 m/55.0 s ! 2.27 m/s. You may have traveled at various speeds during the walk. Nei-
ther average velocity nor average speed provides information about these details.

Average speed !

total distance

total time

! $

x/$t

▲

PITFALL PREVENTION

2.1 Average Speed and

Average Velocity

The magnitude of the average ve-
locity  is  not the  average  speed.
For  example,  consider  the
marathon runner discussed here.
The magnitude of the average ve-
locity  is  zero,  but  the  average
speed is clearly not zero.

Quick Quiz 2.1

Under which of the following conditions is the magnitude of

the average velocity of a particle moving in one dimension smaller than the average
speed over some time interval? (a) A particle moves in the

x direction without revers-

ing. (b) A particle moves in the # x direction without reversing. (c) A particle moves in
the

x direction and then reverses the direction of its motion. (d) There are no con-

ditions for which this is true.

Average velocity

Example 2.1 Calculating the Average Velocity and Speed

C H A P T E R 2 • Motion in One Dimension

2.2 Instantaneous Velocity and Speed

Often we need to know the velocity of a particle at a particular instant in time, rather
than the average velocity over a finite time interval. For example, even though you
might want to calculate your average velocity during a long automobile trip, you would
be especially interested in knowing your velocity at the instant you noticed the police
car parked alongside the road ahead of you. In other words, you would like to be able
to specify your velocity just as precisely as you can specify your position by noting what
is happening at a specific clock reading—that is, at some specific instant. It may not be
immediately obvious how to do this. What does it mean to talk about how fast some-
thing is moving if we “freeze time” and talk only about an individual instant? This is a
subtle point not thoroughly understood until the late 1600s. At that time, with the in-
vention of calculus, scientists began to understand how to describe an object’s motion
at any moment in time.

To see how this is done, consider Figure 2.3a, which is a reproduction of the graph

in Figure 2.1b. We have already discussed the average velocity for the interval during
which the car moved from position ! to position " (given by the slope of the dark
blue line) and for the interval during which it moved from ! to & (represented by
the slope of the light blue line and calculated in Example 2.1). Which of these two
lines do you think is a closer approximation of the initial velocity of the car? The car
starts out by moving to the right, which we defined to be the positive direction. There-
fore, being positive, the value of the average velocity during the ! to " interval is
more representative of the initial value than is the value of the average velocity during
the ! to & interval, which we determined to be negative in Example 2.1. Now let us
focus on the dark blue line and slide point " to the left along the curve, toward point
!

, as in Figure 2.3b. The line between the points becomes steeper and steeper, and as

the two points become extremely close together, the line becomes a tangent line to
the curve, indicated by the green line in Figure 2.3b. The slope of this tangent line

▲

PITFALL PREVENTION

2.2 Slopes of Graphs

In any graph of physical data, the
slope represents  the  ratio  of  the
change  in  the  quantity  repre-
sented on the vertical axis to the
change  in  the  quantity  repre-
sented on the horizontal axis. Re-
member that a slope has units (un-
less  both  axes  have  the  same
units).  The  units  of  slope  in
Figure  2.1b  and  Figure  2.3  are
m/s, the units of velocity.

Find the displacement, average velocity, and average speed
of the car in Figure 2.1a between positions ! and &.

Solution From the position–time graph given in Figure
2.1b, note that x

30 m at t

0 s and that x

! #

53 m

at t

50 s. Using these values along with the definition of

displacement, Equation 2.1, we find that

This result means that the car ends up 83 m in the nega-
tive  direction  (to  the  left,  in  this  case)  from  where  it
started.  This  number  has  the  correct  units  and  is  of  the
same  order  of  magnitude  as  the  supplied  data.  A
quick look at Figure 2.1a indicates that this is the correct
answer.

It is difficult to estimate the average velocity without

completing the calculation, but we expect the units to be
meters per second. Because the car ends up to the left of
where we started taking data, we know the average velocity
must be negative. From Equation 2.2,

83 m

x ! x

! #

53 m # 30 m !

We cannot unambiguously find the average speed of the

car from the data in Table 2.1, because we do not have infor-
mation  about  the  positions  of  the  car  between  the  data
points.  If  we  adopt  the  assumption  that  the  details  of  the
car’s position are described by the curve in Figure 2.1b, then
the  distance  traveled  is  22 m  (from  ! to  ")  plus  105 m
(from " to &) for a total of 127 m. We find the car’s average
speed for this trip by dividing the distance by the total time
(Eq. 2.3):

2.5 m/s

Average speed !

127 m

50 s

1.7 m/s

53 m # 30 m

50 s # 0 s

83 m

50 s

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