Just as distance and displacement have distinctly
different meanings (despite their similarities), so do
speed and velocity.
Speed is a scalar
quantity which refers to "how fast an object is
moving." A fast-moving object has a high speed while a
slow-moving object has a low speed. An object with no
movement at all has a zero speed.
Velocity is a
vector quantity which refers to
"the rate at which an object changes its position."
Imagine a person moving rapidly - one step forward and
one step back - always returning to the original starting
position. While this might result in a frenzy of
activity, it would also result in a zero velocity. Because the
person always returns to the original position, the
motion would never result in a change in position. Since
velocity is defined as the rate at which the position
changes, this motion results in zero velocity. If a
person in motion wishes to maximize his/her velocity, then
that person must make every effort to maximize the amount
that he/she is displaced from his/her original position.
Every step must go into moving that person further from
where he/she started. For certain, the person should
never change directions and begin to return to where
he/she started.
Describing Speed and Velocity
Velocity is a vector quantity. As
such, velocity is "direction-aware." When evaluating the
velocity of an object, you must keep track of its direction.
It would not be enough to say that an object has a
velocity of 55 mi/hr. You must include direction
information in order to fully describe the velocity of
the object. For instance, you must describe an object's
velocity as being 55 mi/hr, east. This is one of
the essential differences between speed and velocity.
Speed is a scalar and does not keep track of
direction; velocity is a vector and is
direction-aware.
The task of describing the direction
of the velocity vector is easy! The direction of the
velocity
vector is the same as the direction in which an
object is moving. It does not matter whether the object
is speeding up or slowing down, if the object is moving
rightwards, then its velocity is described as being
rightwards. If an object is moving downwards, then its
velocity is described as being downwards. Thus an airplane
moving towards the west with a speed of 300 mi/hr has a
velocity of 300 mi/hr, west. Note that speed has no
direction (it is a scalar) and that velocity is simply the
speed with a direction.
Average Speed and Average Velocity
As an object moves, it often undergoes changes in
speed. For example, during an average trip to school,
there are many changes in speed. Rather than the
speedometer maintaining a steady reading, the needle
constantly moves up and down to reflect the stopping and
starting and the accelerating and decelerating. At one
instant, the car may be moving at 50 mi/hr and at another
instant, it may be stopped (i.e., 0 mi/hr). Yet during
the course of the trip to school the person might average a speed of
25 mi/hr.
The average speed during the course of a motion is
often computed using the following equation:
Meanwhile, the average velocity is often computed
using the equation:
Example
The following problem will test your
understanding of these definitions:
While on vacation, Lisa Carr traveled a
total distance of 440 miles. Her trip took 8
hours. What was her average speed?
To compute her average speed, simply divide the
distance of travel by the time of travel.
That was easy! Lisa Carr averaged a
speed of 55 miles per hour. She may not have been
traveling at a constant speed of 55 mi/hr. She
undoubtedly, was stopped at some instant in time (perhaps
for a bathroom break or for lunch) and she probably was
going 65 mi/hr at other instants in time. Yet, she
averaged a speed of 55 miles per hour.
Instantaneous Speed
Since
a moving object often changes its speed during its
motion, it is common to distinguish between the average
speed and the instantaneous speed. The distinction is as
follows:
Instantaneous
Speed - speed at any given instant in time.
Average Speed -
average of all instantaneous speeds; found simply by a
distance/time ratio.
You might think of the instantaneous speed as the
speed which the speedometer reads at any given instant in
time and the average speed as the average of all the
speedometer readings during the course of the trip.
Constant Speed
Moving objects don't always travel with erratic and
changing speeds. Occasionally, an object will move at a
steady rate with a constant speed. That is, the object
will cover the same distance every regular interval of
time. For instance, a cross-country runner might be
running with a constant speed of 6 m/s in a straight
line. If her speed is constant, then the distance
traveled every second is the same. The runner would cover
a distance of 6 meters every second. If you measured
her position (distance from an arbitrary starting point)
each second, you would notice that her position was
changing by 6 meters each second. This would be in
stark contrast to an object which is changing its speed.
An object with a changing speed would be moving a
different distance each second. The data tables below
depict objects with constant and changing speeds.
Example
Now let's try a more difficult
case by considering the motion of that
physics teacher again. The physics teacher walks 4
meters East, 2 meters South, 4 meters West, and finally 2
meters North. The entire motion lasted for 24 seconds.
Determine the average speed and the average velocity.
The physics teacher walked a distance
of 12 meters in 24 seconds; thus, her average speed was
0.50 m/s. However, since her displacement is 0 meters,
her average velocity is 0 m/s. Remember that
displacement refers to the
change in position and that velocity is based upon this
position change. In this case of the teacher's motion,
there is a position change of 0 meters and thus an
average velocity of 0 m/s.
Exercise 1
Here is an exercise similar to one
seen before in the discussion of distance
and displacement. The diagram below shows the
position of a cross-country skier at various times. At
each of the indicated times, the skier turns around and
reverses the direction of travel. In other words, the
skier moves from A to B to C to D. Use the diagram to
determine the average speed and the average velocity of
the skier during these three minutes. When finished,
depress the mouse on the pop-up menu to see the
answer.
Exercise 2
Seymour
Butz views football games from under the bleachers. He
frequently paces back and forth to get the best view.
The diagram below shows several of Seymour's positions
and times. At each marked position, Seymour makes a
"U-turn" and moves in the opposite direction. In other
words, Seymour moves from position A to B to C to D. What
is Seymour's average speed and average velocity? Depress
the mouse on the pop-up menu below to see the answer.
In conclusion, speed and velocity are
kinematic quantities which have distinctly different
definitions. Speed, a scalar
quantity, is the distance (a
scalar quantity) per time ratio. Speed is ignorant of
direction. On the other hand, velocity is
direction-aware. Velocity, a vector
quantity, is the rate at which the position changes.
It is the displacement or
position change (a vector quantity) per time ratio.
