Many times predictions are made based on our previous experiences; for
example, how long will a car trip from Dallas to Houston take? Since the distance
between the two cities is about 239 miles, such a trip should take about
4½ hours under normal traffic conditions. If
in the first hour of driving only 30 miles are covered, what will the
total time of the trip be? All these predictions are made, most of the
time, based on previous experiences. Most people can make these
predictions even before taking a formal course that include basic
classical mechanics. In fact, the predictions are based on a simple
proportion. The proportion is based on the ratio between the distance
traveled and the time taken. This ration is the foundation for defining
speed and velocity. Thus, these ratios are defined with the purpose
of facilitating predictions about motion.
In physics,
a definition is a peoples (scientist) made concept that turn out to be
useful for understanding experimental
results.
Definition of Average Speed,
.
The average speed is a scalar quantity. Only the magnitude (value) of the
speed is necessary for completely defining this quantity.
The bar over the letter v
is the symbol used for representing average. Any quantity with the bar on top
will be an average quantity.
The previous relation is an average value because the time taken can be long.
Consider a person walking on a city street for several minutes, does the person
maintain a unique walking rate? Since the answer to this question in most cases
is no, the definition is really about average. The average speed is the rate at
which distance is traveled in the time taken.
Units
The units of speed in the MKS system are the
. There are other
units such as feet/second, miles/hour, or Kilometer/hour but the one used in
these notes are the meter/second.
Definition of Average Velocity,
.
The average velocity is a vector quantity. Thus, both the magnitude and
direction must be given in order to determine the average velocity. In this
case, the vector nature of velocity comes from the
displacement, which is also a
vector. The notation given below much a one dimensional system. However, the
notation given can be extended to multiple dimensions. To simplify the
understanding of vector, graphically, they are represented by arrows.
The quantities represented in the last expression are also
shown in the diagram. Again, since the time taken or the change in time are
finite (not very small), the resulting velocity is an average velocity. The
average velocity is the rate at which the displacement change with the time
taken.
Comparing average speed with the average velocity it can be
said that most of the differences comes from the difference between distance
traveled (speed) and displacement (velocity); for example, the distance traveled
can only be positive while the displacement can be positive or negative.
Consequently, if there is actual physical motion, the average speed will not be
zero. For the same case, the average velocity can be zero when the motion
include displacement in opposite direction that can result in zero displacement.
A tourist walk through New York City following the
blue path of the figure, starting at the corner of W 42^{nd} St.
and Park Ave. he walks uptown along Park Ave. After 2½
hours he is at the corner of Park Ave. and W 110^{th} St.
walking East towards Frederick Douglas Ave. (Central Park W). He makes
it to that corner in ½ a hour. Then, he continue walking toward downtown
along Central Park W (8^{th} Ave.) arriving at the corner of
W 42^{nd} St and Central Park W (8^{th}
Ave.) in an additional 3¾ hours. Before returning to W 42^{nd}
St. and Park Ave, he decides to calculate the distance traveled, the
displacement, the average speed, and his average velocity. After an
additional hour, the tourist is back at the corner of W 42^{nd}
St. and Park Ave.
The scale shown in the picture is 1.0 Km. Thus,
the magenta piece correspond to 0.25 Km.
Before the tourist return to W 42^{nd}
St. and Park Ave. and using the scale provided in the picture, find
the distance traveled by the tourist
a)
13 Km
N
b)
12 Km
c)
11 Km
d)
1 Km
e)
None of the above.
In the same case at the previous question, find the displacement
of the tourist
a)
13 Km
b)
12 Km
c)
11 Km
N
d)
1 Km
e)
None of the above.
Calculate the average speed of the tourist for the walk from
W 42^{nd} St. and Park Ave to
W 42^{nd} St and Central Park W (8^{th}
Ave.) when following the path marked in the map along the Central Park
area,
N
a)
1.78 Km/H
b)
0.15 Km/H
c)
1.93 Km/H
d)
1.98 Km/H
e)
None of the above.
For the same path as in the calculation of the average speed,
calculate the average velocity
a)
1.78 Km/H
N
b)
0.15 Km/H
c)
1.93 Km/H
d)
0.17 Km/H
e)
None of the above.
Calculate the average speed for the completed closed trip (back to
the starting point,
W 42^{nd} St. and Park Ave)
a)
0 Km/H
b)
0.15 Km/H
c)
1.92 Km/H
N
d)
1.68 Km/H
e)
None of the above.
Calculate the average velocity for the completed closed trip (back
to the starting point,
W 42^{nd} St. and Park Ave)
N
a)
0 Km/H
b)
0.15 Km/H
c)
1.92 Km/H
d)
1.68 Km/H
e)
None of the above.
Units
The units of average velocity are the same as the units of average speed.
Conceptually, the instantaneous speed is the reading
of the speedometer. The reading of the speedometer changes continuously
with the changes in speed. The value of the speed is only valid during
the instant when the actual reading is made. The instantaneous velocity
is the reading of the speedometer, for the magnitude, and the reading of
a compass for the direction. Thus, the
instantaneous speed is the magnitude of the instantaneous velocity.
The applet below shows a
car starting from rest traveling from the origin, O, to a final
destination at F where the car is stopped. The car speedometer will
constantly change reading because of the change in speed required for
the motion. Can we guess the reading of the car
speedometer when the car is crossing the light pole? The first
draw of the applet shows a method based on our previous definitions. To
obtain a prediction, times and displacements are measured.
The reading of a watch are taken at two different
instants, t_{i} and
t_{f}; at those
times, the car displacement can be identified by two points,
x_{i} and
x_{f}. Thus, the
prediction for the reading of the speedometer can be obtained from the
ratio between the displacement and the time taken, .
It can be expected that because of the nature of the
motion and the simplicity of the measurement that this prediction will
be off the actual reading of the speedometer. Can
this prediction be improved?
A direct form of improving this prediction is by
shortening the time involved in the previous calculation . In this case the new prediction is calculated from:
Since the time involved between the readings is
smaller, the moving conditions for the car will change less. Thus, it is
expected that the second prediction will be a more accurate value of the
actual reading of the speedometer.
A even better prediction can be obtained if the time
between readings is reduce even further; for example, reduce the time
taken to the time need by the car to cross the light pole
. In this case, it is difficult to measure the time taken
by the car for crossing the light pole. However, assuming that these
times can be measured accurately, the prediction of the reading of the
speedometer obtained from these data will be very close to the actual
reading of the speedometer,
In conclusion, the quality of the prediction improves
by shortening the time. The ideal situation will occurs when the time
taken is closed to zero. Notes that small time taken also implies small
displacement. Thus, the prediction are obtained from the ratio between
two very small numbers. The ratio between two small numbers can be a
large or small number; for example, the result is a large number for a
ratio like 0.03 m/0.001 s = 30 m/s.
In mathematical notation, the previous processes is reduced to the following
expression:
Notice that the bar for average is not used in the notation anymore, in the
limit, the velocity goes from average to instantaneous. In this notation,
, which reads
as the limit when delta t goes to zero,
representsthe process of considering shorter and shorter time taken (Dt
smaller and smaller) until the time taken can be neglected. For algebra based
classes, the previous expression is just a notation to represent the process
of shortening the time explained above.
Graphical Meaning of Average and Instantaneous
Velocity
Motion can be represented by two dimensional graphs. In particular, the graph
of the displacement (not distance) versus the time can be very helpful for
understanding the concepts of average velocity and instantaneous velocity.
The sequence of drawing on the left represents a possible motion
of an object. The motion is represented in a graph of the displacement
versus the time, Graph I
. In the horizontal axis the time
is represented while the vertical axis represents the displacement. The
graph can not represent the distance traveled because distances traveled
are always increasing with time and not decreasing as it is the case
here between 2 seconds and 4 seconds. How can be
calculated the average velocity of the object between two points,
i and
f ? See the second graph,
. For those two points, the horizontal and vertical
coordinates can be read at the axis. Thus, the coordinates for the point
labeled i are (t_{i}
, x_{i}), Graph III
of the sequence
. And the coordinates of the point labeled
f are (t_{f}
, x_{f}),
Graph IV of the sequence
. The average velocity between the two points is
calculated with the formula,
The previous expression is also the slope of the line
between the two points labeled i
and f in Graph V of
the sequence
. The next graphs of the
sequence show the steps necessary to
calculate the slope. The slope of a line is the
rise over the run. In this case, the rise is shown in Graph VI
. The run of the line is presented in Graph VII
.
Finally, the slope is presented in the Graph VIII
.
Therefore, the average velocity between two instants
of time is the slope of the line between the two points of the curve
representing the object displacement in a graph of the displacement
versus the time.
Since the instantaneous velocity is obtained when the
change in time gets smaller and smaller; graphically, this is
represented when the points on the graph are getting closer and closer
to each other. Thus, the average velocity calculated as the slope of the
line between two points becomes the slope of the line for two points
very closed to each other. The extreme case,
Dt → 0,
corresponds to the case when the line is tangent to the curve at
the particular point. Therefore, the instantaneous velocity of a mobile at any time is
equal with the slope of the line tangent to the curve at the
corresponding time.
The calculation of the slope for
a line tangent to a point can be simulate using the applet above. This
applet represents the first few seconds for the graph of the motion of a
mobile. As always, the horizontal axis is the time, while the vertical
axis is the displacement. The position of the vertical axis can be
change by dragging the axis from the arrow head at the top. Similarly,
the location of the horizontal axis can be changed by dragging the axis
from the right most arrow head of the axis. Different points of the
graph can be selected by clicking on the graph at the selected location. This applet can be used for calculating the slope of
a line between two points in the graph. You can do so by clicking at two
points of the graph and then clicking on the v
Average bottom of the applet. If you check the
Instantaneous velocity check box of the
applet, a third point on the graph can be used for calculating the
instantaneous velocity at the middle point, among the three points in
the graph, when pressing the Run bottom of
the applet.
The simulation represent the process from where the
instantaneous velocity can be calculated graphically. The results
obtained do not pretend to be accurate but rather an illustration of the
conceptual approach used for defining the instantaneous velocity at a
given point. A more accurate result for the instantaneous velocity
can be obtained by manually moving the line after the simulation is
concluded. This is achieved by placing a point over the final line and
dragging it to best fit the tangent to the second point.