Lesson 1 - Average Speed as a Weighted Average
Average Speed as a Weighted Average explores the difference
between the instantaneous and average speed of an object by
varying the speed of a car on an elliptical track.
Prerequisites
Students should be familiar with the definition of average speed as the
ratio of distance traveled over time elapsed.
Learning Outcomes
Students will be able to use the concept of time-weighted average of
speed to calculate the average speed.
Instructions
Students should understand the applet functions that are described in
Help and ShowMe. The applet should be open. The step-by-step
instructions in this lesson are to be carried out in the applet. You may
need to toggle back and forth between instructions and applet if your
screen space is limited.
Contents
1. The Average of Two Speeds
2. Average Speed as a Time-Weighted Average
1. The Average of Two Speeds
Suppose a car goes around an oval track at a speed of 20 m/s. Upon
completing the first round, the car increases its speed very quickly
(let's assume instantaneously to make the calculation easier) to
40 m/s and then goes around the track the second time at 40 m/s.
What is the car's average speed over the two rounds?
It may be tempting to say 30 m/s. After all, the average would be
(20 m/s + 40 m/s) / 2 = 30 m/s. However, that would not be the right
answer because average speed is not defined as an arithmetic mean of
multiple speeds.
Average Speed is equal to the distance traveled divided by the
time elapsed.
Expressed as an equation,
(1)
Quantity
Symbol
SI Unit
vav
m/s
distance
d
m
time interval
Δt
s
average speed
Before we use equation (1) we will use the applet to determine the
average speed "experimentally".
Simulate the car's motion with the applet by setting the speed to
20 m/s, letting the car complete one round, pausing the motion,
resetting the speed to 40 m/s, and letting the car complete the second
round and pausing the motion again. You may not be able to pause
the motion exactly after the car has completed a round, which will
introduce a slight error.
Sketch the car's average speed vs. time after the car has
completed the second round by selecting the Speed vs. Time
checkbox. Observe that the average speed changes continually
during the second round, but that at no time during that round it is
equal to 30 m/s. The average speed is always less than 30 m/s,
and you should find that upon completion of the second round it is
around ____________ m/s.
Record the total accumulated time from the data table (click
total time: ___________ s
Average Speed vs. time
).
Reset the applet. Set the car's speed to 20 m/s and pause the
motion after it has completed one round.
Display the data table and record the accumulated time from Bin 9
in the Data table.
time = _______ s
Sketch the graph of speed vs. time graph and calculate the area
under the graph for the first round. The area of a rectangle is equal
to height x base.
area = ____________
units of the area = ___________
20 m/s Speed vs. time
Reset the applet. Set the car's speed to 40 m/s and pause the
motion after it has completed one round.
Display the data table and record the accumulated time from Bin 9
in the data table.
time = _______ s
Sketch the graph of speed vs. time.
Again, determine the area (notice that the rectangle is twice as
high, but only half as wide).
area = ____________
units of the area = ___________
40 m/s Speed vs. Time
Is the area of the second graph equal to the area of the first? If so,
what does this mean?
Verify that the area under the average speed graph from Exercise 1 is
equal to the distance traveled by the car when going around the oval
twice. Thus, the average speed line is drawn at a level such that the
definition above for average speed is satisfied.
area (distance) of average speed graph = _____________ m (re: use
the total time and average speed recorded in Exercise 1)
area of 20 m/s speed graph + area of 40 m/s speed graph =
____________ m (re: Exercise 2 + Exercise 3)
Compare the two distances - they should be nearly identical.
Example problem: Calculate the average speed for the two
rounds assuming a speed of 20 m/s was maintained for exactly one
round and a speed of 40 m/s for the second round. The distance
(d) around the oval is not given because it is not needed. Work
with the symbol (d) for this distance in your calculation. (If you
find this difficult, use d = 500 m. The value of the average speed
should not depend on what value for d you assume.)
Solution: We need the total distance
traveled and the total time elapsed. The total
distance traveled is 2d. The time elapsed
during the first round is t = d/(20 m/s). The
time elapsed during the second round is t =
d/(40 m/s). Thus, the total time T elapsed
is:
This gives for the average speed:
Calculate the average speed for the two rounds assuming a speed of
15 m/s was maintained for exactly one round and a speed of 45 m/s
for the second round. Verify your answer using the applet.
2. Average Speed as a Time-Weighted Average
In the previous section, you calculated the average speed for a motion
where two different speeds were maintained over equal distances.
What if the distances are not equal? A more general expression for the
average speed will now be derived that applies in all cases where a
motion is performed at two different speeds.
Let the two speeds be denoted by v1 and v2, the distances traveled
at these speeds by d1 and d2, and the times elapsed while these
speeds are maintained by t1 and t2 respectively. The total distance
traveled is d1 + d2 and the total time T elapsed is t1 + t2. With
that, the definition of average speed gives equation (2).
(2)
In equation (2) for the average speed, each speed is multiplied by a
factor equal to the fraction of the total time during which the speed is
maintained. This factor is a called a weighting factor and this kind of
an average is called a time-weighted average.
Equation (2) requires the times and speeds to be known. If instead of
the times, the distances are known, equation (2) becomes:
(3)
Is it ever true that the average speed over a motion that consists of
two different speeds v1 and v2 is equal to the arithmetic mean (v1 +
v2) / 2? If so, under what special condition would this be true?
Only when the factors (t1/T) and (t2/T) in equation (2) are both equal
to 1/2, the arithmetic mean will give the average speed. This is the
case when the times t1 and t2 elapsed during which the two speeds are
maintained are equal. This is not the case when the two speeds are
maintained over equal distances, as was the case in Exercise 1. The
times taken to cover the same distance will be different when the car
is moving at different speeds.
Example problem: A car travels around an oval track at a speed
of 20 m/s and then a second time at 40 m/s. Calculate the car's
average speed for the two rounds using equation (2).
Solution: Since the car goes twice as fast
the second time around, its travel time for
the second round is half that for the first
round. Therefore,
Substituting the time factors into equation (2)
gives:
Reset the applet. Set the car's speed to 15.0 m/s. Make sure both the
Speed vs. Time graph and the Accumulated Time vs. Speed graph are
not displayed. Do display the Data box, and scroll its display so that
Bins 7 and 14 are simultaneously on display. These bins display the
times associated with the speeds 15 m/s and 30 m/s.



Play the car's motion, and Pause it when the Accumulated Time
in Bin 7 is 10.0 s or close to it.
Then set the car's speed to 30.0 m/s.
Play the car's motion, and Pause it when the Accumulated Time
in Bin 14 is 30.0 s or close to it.
Use equation (2) to calculate the average speed for the actual times
elapsed in your experiment, which will be close to 10 s and 30 s for
the two speeds. Compare your result to the "Average Speed (exact
value)" at the bottom of the Data box.
Sample calculation. Figure 1 below shows values similar to those
that you might obtain.
With t1 = 10.02 s , t2 = 30.18 s , and the total time T elapsed
equal to 10.02 + 30.18 = 40.20 s , equation (2) gives:
This is the value shown at the bottom of the Data box in Figure 1.
Figure 1
Show your times and calculations here.
t1 = _______ s
t2 = _______ s
T = t1 + t2 = __________s
Bin Speed Average: In addition to the exact value of the average
speed (26.26 m/s), the applet also calculates a bin speed average,
which in this case comes out to 25.51 m/s. This value is obtained as
follows.
The applet divides the available speed range from 0 to 50 m/s into
bins of equal size. At present, with the number of bins equal to 25,
each bin has a width of 50/25 = 2 m/s. The bins are numbered 0 to
24. Bin 7 covers the interval (14.0,16.0] m/s. This interval is open at
the left and therefore does not include 14.0 m/s, but is closed on the
right and does include 16.0 m/s.
The bin speed, the speed at the midpoint of a bin, is representative
for all speeds in that bin. Bin 7 has 15 m/s at its midpoint, so
15 m/s is the bin speed. Bin 14 covers the interval (28.0,30.0] m/s
and has the bin speed 29.0 m/s. The bin speed average is the
time-weighted average bin speed calculated according to equation
(2). In the present case, the bin speed average is
Continuing from Exercise 7, if you display the Accumulated Time
vs. Speed graph it will look similar to Figure 2.
Figure 2
The bin speed of 15.0 m/s is shown with an accumulated time of
10 s and the bin speed of 29.0 m/s with an accumulated time of
30 s. The bin speed average of 25.51 m/s is displayed below the
speed axis.
The two columns in Figure 2 are proportional to the time
weightings given to the two bin speeds shown in the calculation
above. The bin speed average is a kind of "center of weight" for
the two bin speeds that are contributing to the average. Note that
the bin speed average is quite a bit closer to the bin speed with the
larger time weighting. It is not in the middle between the two
speeds.
Why Bins and Bin Speeds?
We need to sort speeds into finite-sized bins in order to be able to
define a time-weighted average of speed for continuously variable
speed. If only discrete speeds are involved in a situation, as in the
examples dealt with here, we can work with only these speeds and do
not need bins. However, if the speed is varying continuously, i.e., not
jumping by discrete amounts, sums like those in equation (2) need to
be replaced by an integral. An integral is the limit of a calculation that
uses bins and where one lets the bin size approach zero.
Finite bins introduce an error. In Examples 2 and 3, the actual speed
of 30.0 m/s is replaced by the bin speed of 29.0 m/s. However,
making the bins smaller will tend to make this kind of error smaller as
well. The error vanishes in the limit of zero bin size.
Reset the applet. Set the car's speed to 18.0 m/s. Make sure both the
Speed vs. Time graph and the Accumulated Time vs. Speed graph are
not displayed. Do display the Data box, and scroll its display so that
Bins 8 and 11 are simultaneously on display. These bins display the
times associated with the speeds 17 m/s and 23 m/s.



Play the car's motion, and Pause it when the Accumulated Time
in Bin 8 is 10.0 s or close to it.
Then set the car's speed to 23.0 m/s.
Play the car's motion, and Pause it when the Accumulated Time
in Bin 11 is 30.0 s or close to it.
a. Use equation (2) to calculate the average speed for the actual
times elapsed in your experiment, which will be close to 10 s and
30 s for the two speeds. Compare your result to the "Average
Speed (exact value)" at the bottom of the Data box.
b. Now use equation (2) to calculate the average bin speed in your
experiment. Compare your result to the "Bin Speed Average" at
the bottom of the Data box.
Suppose a motion involves three different speeds. Write down an
expression for the average speed as a time-weighted average of the
three speeds analogous to equation (2) for the time-weighted average
of two speeds.
Summary
If you need to calculate the value of the average speed, you cannot go
wrong if you use the definition of average speed as distance traveled
divided by time elapsed. Then why should one know about timeweighted averages and equations like equation (2) at all?
Knowing how to calculate an average speed as a time-weighted
average of speed provides a deeper understanding of the concept of
average speed. Equation (2) makes it clear what kind of an average is
involved in calculating an average speed, while the ratio of distance
traveled over time elapsed does not.
Moreover, the concept of time-weighted average of a quantity is a
general concept that does not apply only to speed. For example, one is
often interested in the average temperature over some time period,
say, one month. This is a time-weighted average of temperature, and
this average cannot be calculated as the ratio of the change in some
quantity divided by the time elapsed. One has to use the basic
definition of a time-weighted average to calculate average
temperature, i.e., to split the possible temperature range into equalsized bins, measure the amount of time during which the value of the
temperature was in each bin, etc.
Physics 20-30 v1.0
©2004 Alberta Learning (www.learnalberta.ca)
Last Updated: June 16, 2004