Lesson 1 - Average Speed as a Weighted Average



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.

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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.

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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, |

|[pic](1) |

|Quantity |

|Symbol |

|SI Unit |

| |

|average speed |

|vav |

|m/s |

| |

|distance |

|d |

|m |

| |

|time interval |

|Δt |

|s |

| |

Before we use equation (1) we will use the applet to determine the average speed "experimentally".

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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 [pic]). |

|total time: ___________ s |

|Average Speed vs. time |

|[pic] |

|[pic] |

|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 |

|[pic] |

|[pic] |

|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 |

|[pic] |

|Is the area of the second graph equal to the area of the first? If so, what does this mean? |

[pic]

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: | |

|[pic] | |

|This gives for the average speed: | |

|[pic] | |

[pic]

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). |

|[pic](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:

[pic](3)

[pic]

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, | |

|[pic] | |

|Substituting the time factors into equation (2) gives: | |

|[pic] | |

[pic]

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: |

|[pic] |

|This is the value shown at the bottom of the Data box in Figure 1. |

|[pic] |

|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 |

|[pic] |

| |

|Continuing from Exercise 7, if you display the Accumulated Time vs. Speed graph it will look similar to Figure 2. |

|[pic] |

|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.

[pic]

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.

[pic]

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 time-weighted 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 equal-sized bins, measure the amount of time during which the value of the temperature was in each bin, etc.

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Physics 20-30 v1.0

©2004 Alberta Learning (learnalberta.ca)

Last Updated: June 16, 2004

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