Conference of Asian Science Education,February 20-23, 2008
Kaohsiung, Taiwan
A Study on the Misconceptions of Average
Velocity from Teaching and Learning
Approaches
Yun-Ju Chiu
General Education Center, Chang Gung University, Taiwan
[email protected]
Abstract: This paper addresses some factors which will make students confused in class
when they are learning the definition of the average velocity. I will also explain how to use
suitable examples to help students understand the concepts of average velocity.
Keywords: concepts, velocity, speed, kinematics, textbook
Introduction
In Physics, Several quantities are associated with the phrase “how fast”, they are speed
and velocity, average velocity and instantaneous velocity, average speed and average
velocity. For Physics teachers, these are important concepts in teaching kinematics. Every
textbook uses examples to help students distinguish these definitions.
What is the difference between speed and velocity? For students who have learned
kinematics, the natural response is “speed does not include direction, but velocity does.
Speed is the magnitude of velocity.”
Are the magnitudes of speed and velocity really the same? It depends on the
situation. Velocity is associated with displacement, but speed is associated with distance
traveled. In some situations, displacement does not equal distance traveled. However, for
most students they have forgotten the definitions, they just know these two terms are
related with how fast an object moves. Moreover, these two terms are not both used in
daily language. One is usually used in daily life, and the other is seldom heard outside of
textbooks.
The mismatch of daily language and textbook language
1. The origin of velocity
According to the etymology of the word velocity, it comes from 15th century Latin.
Science literature of the 11th ~18th century were written in Latin. This is why we represent
velocity and speed both with v.
2. Speed and velocity, which is more familiar to people?
For people who speak English, speed is commonly used in daily language, and velocity is
seldom used outside of textbooks. But for people who speak Chinese, it is contrary.
In Taiwan, we use the Chinese word 速度(su du) in our spoken language, but the
Chinese textbooks translate 速度(su du) with velocity, which is seldom used in English
daily life. In the same way, the Chinese textbooks translate speed with the Chinese word
速率(su lu), which is seldom used in Chinese daily life. That is to say, we use the Chinese
word 速度(su du) in our spoken language to mean speed but in our textbooks it means
velocity. Therefore, the concept taught in elementary school is speed(distance divided by
time interval), but teachers call it 速度(su du).
This is an interesting problem. In Taiwan’s daily life, we say速度(su du),but its
definition is speed (速率,su lu) in textbooks. No wonder students get confused in class by
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Conference of Asian Science Education,February 20-23, 2008
Kaohsiung, Taiwan
the mismatch of daily language and textbook language.
How do textbooks teach speed and velocity?
1. Definitions of average speed and velocity
The average speed and velocity for a finite time interval are defined as
distance traveled
displacement
average speed =
average velocity =
time interval
time interval ,
,
Textbooks usually give some examples to help students to learn these two
definitions in teaching one-dimensional motion.
2. Examples used of one-dimensional motion
2.1 Example 1: Problem with turning points
A bird flies at 10 m/s for 100m. It then turns around and flies at 20 m/s for 15s (Fig.1).
Find its average speed and average velocity.
In this familiar example, the dissimilarity of speed and velocity is emphasized.
Students must notice the difference comes from the different definitions of displacement
and distance. The result of this example is “the magnitude of velocity does not equal
speed.”
2.2 Example 2: The yo-yo problem
A boy throws a yo-yo with initial speed 1 m/s and the yo-yo comes back with the same
speed in 2 seconds. What is yo-yo’s average velocity?
This is a question from the national entrance examination of winter 2004. The answer
is zero because the displacement is zero. If the information of the length of string is given,
for example the length is 1m, and then we can calculate the average speed. The answer is
1m/s. The magnitudes of average velocity and average speed are different, the same as in
Example 1.
2.3 Example 3: The rolling down problem
A marble rolls down an inclined aluminum track as in Fig.2, if c is the midpoint of a and b,
does the average velocity from point a to b equal the velocity of point c?
In my previous research (Chiu, 2005), I discovered that over 50% of the freshmen
believed that the average velocity between two points equals the instantaneous velocity of
the midpoint. This misconception is not easy to break.
2.4 Example 4: Presentation of x versus t graph
The motion of a particle is often presented in graphical form. In this example as in Fig.3,
the x versus t graph of a particle whose velocity is not constant. The slope of the straight
line joining two points on the curve is the average velocity for that time interval.
The graph in Fig.3 is abstract and confusing for most students. Fig.3 depicts the
straight-line motion of a particle, but the curve looks like it’s moving in a two-dimensional
plane because the motion is not at a constant speed. This is one of the difficulties students
encounter when learning about kinematics (McDermott, Rosenquist and Zee, 1987).
Fig.1 Problem with a turning point
Fig.2 The rolling down problem
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Fig.3 The x versus t graph
Conference of Asian Science Education,February 20-23, 2008
Kaohsiung, Taiwan
3. Examples used of two-dimensional motion
3.1 Example 5: Traveling from Taipei to Hualien
It takes 3 hours to travel from Taipei to Haulien by train as in Fig.4. What is the average
velocity and average speed of the train respectively?
According to the definitions, the average velocity equals the displacement (117 km)
divided by 3 hours. The answer is 39 km/hr. By the way, don’t forget the direction! Is the
answer meaningful? Does the answer represent how fast the train is going?
Because the distance along the path from Taipei to Haulien is 196km, the average
speed (65.33 km/hr) is meaningful. In this example, the magnitudes of average velocity
and speed are different obviously. Which can represent how fast the train is going, 39
km/hr or 65.33 km/hr?
This example presents a problem. The definition of average velocity is not
applicable in our daily lives. Also its magnitude does not represent how fast the train is
going.
3.2 Example 6: Running in a playground
If somebody runs along a course and its total length is 400m (Fig.5). If it took the runner
40 seconds from point A to B running the course counter clockwise. What is the average
velocity and average speed of the runner respectively?
According to the definitions, the average speed equals 5m/s and the magnitude of
average velocity is 2m/s. A Taiwanese textbook emphasized the differentiation of the two
concepts by using this example. It stated “the magnitude of average velocity does not
necessarily equal to the magnitude of average speed.” The statement is not wrong, but I
think it confuses students.
Don’t you feel a little strange? What’s this all mean? In daily life, do you care about
average velocity which equals displacement divided by time interval?
3.3 Example 7: The second hand problem
Another commonly used example is the second hand problem, for example a clock has a
10 cm length second hand. What is the average velocity of the pinpoint when it goes from
10 to 12? The answer is 1cm/s owing to displacement of the pinpoint divided by the time
interval.
The purpose of this example is to help students to understand the definition of
average velocity. Nevertheless, if we alter the example with going from 9 to 12 or 6 to 12,
the answer will be different. Can we interpret it to the clock’s speed changing?
If we say the magnitude of average velocity does not represent how fast the second
hand is going, then it is not consistent with our textbooks. What is the main point of the
problem?
Fig.4 Traveling from Taipei to Hualien
Fig.5 Running in a playground
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Conference of Asian Science Education,February 20-23, 2008
Kaohsiung, Taiwan
Understanding instantaneous velocity as a limit
1. The concept of instantaneous velocity
The concept of instantaneous velocity is the basis to learn acceleration. They are the key
concepts to learn concepts such as free fall, projectile motion, circular motion, simple
harmonic motion and so on. To solve these problems Newton presented the concept of
limit to conduce the development of calculus in the 17th century.
The concept of instantaneous velocity is better developed through a series of
experiments and exercises involving non-uniform motion apposed to uniform motion.
According to the definition, instantaneous velocity is indeed the same as average
velocity in a minute time interval. They are both defined with displacement divided by
time interval. To help students understand the concepts of instantaneous velocity and
average velocity, we must teach them understand the shorter the time interval is, the closer
the amount of displacement and distance is.
The motion of the pinpoint is uniform circular motion. The magnitude of the
instantaneous velocity is indeed the magnitude of the average speed, which equals the
circumference divided by 60 seconds. The example we use must let students understand
that the lengths of the distance and the displacement of 11 to 12 will be closer than that of
10 to 12. Therefore, I think the second hand problem is a good example if used properly.
But it is ridiculous to ask students what the average velocity of longer time intervals are
such as 9 to 12, this doesn’t help the students.
2. Two types of limit
The concept of limit when considering instantaneous velocity can be categorized into two
parts. One is the limit of distance; the other is the limit of time interval.
In a two-dimensional problem, such as the second hand problem, if we keep
dividing distance into shorter sections, we can substitute distance with displacement. This
concept is not difficult to comprehend.
To apprehend the limit concept of time interval is more subtle because it is quite far
removed from direct observation or experience. The concept of minute time has puzzled
mathematicians throughout the history of the development of Calculus. Therefore, the
problem is, how do we let students understand the limit concept of instantaneous velocity
and avoid teaching Calculus directly. It is a problem that teachers should consider deeply.
3. Difficulties understanding just by reciting definitions
The definition of average velocity will not contradict the statement about the magnitudes
of average velocity and average speed in textbooks if the time interval is short. Therefore,
the instantaneous velocity is indeed the average velocity in a minute time interval. Maybe
students can repeat this definition from memory, but this doesn’t mean that they really
understand the concept in examples such as 3 and 4.
Implications for instruction
It seems that the definition of average velocity doesn’t correspond in our daily lives. It is
important in Physics to define it with displacement divided by time interval. It looks
strange because the unsuited examples that we mentioned previously make it strange.
The displacement will be very close to the distance when the time interval is divided
into very narrow sections. Therefore, the aim to teach the definition of average velocity is
to help students to understand the definition of instantaneous velocity. The definition of
average velocity is essential when learning kinematics. Furthermore, it is a very basic
concept in Calculus.
The definition of average velocity is not significant when the distance or time
interval is long. Therefore, the examples that emphasis the difference between the distance
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Conference of Asian Science Education,February 20-23, 2008
Kaohsiung, Taiwan
and the displacement is not suitable when teaching the definition of average velocity. We
must use some examples to help students understand what situations the distance will
approach to the displacement.
From what I’ve seen in Taiwan, many teachers do not think these examples stated
previously are unsuitable. They let their students practice them again and again because
these examples are in the textbooks and have appeared in the national entrance
examinations such as Example 2.
Textbooks teach one dimensional concepts first followed by two dimensional
concepts. Also textbooks usually introduce the concept of velocity and speed in the section
pertaining to one dimensional concepts. However, I think introducing the concept of
average velocity and speed is more suitable for two-dimensional examples with minute
time interval such as Example 7 is better than one dimensional examples. After they
understand the approaching concept between displacement and distance, students will
have less difficulty learning the limit concept of time interval such as Example 3 and 4.
Conclusion
To teach the concept of average velocity, teachers must understand the mismatch of daily
language and textbook language will make students confused. Teachers must select
suitable examples and strategies to help their students.
Is the difference between speed and velocity only defined by whether direction is
involved or not? Are the magnitudes of velocity and speed the same after all? These
questions will be easier to understand by categorizing motions into three types: one-way
straight motion, back and forth one-dimensional motion or two-dimensional motion.
In one-way straight motion, it makes no difference what the magnitudes of average
velocity and average speed are. So it is meaningless to teach this concept using one-way
straight motion examples.
You can ask the average speed of a back and forth one-dimensional motion, but it is
not suitable to ask the average velocity in this situation. If this situation is used to
emphasize average velocity to students, it will confuse them.
Two-dimensional examples are suitable to help students understand what situations
the distance will approach to the displacement. However, to emphasis the difference
between the distance and the displacement is not suitable in examples such as 5 and 6.
This paper does not delve in the case of the limit concept of examples 3 and 4.
These problems are perhaps even more problematic when it comes to teaching students
and further research needs to be done on how to approach these difficulties which we will
not discuss in this particular paper.
Acknowledgments
I deeply appreciate the support and grants (Grant NSC 95-2511-S-182-001-) I have
received from the National Science Council.
References
[1] Chiu,Y.J., (2005). "The misunderstanding of velocity and speed", Proceedings of the
conference on Physics teaching and demonstration 2005, Hsinchu, Taiwan.
[2] McDermott, L.C., M.L. Rosenquist, and E.H. van Zee, (1987). "Student difficulties in
connecting graphs and physics: Examples from kinematics," Am. J. Phys.55 (6) 503.
[3] Rosenquist, M.L. and L.C. McDermott, (1987). "A conceptual approach to teaching
kinematics," Am. J. Phys. 55 (5) 407.
[4] Trowbridge, D.E. and L. C. McDermott, (1980). "Investigation of student
understanding of the concept of velocity in one dimension," Am. J. Phys. 48 (12) 1020.
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