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Chapter 3
Vectors and Motion in Two
Dimensions
Recommended class days: 2
Background Information
Surveys (Knight, 1995) have found that only about one-third of students in a typical introductory
calculus-based physics class are knowledgeable enough about vectors to begin the study of
Newtonian mechanics. Another one-third have partial knowledge of vectors (e.g., a student who can
add vectors graphically but isn’t familiar with vector components), while the final one-third have
essentially no useful knowledge of vectors. Surveyed students who were repeating the course
generally displayed major gaps in their knowledge of vectors, and this was likely a contributing
factor to their previous failure of the course. We shouldn’t expect the results to be any better for
students in the algebra-based course.
Students who can successfully add and subtract vectors are still often confused as to just what a
vector is. When posed the open-ended question “What is a vector?” they may respond with “A
vector is a force” or some similar answer. These students may have difficulty recognizing velocity
or acceleration as vector quantities.
Although students have used Cartesian coordinate systems throughout high school, many have a
hard time interpreting a statement such as “A vector points in the negative x-direction.” These
students are especially prone to making sign errors when decomposing vectors into components.
Vectors are such an important tool, and the lack of student understanding is so problematic, that
we spend a good deal of Chapter 3 explicitly addressing how to work with vectors. After this, we
move on to two-dimensional kinematics—a good topic to pair with a treatment of vectors, as it
gives students the chance to immediately apply their knowledge of vectors to real physical
situations without introducing many new physics concepts.
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When you treat motion in two dimensions, you will find that the conceptual misunderstandings
that bedevil students when dealing with dynamics problems will begin to reveal themselves. For
example, when dealing with projectile motion, students are prone to think that there is some sort of
agent that “pushes” an object along its trajectory. They may be quite reluctant to analyze the motion
as independent horizontal and vertical motions. It is worthwhile to take this opportunity to show
that the only acceleration in projectile motion is the acceleration of gravity. We find it useful to
spend a good amount of time on the conceptual background to two-dimensional motion, with videos
and/or demonstrations illustrating the independence of the two components of the motion, and the
fact that gravity and gravity alone determines the acceleration. The conceptual leap that the students
need to make will serve them well when they begin to consider forces in Chapters 4 and 5.
Student Learning Objectives
In covering the material of this chapter, students will learn to
• Work with vectors, coordinate systems, and components.
• Analyze a motion diagram in two dimensions, adding acceleration vectors.
• Use vectors to understand motion on a ramp and relative motion.
• Solve problems involving projectiles following parabolic paths.
• Understand basic circular motion variables and the concept of acceleration in circular motion.
Pedagogical Approach
The use of vectors will be an important tool for solving problems in the next several chapters, so it’s
worthwhile spending a bit of time thinking about how to work with vectors, coordinate systems, and
components. A whole chapter devoted to vectors and how to work with them might be a good idea
in theory, but this student group would find it quite uninteresting, and coming so early in the course,
it would certainly confirm their worst fears about a physics course. Rather than treat this important
mathematics technique in isolation, we introduce vectors in a chapter that furthers understanding of
kinematics. By introducing the concepts of vectors just before a treatment of motion on a ramp,
relative velocity, and motion in two dimensions, we give context to the discussion and a chance for
students to immediately apply the techniques they have learned in real, practical problems.
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The chapter starts with just enough information about vectors for students to get through
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Newton’s laws. An explicit vector notation with arrows (e.g., F ) is used throughout the text, rather
than the more traditional boldface notion (e.g., F). Students seem to pay little attention to the
boldface type, and we’ve found that they handle vectors better when the text uses the same notation
that the instructor and students use in handwritten work. When components are not required, vectors
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are often written as (magnitude, direction)—for example, F  (50 N, 30° north of east). Students
find this a logical and useful notation, and it is used throughout the entire text.
Another notation difficulty in most textbooks is the lack of distinction between v as the
magnitude of the velocity vector (i.e., speed, with non-negative values only) and v as the component
of the velocity vector (a signed quantity) in one-dimensional problems. The same is true for
acceleration a and force F. Experienced scientists interpret the symbol properly by recognizing the
context, but this is a major source of confusion to beginners.
We’ve dealt with this by always writing components explicitly, such as v x and a y , even for
one-dimensional motion. This notation was introduced in Chapter 2. Thus the quantities v, a, and F
are unambiguously the magnitudes of vectors. Although this notation can become slightly
cumbersome when other subscripts are needed (with notation such as (v1x )f for the final velocity of
object 1 along the x-axis) it’s still much preferable to the confusion between components and
magnitudes that otherwise results. Instructors are urged to use a consistent and explicit vector notion
in class and to expect similar use of students.
After introducing vectors, instructors can immediately apply the newly learned techniques to
solving real problems. Motion on a ramp gives a good opportunity to work with vector components;
relative velocity gives a chance to add vectors. In this latter case, the use of vectors is especially
important to conceptual understanding. Projectile motion gives another chance to work with
components.
Chapters 1–3 are as much about introducing skills and techniques for solving problems as they
are about introducing physics concepts, so it is worthwhile to stress the problem-solving strategies,
and to use them in solving the relatively straightforward problems in these chapters. This will be
excellent practice for the students before they meet the more complex problems in the coming
chapters, where a thorough, thoughtful approach to problem solving is an absolute must. Solving
projectile motion problems is a good chance to stress good problem-solving techniques,
incorporating full visual overviews, and advising students to do the same, as always avoiding “plug
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and chug.” Chapter 3 includes equations for the range and time of flight for projectile motion
(mainly so that we can show that 45° is the optimal angle for the maximum range) but we
deemphasize their use in solving problems—and you should as well.
We have chosen to treat projectile motion before Newton’s Laws, so we can’t invoke Newton’s
first law to explain why the horizontal component of the motion continues unimpeded. We will treat
circular motion in more detail later, but it is introduced here, again, before Newton’s Laws. Having
students begin to describe more complex motions will start them wondering about the causes of
motion, and what brings about a change in motion. This is a good way to end Chapter 3, and an
excellent preparation for the discussion of forces and Newton’s Laws in the coming chapters.
Suggested Lecture Outlines
Chapter 3 isn’t a particularly short chapter, but it is possible to treat it effectively in 2 days. A good
deal of the chapter is devoted to vectors, but if students read this material before class it isn’t
necessary or advisable to spend time lecturing on vectors. It is much more effective to spend class
time practicing using vectors. After students have done some problems dealing with vectors and
components, you can quickly move on to the kinematics topics of the chapter, giving them a chance
to practice vector techniques in context.
If your time is short, there are some efficiencies possible. Some instructors may choose to save
the material on circular motion for Chapter 6. It is also possible to skim over relative motion, saving
the bulk of this discussion until you treat relativity.
Another possibility is to expand the treatment of relative motion, extending it to include the
ideas of special relativity at this point in the course. This lets you introduce a modern physics topic,
one of great interest to students, at a very early stage. This is probably a very good idea. Though
few of us do it, we all “know” that we should endeavor to include more modern, interesting topics
as early as possible in our courses. (This suggestion has been one very consistent outcome of
physics education reform efforts of recent years.) If you choose to do this, the only major topic of
the relativity chapter that you won’t be able to include is energy. Relativistic energy could then be
incorporated into your treatment of energy, when you come to Chapter 10.
DAY 1: Vectors and first applications. Students have had some introduction to vectors in
Chapter 1, though you may have chosen to wait on much of the vectors material until the full
treatment in this chapter. You should be sure that your students read the text material on vectors
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before class so that you can spend some time reviewing this material before moving on to
applications of vectors.
There is not too much that you can say that is not covered in the text; instead, you should spend
class time practicing vector problems with the students. If your students have difficulties with the
details (as ours often do) you might also want to give a day in the lab or recitation for supervised
work on vector concepts.
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Clicker Question: Here are two vectors, P and Q :
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The following vectors represent different possible combinations of P and Q :
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1) Which of the above vectors best represents the vector sum P  Q ?
r r
2) Which of the above vectors best represents the difference P  Q ?
r r
3) Which of the above vectors best represents the difference Q  P?
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4) Which of the above vectors best represents the vector sum Q  2 P?
Much of the work students will do with vectors involves components, so it’s worthwhile to
review the notion of a coordinate system and the idea of components.
Clicker Question: What are the x- and y-components of the following vector?
A. 3, 2
B. 2, 3
C. 3, 2
D. 2, 3
E. 3, 2
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Clicker Question: What are the x- and y-components of the following vector?
A. 3, 1
B. 3, 4
C. 4, 3
D. 4, 3
E. 3 , 4
Clicker Question: The following vector has length 4.0 units. What are the x- and y-components?
A. 3.5, 2.0
B. 2.0, 3.5
C. 3.5, 2.0
D. 2.0, 3.5
E. 3.5, 2.0
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Clicker Question: The following vector has length 4.0 units. What are the x- and y-components?
A. 3.5, 2.0
B. 2.0, 3.5
C. 3.5, 2.0
D. 2.0, 3.5
E. 3.5, 2.0
You should conclude your treatment of vectors by asking students to find the components of
vectors parallel and perpendicular to a tilted line. Even students already familiar with vectors find
this difficult, but it’s clearly a prerequisite to working successfully with forces on inclined planes.
Example: The following vectors have length 4.0 units. For each vector, what is the component
parallel to the ramp?
Example: The following vectors have length 4.0 units. For each vector, what is the component
perpendicular to the ramp?
After a review of vector concepts, you can move on to some applications that require vectors.
The book moves into a discussion of vectors on motion diagrams, a nice connection to the extensive
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use of motion diagrams in Chapter 2. A brief discussion and a question to test understanding are in
order.
Example: The diagram below shows two successive positions of a particle; it’s a segment of a full
motion diagram.
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Which of the following vectors best represents the acceleration between vi and vf ?
After a discussion of acceleration vectors on motion diagrams, the chapter moves on to a quick
treatment of motion on a ramp. The basic idea of motion on a ramp—that the constraint of the ramp
limits the acceleration to the component parallel to the ramp—is quite straightforward. You can
present this quickly, then move to questions and examples. When solving examples, be certain to
do a full visual overview of the problem. This is the time to teach effective problem-solving
strategies. Emphasize the use of vectors in solving the problems, noting the angle of the slope and
showing how to find components.
Example: In the Soapbox Derby, young participants build cars with very low-friction wheels in
which they roll down a hill. Cars racing on the track at Akron’s Derby Downs, where the national
championship is held, begin on a 55 ft section of the track that is tipped 13° from the horizontal.
a) What is the maximum possible acceleration of a car moving down this stretch of track?
b) If a car starts from rest and accelerates at this rate for the full 55 ft, how fast will it be
moving?
Example: A new ski area has opened that emphasizes the extreme nature of the skiing possible on
its slopes. Suppose an ad intones “Free fall skydiving is the greatest rush you can experience . . . but
we’ll take you as close as you can get on land. When you tip your skis down the slope of our
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steepest runs, you can accelerate at up to 75% of the acceleration you’d experience in free fall.”
What angle slope could give such an acceleration?
After motion on a ramp, you can finish the day with a discussion of relative motion—a good
exercise in adding and subtracting vectors. This is a topic that doesn’t need too much discussion;
you can move along quickly to working examples.
Example: An airplane pilot wants to fly due west from Spokane to Seattle. Her plane moves
through the air at 200 mph, but the wind is blowing 40 mph due north. In what direction should she
point the plane—that is, in what direction should she fly relative to the air?
Example: A skydiver jumps out of an airplane 1000 m directly above his desired landing spot. He
quickly reaches a steady speed, falling through the air at 35 m/s. There is a breeze blowing at 7 m/s
to the west.
a) At what angle with respect to vertical does he fall?
b) When he lands, what will be his displacement from his desired landing spot?
DAY 2: Motion in two dimensions. Chapter 3 only deals with two examples of motion in two
dimensions: projectile motion and circular motion. Circular motion is a much more interesting and
rich topic, but it gets its own chapter, Chapter 6, so you can spend the bulk of the day on projectile
motion.
Your treatment of projectile motion should focus on the independence of the vertical and
horizontal components of the motion. This is a hard concept for students to grasp, as we noted
above. Some students may take a good deal of convincing. We have found that video is a very
helpful tool; a video of two objects, one launched horizontally and one simply dropped (a video
version of the picture in the text) is quite effective, as students can see that the two objects fall in
exactly the same manner. A demonstration can help as well.
It can be hard to come up with projectile motion problems that are of interest to the students in
this class. It’s easy to come up with problems involving balls and bullets, but some creativity is
useful to teach the topic in a manner that brings in other areas. The chapter mentions Hollywood
stunts in two places; showing an appropriate video clip in class (and it’s easy to find examples of
jumps) is a good way to engage the students.
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Example: In the movie Road Trip, some students are seeking to jump a car across a gap in a
bridge. One student, who professes to know what he is talking about (“Of course I’m sure—with
physics, I’m always sure.”), says that they can easily make the jump. He gives the following data:
• The car weighs 2100 pounds, with passengers and luggage.
• Right before the gap, there’s a ramp that will launch the car at an angle of 30°.
• The gap is 10 feet wide.
He then suggests that they should drive the car at a speed of 50 mph in order to make the jump.
a) If the car actually went airborne at a speed of 50 mph at an angle of 30° with respect to the
horizontal, how far would it travel before landing?
b) Does the mass of the car make any difference in your calculation?
In addition to Hollywood stunts, biology content is a good way to add interest. Horizontal jumps
are a good example of projectile motion.
Example: A grasshopper can jump a distance of 30 in (0.76 m) from a standing start.
a) If the grasshopper takes off at the optimal angle for maximum distance of the jump, what is
the initial speed of the jump?
b) Most animals jump at a lower angle than 45°. Suppose the grasshopper takes off at 30° from
the horizontal. What jump speed is necessary to reach the noted distance?
The treatment of circular motion in Chapter 3 is reasonably basic, as it will be covered more
thoroughly in Chapter 6. You can quickly cover the basic variables and describe the nature of the
acceleration for uniform circular motion. The idea that there is an acceleration even though the
speed is constant is a tricky one, and takes some care to explain. A demonstration can be an
effective tool to make this point.
Demonstration: Accelerometer spun in a circle. A thin rectangular plastic tank with colored
liquid inside makes a good accelerometer. Such tanks are easily obtained from educational supply
houses. The liquid moves in a direction opposite the acceleration. If you hold the tank and begin
accelerating forward, the liquid level will tilt backward. Now, if you walk in a circle and hold the
tank along a radial line, the liquid will tip as well, toward the outside of the circle. This is quite
visible, and makes the direction of the acceleration easily determined. This is a very effective way
to convince the students that the acceleration is toward the circle’s center.
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A few simple examples make a good end to the day.
Example: Old vinyl records are 12" in diameter, and spin at 33⅓ rpm when played. What’s the
acceleration of a point on the edge of the record?
Example: Two friends are comparing the acceleration of their vehicles. Josh owns a Ford
Mustang, which he clocks as doing 0 to 60 mph in a time of 5.6 seconds. Josie has a Mini Cooper
that she claims is capable of a higher acceleration. When Josh laughs at her, she proceeds to drive
her car in a tight circle at 13 mph. Which car experiences a higher acceleration?
Other Resources
In addition to the specific suggestions made above in the daily lecture outlines, here are some other
suggestions for demonstrations, examples, questions, and additional topics that you could weave
into your class time.
Suggested Demonstrations
Good demonstrations are a wonderful complement to a good lecture. Students need to see
something real, to take a mental break, and to be reminded that the seemingly abstract concepts they
are seeing in class have applications in the real world. There are some wonderful, effective
demonstrations that can be done for the material of this chapter. Here are some of our favorites:
Dropped ball, launched ball. Students have great difficulty understanding and accepting that
the vertical motion of an object is independent of its horizontal motion. A great way to drive this
point home is to compare the time from launch to striking the floor of a ball launched horizontally
and one simply dropped. It’s easy to find demonstration devices that drop one ball and launch
another simultaneously; the single “click” when both balls hit the floor at the same instant makes
the point about the independence of the two motions rather convincingly.
Monkey and hunter. The ultimate extension of the above demo is the classic “monkey and
hunter” experiment, in which a ball is launched at a target that is dropped at the exact instant that
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the ball is fired. If the ball is aimed at the target, the ball will strike the target, even though both are
falling.
Running 700 mph. This is more of a joke that helps make a concept memorable than a true
demonstration. You can claim to be able to run at 700 mph, and then simply jog at a modest speed
toward the east. Your motion plus the eastward motion of the surface of the earth is well over 700 mph.
Sample Reading Quiz Questions
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1. Ax is the __________ of the vector Ax .
A. magnitude
B. x-component
C. direction
D. size
E. displacement
2. The acceleration vector of a particle in projectile motion:
A. points along the path of the particle
B. is directed horizontally
C. vanishes at the particle’s highest point
D. is directed down at all times
E. is zero
3. The acceleration vector of a particle in uniform circular motion:
A. points tangent to the circle, in the direction of motion
B. points tangent to the circle, opposite the direction of motion
C. is zero
D. points toward the center of the circle
E. points outward from the center of the circle
One Step Beyond: Broad Jumps
“And when it come to fair and square jumping on a dead level, [that frog] could get over more
ground at one straddle than any animal of his breed you ever see. Jumping on a dead level was his
strong suit, you understand; and when it come to that, Smiley would ante up money on him as long
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as he had a red. Smiley was monstrous proud of his frog, and well he might be, for fellers that had
traveled and been everywheres, all said he laid over any frog that ever they see.”
—Mark Twain, The Celebrated Jumping Frog of Calaveras County
In the previous chapter of this guide, we looked at the vertical leaps of which animals are
capable, and we noted that all animals that can jump can achieve approximately the same vertical
leap. We might expect the same rough equality to hold for horizontal leaps—broad jumps—and,
indeed, it does.
Humans are rather poor jumpers, horizontally as well as vertically. In Chapter 2, we noted that a
good jumper can achieve a high jump of 2.0 m, but this merely means that the jumper can get over a
bar at a height of 2.0 m; the center of mass doesn’t rise this high. A better estimate of how high
humans can jump comes from basketball, where players shoot for the highest vertical leap—the
distance off the ground. Spud Webb, who made history by winning a slam dunk competition though
he was just over 5½ feet tall, had a vertical leap of over 1.1 m.
This corresponds to a takeoff speed of 4.6 m/s. A good calculation for your students would be to
figure how far a man could jump at this initial speed if he jumped at the optimal 45° angle. This
speed and angle would lead to a standing horizontal jump of 2.2 m.
In modern times, broad jumpers get a running start, but in the early 1900s, Olympic athletes did
compete in a standing broad jump. An excellent jumper could achieve a distance of 3.5 m by
jumping at a shallow angle and landing with the center of mass lower than at the start. (A horizontal
jump at 4.6 m/s during which the center of mass drops by 1.0 m (starting the jump with arms up and
legs extended and finishing with arms down and legs bent) leads to a 2.1 m distance—a good
calculation to do with your students.)
Frogs, celebrated jumpers, can do better than humans in this respect. Perhaps it is only that more
data is available for frogs than for other animals, thanks to Mark Twain, but frogs currently hold the
record among animals for the standing (crouching?) broad jump. The record bullfrog leap at the
Calaveras County Fair and Jumping Frog Jubilee was set in 1986 by Rosie the Ribeter, who leaped
an impressive 6.5 m. This is quite a bit more than humans can do, but the difference between us,
one of the poorer jumpers among jumping animals, and bullfrogs, the best, is only a factor of 3.
How does the bullfrog’s jumping ability compare to the excellent jumpers discussed previously?
This makes a good calculation for your students. Assuming a 45° takeoff, a 6.5 m broad jump
corresponds to a takeoff speed of 8.0 m/s. (Frogs have much smaller bodies than humans, and are
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nicely approximated as a particle. The change in center of mass position doesn’t add much to the
jump.) If a frog did a vertical leap at this speed (which they don’t, and probably can’t) they could
jump to a height of over 3.2 m—impressive, but only slightly more than the springbok and galago
we saw in previous chapter.
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