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Kinematics
9 Vectors, Scalars, and Coordinate Systems
Learning Objectives
Define and distinguish between scalar and vector quantities.
Assign a coordinate system for a scenario involving one-dimensional motion.
In our exploration of motion, it’s important to distinguish between quantities that include direction and those that do not. This distinction helps us describe motion clearly—especially when dealing with displacement, velocity, and force.
Displacement vs. Distance
Displacement includes both magnitude and direction. It tells us how far and in what direction an object has moved from its starting position. In contrast, distance is only concerned with the total length of the path traveled—not the direction. For example, if a mouse runs 2 meters forward and 2 meters back to its starting point, the total distance is 4 meters, but the displacement is 0 meters.
Displacement is a vector quantity. Distance is a scalar quantity.
Figure 8.1: This Eclipse Concept jet’s movement can be described either by the total distance it has traveled or by its displacement. If we choose motion to the left as the positive direction (to match the plane’s forward path), the displacement vector points left. In other contexts, we may choose right or upward as positive depending on what simplifies the analysis. (credit: Armchair Aviator, Flickr)
What Are Vectors and Scalars?
A vector is any quantity that has both a magnitude (a size or value) and a direction. Examples include:
Displacement
Velocity (e.g., 90 km/h east)
Acceleration
Force
Vectors are often represented visually with arrows. The arrow’s length corresponds to the magnitude of the vector, and its direction shows the direction of the quantity.
In contrast, a scalar is a quantity that has only magnitude, with no direction. Examples include:
Distance
Speed
Mass
Temperature (e.g., 20°C or even −20°C)
Time
Energy
Scalars are not represented with arrows.
Coordinate Systems for One-Dimensional Motion
To describe motion in a single direction (like along a straight line), we choose a coordinate system. This system defines what we consider positive and negative directions.
Typically, right (horizontal) or up (vertical) is taken as positive.
Left or down is then negative.
But the choice is flexible—what matters is that the convention is clear and consistent throughout a problem.
For example, in biological experiments measuring blood flow in a vessel, you might define the flow direction toward the heart as positive. The key is to define it explicitly.
In some cases, however, as with the jet in Figure 8.1, it can be more convenient to switch the positive and negative directions. For example, if you are analyzing the motion of falling objects, it can be useful to define downwards as the positive direction. If people in a race are running to the left, it is useful to define left as the positive direction. It does not matter as long as the system is clear and consistent. Once you assign a positive direction and start solving a problem, you cannot change it.
Figure 8.2: In many problems, upward or rightward motion is defined as positive, and downward or leftward as negative. But the choice is up to the problem-solver—as long as the rule is followed consistently.
Check Your Understanding
A person jogs at a steady speed of 3 m/s around a circular path. Is speed a scalar or vector?
Answer: Speed is a scalar quantity because it has magnitude only. Even though the person is changing direction while running in a circle, the speed remains constant. If this were a vector, such as velocity, it would change whenever the direction changed.
Section Summary
A vector has both magnitude and direction. Examples: displacement, velocity, force.
A scalar has magnitude only. Examples: distance, speed, time, mass.
In one-dimensional motion, we use a plus or minus sign to indicate direction (e.g., left vs. right, up vs. down).
Displacement and velocity are vectors, while distance and speed are scalars.
Conceptual Questions
A student writes, “A bird that is diving for prey has a speed of
[latex]-\mathit{\text{10}}\phantom{\rule{0.25em}{0ex}}m/s[/latex].” What is wrong with the student’s statement? What has the student actually described? Explain.
Acceleration is the change in velocity over time. Given this information, is acceleration a vector or a scalar quantity? Explain.
A weather forecast states that the temperature is predicted to be [latex]-5ºC[/latex] the following day. Is this temperature a vector or a scalar quantity? Explain.
