5 Steps to a 5: AP Physics C - Greg Jacobs 2019

A bit about vectors
Review the knowledge you need to score high

IN THIS CHAPTER

Summary: Understand the difference between scalars and vectors, how to draw vectors, how to break down vectors into components, and how to add vectors.

Key Ideas

Scalars are quantities that have a magnitude but no direction—for example, temperature; in contrast, vectors have both magnitude and direction—for example, velocity.

Vectors are drawn as arrows; the length of the arrow corresponds to the magnitude of the vector, and the direction of the arrow represents the direction of the vector.

Any vector can be broken down into its x- and y-components; breaking a vector into its components will make many problems simpler.

Relevant Equations

Note: this assumes that θ is measured from the horizontal. These equations are not on the equation sheet, but should be memorized.

Scalars and vectors are easy. So we’ll make this quick.

Scalars

Scalars are numbers that have a magnitude but no direction.

Magnitude: How big something is

For example, temperature is a scalar. On a cold winter day, you might say that it is “4 degrees” outside. The units you used were “degrees.” But the temperature was not oriented in a particular way; it did not have a direction.

Another scalar quantity is speed. While traveling on a highway, your car’s speedometer may read “70 miles per hour.” It does not matter whether you are traveling north or south, if you are going forward or in reverse: your speed is 70 miles per hour.

Vector Basics

Vectors, by comparison, have both magnitude and direction.

Direction: The orientation of a vector

An example of a vector is velocity. Velocity, unlike speed, always has a direction. So, let’s say you are traveling on the highway again at a speed of 70 miles per hour. First, define what direction is positive—we’ll call north the positive direction. So, if you are going north, your velocity is +70 miles per hour. The magnitude of your velocity is “70 miles per hour,” and the direction is “north.”

If you turn around and travel south, your velocity is −70 miles per hour. The magnitude (the speed) is still the same, but the sign is reversed because you are traveling in the negative direction. The direction of your velocity is “south.”

IMPORTANT: If the answer to a free-response question is a vector quantity, be sure to state both the magnitude and direction. However, don’t use a negative sign if you can help it! Rather than “−70 miles per hour,” state the true meaning of the negative sign: “70 miles per hour, south.”

Graphic Representation of Vectors

Vectors are drawn as arrows. The length of the arrow corresponds to the magnitude of the vector—the longer the arrow, the greater the magnitude of the vector. The direction in which the arrow points represents the direction of the vector. Figure 9.1 shows a few examples:

Figure 9.1 Examples of vectors.

Vector A has a magnitude of 3 meters. Its direction is “60 degrees above the positive x-axis.” Vector B also has a magnitude of 3 meters. Its direction is “β degrees above the negative x-axis.” Vector C has a magnitude of 1.5 meters. Its direction is “in the negative y-direction” or “90 degrees below the x-axis.”

Vector Components

Any vector can be broken into its x- and y-components. Here’s what we mean:

Place your finger at the tail of the vector in Figure 9.2 (that’s the end of the vector that does not have a on it). Let’s say that you want to get your finger to the head of the vector without moving diagonally. You would have to move your finger three units to the right and four units up. Therefore, the magnitude of left—right component (x-component) of the vector is “3 units” and the magnitude of up—down (y-component) of the vector is “4 units.”

Figure 9.2 Breaking vectors into x- and y-components.

If your languages of choice are Greek and math, then you may prefer this explanation:

Given a vector V with magnitude v directed at an angle θ above the horizontal,

You may want to check to see that these formulas work by plugging in the values from our last example.

Exam tip from an AP Physics veteran:

Even though the vector formulas in the box are not on the equations sheet, they are very important to memorize. You will use them in countless problems. Chances are, you will use them so much that you’ll have memorized them way before the AP exam.

—Jamie, high school senior

Adding Vectors

Let’s take two vectors, Q and Z, as shown in Figure 9.3a.

Figure 9.3a Two vectors.

Now, in Figure 9.3b, we place them on a coordinate plane. We will move them around so that they line up head-to-tail.

Figure 9.3b Vectors on a coordinate plane.

If you place your finger at the origin and follow the arrows, you will end up at the head of vector Z. The vector sum of Q and Z is the vector that starts at the origin and ends at the head of vector Z. This is shown in Figure 9.3c.

Figure 9.3c Adding vectors.

Physicists call the vector sum the “resultant vector.” Usually, we prefer to call it “the resultant” or, as in our diagram, “R.”

How to add vectors:

1. Line them up head-to-tail.

2. Draw a vector that connects the tail of the first arrow to the head of the last arrow.

Vector Components, Revisited

Breaking a vector into its components will make many problems simpler. Here’s an example:

To add the vectors in Figure 9.4a, all you have to do is add their x- and y-components. The sum of the x-components is 3 + (−2) = 1 units. The sum of the y-components is 1 + 2 = 3 units. The resultant vector, therefore, has an x-component of +1 units and a y-component of +3 units. See Figure 9.4b.

Figure 9.4a Adding vectors.

Figure 9.4b Final sum of vectors.

Some Final Hints

1. Make sure your calculator is set to DEGREES, not radians.

2. Always use units. Always. We mean it. Always.

Practice Problems

1. A canoe is paddled due north across a lake at 2.0 m/s relative to still water. The current in the lake flows toward the east; its speed is 0.5 m/s. Which of the following vectors best represents the velocity of the canoe relative to shore?

(A) 2.5 m/s

(B) 2.1 m/s

(C) 2.5 m/s

(D) 1.9 m/s

(E) 1.9 m/s

2. Force vector A has magnitude 27.0 N and is direction 74° from the vertical, as shown above. Which of the following are the horizontal and vertical components of vector A?

3. Which of the following is a scalar quantity?

(A) electric force

(B) gravitational force

(C) weight

(D) mass

(E) friction

Solutions to Practice Problems

1. B—To solve, add the northward 2.0 m/s velocity vector to the eastward 0.5 m/s vector. These vectors are at right angles to one another, so the magnitude of the resultant is given by the Pythagorean theorem. You don’t have a calculator on the multiple-choice section, though, so you’ll have to be clever. There’s only one answer that makes sense! The hypotenuse of a right triangle has to be bigger than either leg, but less than the algebraic sum of the legs. Only B, 2.1 m/s, meets this criterion.

2. A—Again, with no calculator, you cannot just plug numbers in (though if you could, careful: the horizontal component of A is 27.0 N cos 16° because 16° is the angle from the horizontal). Answers B and E are wrong because the vertical component is bigger than the horizontal component, which doesn’t make any sense based on the diagram. Choice C is wrong because the horizontal component is bigger than the magnitude of the vector itself—ridiculous! Same problem with choice D, where the horizontal component is equal to the magnitude of the vector. So the answer must be A.

3. D—A scalar has no direction. All forces have direction, including weight (which is the force of gravity). Mass is just a measure of how much stuff is contained in an object, and thus has no direction.