## 5 Steps to a 5: AP Physics C (2016)

### STEP __4__

### Review the Knowledge You Need to Score High

__CHAPTER__ __ __**9 A Bit About Vectors**

__CHAPTER__ **10 Free-Body Diagrams and Equilibrium**

__CHAPTER__ **11 Kinematics**

__CHAPTER__ **12 Newton”s Second Law, F _{net} = ma**

__CHAPTER__ **13 Momentum**

__CHAPTER__ **14 Energy Conservation**

__CHAPTER__ **15 Gravitation and Circular Motion**

__CHAPTER__ **16 Rotational Motion**

__CHAPTER__ **17 Simple Harmonic Motion**

__CHAPTER__ **18 Electrostatics**

__CHAPTER__ **19 Circuits**

__CHAPTER__ **20 Magnetism**

### CHAPTER 9

### A Bit About Vectors

**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

**also has a magnitude of 3 meters. Its direction is “â degrees above the negative**

*B**x*-axis.” Vector

**has a magnitude of 1.5 meters. Its direction is “in the negative**

*C**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 è 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

**, as shown in**

*Z*__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

**and**

*Q***is the vector that starts at the origin and ends at the head of vector**

*Z***. This is shown in**

*Z*__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:**

- Line them up head-to-tail.
- 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**

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

**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***?**

*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

**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.**

*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.

**CHAPTER 9**

**Vectors**

** 1 .** Five physics students are asked to add the vectors

*P*and

*Q*to find the resultant vector

*R*. The answers from each student are in the following responses. Which student DID NOT add the vectors correctly?

(A)

(B)

(C)

(D)

(E)

** 2 .** A pilot flies 180 m/s at an angle 30° west of due south as shown. How quickly is the pilot traveling west?

(A) 90 m/s

(B) 104 m/s

(C) 156 m/s

(D) 208 m/s

(E) 360 m/s

__Questions 3 and 4 refer to the following information:__ A fisherman wishes to pilot his boat directly north across a river to a destination marked by an X on the figure. The river has a westward current of 3 m/s, and the boat can travel 5 m/s in still water. The fisherman points his boat at a heading angle of θ into the current as shown and begins his journey.

** 3 .** What heading angle θ should the fisherman use to travel directly north across the river?

(A) 30°

(B) 31°

(C) 37°

(D) 53°

(E) 59°

** 4 .** What is the boat”s resultant velocity while traveling due north across the river?

(A) 2.0 m/s

(B) 3.0 m/s

(C) 3.8 m/s

(D) 4.0 m/s

(E) 5.8 m/s

** Answers**

__1__ .**E** —Vectors should be added head to tail. *Note:* Student B simply showed the correct resultant vector *R* . Student C chose to break vector *P* into its *x* and *y* components and add them to the vector *Q* .

__2__ .**A** —The westward component of the velocity is given by the following equation:

__3__ .**C** —The velocities of the boat and the current must be added to produce a resultant that points directly north as shown. The angle θ is found by using

__4__ .**D** —The velocities of the boat and current are added as shown in the figure, which forms a right triangle. The resultant velocity can be found using the Pythagorean theorem or by realizing that this is a 3-4-5 triangle.