420 Chapter 11

11.311.3

In the hallway of your school, or on ashopping trip, you can make many starts andstops resulting in many changes in position(Figure 1). The result will be that you end upeither at a new position or back where youstarted. Walking back and forth to the fridge,school, or mall involves more than onedisplacement vector. We can investigate theresult of these different segments of your tripby doing some math or by drawing somediagrams. In section 11.1 you drew individualvectors to represent displacement. Now wewant to consider two or more vectors in astraight line — in one dimension. In otherwords, we will restrict ourselves to vectors thatare either in the same direction or in oppositedirections. There are three methods forsolving vector problems. Each of thesemethods has advantages and disadvantages.Once you have learned the methods andlanguage for adding vectors in a straight line,you will work with vectors at an angle(section 11.5).

Vector Diagrams

When you add vector quantities such as displacements, you needto consider both the size and the direction of each quantity beingadded. This can be accomplished by combining vectors in adiagram according to the following “head-to-tail” rule:

Join each vector by connecting the “head” end of one vector tothe “tail” end of the next vector [Figure 2(a)].

Recall from section 11.1 that a vector is a line segmentrepresenting the size and direction of a vector quantity.Therefore, the head-to-tail rule includes the sizes and directionsfor all vectors that are joined together. Now that we have amethod of combining vectors, what is the total or sum? Forvectors, the final sum is called the resultant.

Find the resultant by drawing an arrow from the tail of thefirst vector to the head of the last vector [Figure 2(b)].

The resultant displacement, ∆d�R, is a single displacement thathas the same effect as all of the individual displacements combined.When you combine vectors head to tail and draw the resultantvector, you have a vector diagram [Figures 2(a) and 2(b)].

Adding Vectors Along a Straight Line

head-to-tail

∆d1� ∆d2

�

∆dR�

mm 1 cm 2 3 4 5 6

Figure 1

Shopping quite often involves many different displacement segments.

Figure 2

A displacement vector diagram

(a)(b)

Displacement and Velocity 421

Adding Vectors Using Scale Diagrams

We can add straight-line (one-dimensional) vectors by drawing a vectordiagram and then measuring the distance and direction of the resultantvector (Figure 2). The first vector (∆d��1) is drawn to scale with anappropriate length. The second vector (∆d��2) is then drawn, to the samescale, with its tail to the head of ∆d��1. Any other vectors are added to thediagram in the same way: drawn head to tail. The resultant vector, ∆d��R, isdrawn from the tail of the first vector to the head of the last vector (∆d��2, ifthere are only two vectors). Scale diagrams are not as efficient as othermethods and require that you draw and measure carefully.

Sample Problem 1Anne takes her dog, Zak, for a walk. They walk 250 m [W] and then back215 m [E] before stopping to talk to a neighbour. Draw a vector diagramto find their resultant displacement at this point.

∆d��R = 0.70 cm� × �510cm�m� = 35 m [W]

The resultant displacement for Anne and Zak is 35 m [W].

Summary of Scale Diagram Method

1. State the directions (e.g., with a compass symbol).2. List the givens and indicate what variable is being solved.3. State the scale to be used (e.g., 1 cm = 5 km).4. Draw one of the initial vectors to scale.5. Join the second and additional vectors head to tail and to scale.6. Draw and label the resultant vector.7. Measure the resultant vector and convert the length using your scale.8. Write a statement including both the size and direction of the resultant

vector.

This method may seem tedious and inefficient here, but it will be muchmore useful later in the unit, when you are dealing with vectors that arenot on a straight line.

∆d1 = 250 m�

∆d2 = 215 m� ∆dR�

N

S

EW

∆d�1 = 250 m [W]∆d�2 = 215 m [E]∆d�R = ?

Scale: 1 cm = 50 m

The term “vector” is used inbiology to describe an organismthat transports a disease fromone organism to another. Theanopheles mosquito is the“vector” for malaria.

Did You Know ?

422 Chapter 11

Adding Vectors Algebraically

When we add vectors algebraically, we assign a positive or negative direction tothe value of the quantity, just as we learned in Section 11.1. To decide whichdirection is positive and which is negative, there are several conventions(systems). Some choose the initial direction as positive, while others choosenorth and east as being positive. Use the system indicated by your teacher, butalways state the convention used in each problem that you solve. At the end ofthe calculations, the positive or negative sign needs to be translated back into adirection. Algebraic addition is the most efficient method of adding vectorsalong a straight line.

Sample Problem 2This time, Anne’s daughter, Cathryn, takes Zak for a walk. They leave homeand walk 250 m [W] and then back 175 m [E] before stopping to talk to afriend. What is the resultant displacement at this position?Let [E] be positive and [W] be negative.

∆d��1 = 250 m [W] = –250 m

∆d��2 = 175 m [E] = +175 m

∆d��R = ?

∆d��R = ∆d��1 + ∆d��2

= (–250 m) + (+175 m)

= –75 m

The resultant displacement for Cathryn and Zak is 75 m [W].

Summary of Algebraic Method

1. Indicate which direction is positive and which is negative.2. List the givens and indicate what variable is being solved.3. Write the equation for adding the vectors.4. Substitute numbers (with correct signs) into the equation, and solve.5. Write a statement with your answer (including size and direction).

A Combined Method

Vector addition can be done by combining vector diagrams and algebraicaddition. When these are combined, the inefficiency of drawing scale diagramsis removed but the visual advantage of vector diagrams is retained. The vectordiagram is sketched, representing the relative sizes of the vectors (withoutstating a scale), and labelled appropriately. It is important that you draw thevectors approximately accurately, relative to each other. For example, if onedisplacement is 10 m [N] and a second displacement is 20 m [S], the southvector should be approximately twice the length of the north vector. With theaid of a sketch, the vector addition is done algebraically. The combined methodis particularly useful when vectors at an angle are solved mathematically, whichyou will see in future physics courses. Your teacher may indicate which of thethree methods will be the convention to use in your classroom.

Displacement and Velocity 423

Understanding Concepts1. What is the rule for adding vectors in a vector diagram?

2. What is the rule for drawing the resultant vector in a vectordiagram?

3. A soccer player leaves the bench and runs 25 m [N] and then40 m [S]. Use a scale vector diagram to find the resultantdisplacement. Show the full problem-solving approach.

4. What is the rule for adding vectors algebraically?

5. An actor walks 5 m stage right (east in this case) and then15 m stage left.

(a) Find the resultant displacement algebraically.

(b) Draw a scale vector diagram to find the resultantdisplacement.

(c) Compare your answers and account for any differences.

6. A shopper walks 35 m west, another 17 m west, and then67 m east. Show the full problem-solving approach forfinding the resultant displacement by the combined method.

7. On a paper delivery route, Julia walks from home threeblocks north to collect her newspapers, walks back eightblocks south while delivering the papers, and then returnshome. State Julia’s resultant displacement and position(including the reference point).

8. Prepare a table to list the three problem-solving methodsused in this section and the advantages and disadvantagesof each method.

Making Connections9. Draw a scale vector diagram of a segment of a one-

dimensional journey in your daily life. Use paces for yourdistances. Determine the resultant displacement from thediagram and then test your answer by pacing it out. Howclose is the “paced” value to your calculated resultant value?

Exploring10. If you start in a corner and walk around the perimeter of a

football field (assume 100 m by 40 m), what is yourdisplacement when you are halfway around?

(a) Draw a scale vector diagram to determine yourresultant displacement.

(b) Suggest another new method, not mentioned in thissection, that might be used for solving this problem.

Reflecting11. Why do you think the scale diagram method was

introduced here, even though it is more time consuming?

Sample Problem 3Zak decides to take himself for a walk. He heads 30 m [W], stops, thengoes a farther 50 m [W] before returning 60 m [E]. What is Zak’s resultantdisplacement?

East is the positive direction; west is negative.

∆d��R = ∆d��1 + ∆d��2 + ∆d��3

= (–30 m) + (–50 m) + (+60 m)

= –20 m

The resultant displacement for Zak is 20 m [W].

Summary of Combined Method

1. State which direction is positive and which is negative.2. Sketch a labelled vector diagram—not to scale, but using relative sizes.3. Write the equation for adding the vectors.4. Substitute numbers (with correct signs) into the equation, and solve.5. Write a statement with your answer (including size and direction).

The study of motion in physicsinvolves• kinematics: the “how”

(describing motion — theeffect), and

• dynamics: the “why”(explaining motion — thecause).

In this book you are learning thebasics of the “how” (kinematics)part of motion.

Did You Know ?

50 m 30 m

∆dR �60 m

N

S

EW