College Physics

Problems & Exercises

3.2 Vector Addition and Subtraction: Graphical Methods

Use graphical methods to solve these problems. You may assume
data taken from graphs is accurate to three digits.

1.Find the following for path A inFigure 3.54: (a) the total distance
traveled, and (b) the magnitude and direction of the displacement from
start to finish.

Figure 3.54The various lines represent paths taken by different people walking in a
city. All blocks are 120 m on a side.

2.Find the following for path B inFigure 3.54: (a) the total distance
traveled, and (b) the magnitude and direction of the displacement from
start to finish.

3.Find the north and east components of the displacement for the hikers
shown inFigure 3.52.

4.Suppose you walk 18.0 m straight west and then 25.0 m straight north.
How far are you from your starting point, and what is the compass
direction of a line connecting your starting point to your final position? (If

you represent the two legs of the walk as vector displacementsAand

B, as inFigure 3.55, then this problem asks you to find their sum

R=A+B.)

Figure 3.55The two displacementsAandBadd to give a total displacementR

having magnitudeRand directionθ.

5.Suppose you first walk 12.0 m in a direction20ºwest of north and

then 20.0 m in a direction40.0ºsouth of west. How far are you from

your starting point, and what is the compass direction of a line connecting
your starting point to your final position? (If you represent the two legs of

the walk as vector displacementsAandB, as inFigure 3.56, then this

problem finds their sumR = A + B.)

Figure 3.56

6.Repeat the problem above, but reverse the order of the two legs of the

walk; show that you get the same final result. That is, you first walk leg

B, which is 20.0 m in a direction exactly40ºsouth of west, and then

legA, which is 12.0 m in a direction exactly20ºwest of north. (This

problem shows thatA+B=B+A.)

7.(a) Repeat the problem two problems prior, but for the second leg you

walk 20.0 m in a direction40.0ºnorth of east (which is equivalent to

subtractingBfromA—that is, to findingR′ =A−B). (b) Repeat

the problem two problems prior, but now you first walk 20.0 m in a

direction40.0ºsouth of west and then 12.0 m in a direction20.0ºeast

of south (which is equivalent to subtractingAfromB—that is, to

findingR′′ =B-A= -R′). Show that this is the case.

8.Show that theorderof addition of three vectors does not affect their

sum. Show this property by choosing any three vectorsA,B, andC,

all having different lengths and directions. Find the sumA + B + C

then find their sum when added in a different order and show the result is

the same. (There are five other orders in whichA,B, andCcan be

added; choose only one.)

9.Show that the sum of the vectors discussed inExample 3.2gives the

result shown inFigure 3.24.

10.Find the magnitudes of velocitiesvAandvBinFigure 3.57

Figure 3.57The two velocitiesvAandvBadd to give a totalvtot.

11.Find the components ofvtotalong thex- andy-axes inFigure 3.57.

12.Find the components ofvtotalong a set of perpendicular axes

rotated30ºcounterclockwise relative to those inFigure 3.57.

3.3 Vector Addition and Subtraction: Analytical Methods

13.Find the following for path C inFigure 3.58: (a) the total distance

traveled and (b) the magnitude and direction of the displacement from

start to finish. In this part of the problem, explicitly show how you follow

the steps of the analytical method of vector addition.

CHAPTER 3 | TWO-DIMENSIONAL KINEMATICS 119