Vectors

Pearland ISD Physics

© 2013 Mark Lesmeister/Pearland ISD• This work is licensed under the

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Acknowledgements

Select questions taken from Serway and Faughn, Holt Physics © 2002 by Holt, Rinehart and Winston.

Acknowledgements

Vectors and scalars

• Vectors are quantities that have both a direction and a magnitude.

• Examples include:– Displacement– Velocity– Acceleration– Force

• Quantities that have only a magnitude are called scalars.

Representing vectors

• Vectors can be represented by words.

– “Take your team 2 ‘clicks’ (km) north”– “US Air 45, new course 30o at 500 mph.”

• Vectors can be represented by symbols.– In the text, boldface indicates vectors.– Examples:

tΔx

VaF av

Representing vectors

• Vectors can be represented graphically.– The direction of the arrow is the direction of

the vector. – The length of the arrow tells the magnitude

• Vectors can be moved parallel to themselves and still be the same vector.– Vectors only tell amount and direction, so a

vector doesn’t care where it starts.

Multiplying and dividing by scalars• Multiplying or dividing a vector by a

scalar results in a new vector.– Multiplying or dividing by a positive number

changes the magnitude of the vector but not the direction.

– Multiplying or dividing by a negative number changes the magnitude and reverses the direction.

VECTOR OPERATIONS

Adding vectors

• The sum of two vectors is called the resultant.

• To add vectors graphically, draw each vector to scale.

• Place the tail of the second vector at the tip of the first vector.

• Vectors can be added in any order.• To subtract a vector, add its opposite.

Vector Addition Practice

• A vector of magnitude 30 units is added to a vector of magnitude 50 units. Which of the following could possible be the magnitude of the resultant?

• A) 10 units• B) 15 units• C) 27 units• D) 88 units

Graphical addition of vectors

• A hiker follows the following directions. She goes 7 kilometers at 300 , then goes 5 kilometers in a direction of 3150 . (That’s the same as 450 west of north.)

• A thunderstorm threatens, and she needs to get back to the starting point.

• What path should she follow?• Add these displacements graphically to

find out.

Graphical Vector Addition Practice• Using a protractor, ruler and a piece of

paper (or mini-whiteboard), add the following vectors by making a scale drawing.

• 30 m/s E + 20 m/s at 30o N of East.

• 20 m S + 20 m at 60o W of South.

Adding perpendicular vectors

• Perpendicular vectors can be easily added.

• Use the Pythagorean theorem to find the magnitude of the resultant.

• Use the tangent function to find the direction of the resultant.

Adding Perpendicular Vectors Practice• From Holt Physics, p. 91

– “While following the directions on a treasure map, a pirate walks 45.0 m North, then turns and walks 7.5 m East. What single straight line displacement could the pirate have taken?”

– “Emily passes a soccer ball 6.0 m directly across the field to Kara, who then kicks the ball 14.5 m directly down the field to Luisa. What is the ball’s displacement as it travels between Emily and Luisa?”

Resolving vectors into components.• Any vector can be resolved, that is,

broken up, into two vectors, one that lies on the x-axis and one on the y-axis.

Resolving Vectors

• An arrow is shot from a bow at an angle of 25 degrees above the horizontal, with an initial speed of 45 m/s. Find the horizontal and vertical components of the arrow’s initial velocity.

Resolving Vectors

?

?

25

m/s 45

y

x

v

v

v

m/s 4178.40

)cos(25m/s) 45(

cos

cos

x

x

x

x

v

v

vvv

v

=q 25o

v=45m/s

vx

vy

m/s 1901.19

)(25sinm/s) 45(

sin

sin

y

y

y

y

v

v

vvv

v

Adding non-perpendicular vectors• Resolve each vector into x and y

components, using sin and cos.• Add the x components together to get the

total x component. Add the y component together to get the total y component.

• Find the magnitude of the resultant using Pythagorean theorem.

• Find the direction of the resultant using the inverse tan function.

Adding non-perpendicular vectors• Resolving Vector 1:

Δx1

Δy1

D 1 =

7 k

m

km 5.3)30sin(7)30sin(

)30sin(

11

1

1

xD

D

x

km 6)30cos(7)30cos(

)30cos(

11

1

1

yD

D

y30o

Adding non-perpendicular vectors• Resolving Vector 1: X

(km)Y(km)

D1 3.5 6

D2

DR

Adding non-perpendicular vectors• Resolving Vector 2:

Δx2

Δy2

D2 = 5 km

km 5.3)45cos(5)45cos(22 Dx

km 5.3)45sin(5)45sin(

)45sin(

22

2

2

yD

D

y

Adding non-perpendicular vectors• Resolving Vector 2: X

(km)Y(km)

D1 3.5 6

D2 -3.5 3.5

DR

Adding non-perpendicular vectors• Add the x components

to get the resultant x component.

• Use the Pythagorean Theorem to find the magnitude of the resultant.

• Use the inverse tan function to find the direction of the resultant.

X(km)

Y(km)

D1 3.5 6

D2 -3.5 3.5

DR 0 9.5