VECTOR ADDITION

Chapter 4

Physics Southern Boone County High School

Bill Palmer

Question: An Airplane flies north with an airspeed of 575

mph. If the wind is blowing 30° north of east at 50 mph, what is

the speed of the plane as measured from the ground?

What if the wind blew south of west?

Vectors vs. Scalars

One of the numbers below does not fit in the group. Can you decide which one? Why?

35 ft161 mph

-70° F200 m 30° East of North

12,200 people

Vectors vs. ScalarsThe answer is: 200 m 30° East of North

Why is it different? All the others can be completely

described with only a numerical magnitude. Numbers with that property are called SCALARS.

Numbers that need both magnitude and direction to be described are called VECTORS.

Notation

• Vectors can be shown as arrows. – The length of the arrow describes

the magnitude of the vector.– The direction of the arrow

indicates the direction of the vector…

Adding Vectors

Collinear VectorsThese are vectors that a

either in the same direction or in exactly opposite

directions.

What is the ground speed of an airplane

flying with an air speed of 100 mph into a

headwind of 100 mph?

What is the ground speed of an airplane

flying with an air speed of 100 mph into a

headwind of 100 mph?DRAW THE VECTORS

What is the ground speed of an airplane

flying with an air speed of 100 mph and a tail

wind of 50 mph?DRAW THE VECTORS

Adding Collinear Vectors

When vectors are parallel, just add magnitudes and keep the direction.

Ex: 100 mph east + 50 mph east = 150 mph east

Adding Collinear Vectors

When vectors are antiparallel, just subtract the smaller magnitude from the larger and use the direction of the larger.

Ex: 50 mph east + 40 mph west = 10 mph east

An Airplane flies north with an air speed of 650

mph. If the wind is blowing east at 50 mph, what is the speed of the plane as measured from

the ground?

Adding Perpendicular VectorsWhen vectors are perpendicular, just

sketch the vectors in a HEAD TO TAIL orientation and use right triangle trigonometry to solve for the resultant and direction.

Ex: 50 mph east + 40 mph south = ??

Adding Perpendicular Vectors

Use Pythagorean Theorem to solve for R and Right triangle trig. To solve for θ

R

θ

Adding Perpendicular Vectors

Use the Pythagorean Theorem to solve for R.

R Rx2 Ry

2

Examples

Ex1: Find the sum of the forces of 30 lb south and 60 lb east.

Ex2: What is the ground speed of a speed boat crossing a river of 5mph current if the boat can move 20mph in still water?

Vector Components

•Vectors can be described using their components.

•The Components of a vector are two perpendicular vectors that would add together to yield the original vector.– Components are notated using subscripts.

F

Fx

Fy

An Airplane flies north with an air speed of 575 mph. If

the wind is blowing 30° north of east at 50 mph, what is the speed of the

plane as measured from the ground? What if the wind

blew south of west?

Adding Vectors with Scale Diagrams

• When vectors are not parallel or perpendicular the only way to add them is by drawing a SCALE DIAGRAM

• Add the vectors head to tail.• Measure R and θ with a ruler and

protractor.

Adding Vectors by Components

A

B

Adding Vectors by Components

A B

Transform vectors so they are head-to-tail.

Adding Vectors by Components

A BBy

Bx

Ax

Ay

Draw components of each vector...

Adding Vectors by Components

A BBy

BxAx

Ay

Add components as collinear vectors!

Adding Vectors by Components

A B By

BxAx

Ay

Draw resultants in each direction...

Ry

Rx

Adding Vectors by Components

A B

Combine components of answer using the head to tail method...

Ry

Rx

R

Adding Vectors by Components

Use the Pythagorean Theorem to solve for R and q…

R Rx2 Ry

2

ExamplesFind the sum of the forces…140 lb at 40 deg. North of west and 220 lb at 30 deg north of east…

Comparing Methods

Why is the component method a better method than the scale diagram method?

Challenge: The Strongman...When the strongman suspends the 10 lb telephone book with the rope held vertically the tension in each strand of rope is 5 lbs. If the strongman could suspend the book from the strands pulled horizontally as shown, the tension in each strand would be:

a) about 5 lbs

b) about 10 lbs

c) about 20 lbs

d) more than a million lbs

Formulas for Vector addition

Pythagorean Theorem (right triangle)R2 = A2 + B2

orLaw of Cosines (any triangle)R2 = A2 + B2 – 2ABcosθ See Practice problem at bottom of page 66

To solve vector problems

• Sketch the problem showing the vectors in relationship to each other.• Use the proper math procedure• Check the units. Are they correct?• Does the sign make sense? It should be positive.• Is your answer realistic? Your answer should be greater than any one

of the vectors,

PHYSICS ASSIGNMENT

• Read • Chapter 4• Practice Problems 1-4, page 67

– (answers are in Appendix C, page 780)

• Practice Problems 5-10, page 71

PHYSICS ASSIGNMENT

• Turn-in for a grade: DUE EOP Problems pages 78-79Answers for these are NOT in book.

24-29

6 problems @ 5pts each = 30 pts– Bonus problem on next page = 5 pts

Wild and Crazy Vector Problem

• A hunter was hunting bear and saw a bear exactly 100 m to his south. The wind was blowing from the north at 24 km/hr. He fired a bullet that weighed 5 grams at the bear hitting it but not killing the bear. The hunter walked to the original place where he saw the bear and followed a trail of blood 50 m to the west. The wind stopped blowing. The trail of blood went south for 25 meters, east for 200 meters and then back north for 125 m. The hunter was at the same exact spot where he originally pulled the trigger.

• What color was the bear?