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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
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
Combine components of answer using the head to tail method...
Ry
Rx
R
ExamplesFind the sum of the forces…140 lb at 40 deg. North of west and 220 lb at 30 deg north of east…
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?