Vectors
Scalar:
Vector:
Scalars
Vectors
Wind is blowing 15 m/s East. What is the magnitude of the wind’s velocity? What is the direction?
Vectors are represented by an arrow. The length of the arrow represents ___________________ while the way the arrow points represents direction.
Vectors *need* to be an arrow. Without the arrow, there is no direction!
Drawing Vectors ­ Scaled Vector Diagrams
Use a ruler and the scale given to draw the following vectors 1. 1 cm = 10 m
2. 1 cm = 5 m/s
Draw 30 m East
Draw 20 m/s West
3. 1 mm = 1 m
Draw 25 m East
Determine the scale for the following vectors
4. 1 cm = 5. 6.
1 cm = 1 cm = Vector Direction
A cow runs 10° South from West with a velocity of 6 m/s
Start from West
You are now 0 degrees from West
What are the possible directions you could go from West?
Which way should you go from West?
Make a sketch of the vector 10° South from West, label it #1
Now make a sketch of a vector that is 10° West from South, label it #2. Is this the same as 10° South from West?
8. Sketch a vector that is
a. 20° North from East, label #1
b. 82° East from South, label #2
Vector Addition
when 2+2 ≠ 4 (sometimes)
Equilibrant – Example:
9. Mr. Vigneaux pushes on a desk to the right with a force of 4 Newtons (the unit for force). A disgruntled student works against him by pushing it to the left with a force of 3 Newtons. What is the resultant force?
The student and Mr. Vigneaux decide to work together to push the desk to the right, with the same magnitudes as before. What is the resultant force?
Under what conditions can two vectors create the maximum resultant?
When would they create the minimum resultant?
Example:
10. Two forces, a 25 N force and a 15 N force act concurrently on an object. Which of the following could be the magnitude of the resultant force? (There can be more than one correct answer).
8 N
10 N
18 N
40 N
45 N
52 N
Adding Vectors Mathematically
When two vectors act in the same direction,
When two vectors act in opposite directions,
When two vectors act perpendicular to each other,
11. A plane is flying at 150 m/s due East while a wind blows it at 30 m/s. Find the magnitude of the resultant velocity when the wind is blowing
a. North
b. South
c. East
d. West
Adding Vectors Graphically – Tip to Tail
We can move vectors around so long as we keep the correct magnitude and direction. To determine the resultant from two vectors, we move the beginning (tail) of one vector to the end (tip) of the other vector. The resultant then points from start to finish. Move B to the tip of A, Move A to the tip of B,
then sketch the resultant
then sketch the resultant
How do the results compare? Does it matter which vector we move?
Why am I allowed to do this?
David walks 5 m North and 5 m East.
Moe walks 5 m East and 5 mo' meters North.
How do their positions compare?
We can move vectors in space, as long as we don't change size (magnitude) or direction.
12.
13.
14. Sketch a vector that could be added to vector A to create resultant R
Scaled Vector Diagrams
15. Example: Use a ruler and a scale of 1cm = 2m
A man walks 10m East and then walks 4m West. Draw a scaled vector diagram of his walk. Then draw the resultant. *Remember that vectors add tip­to­tail and the resultant points start­to­finish.* 16. Starting from point A, use a scale of 1cm = 5 N
A force of 25 N East and a force of 10 N South both push on an object. Draw a scaled vector diagram
Draw the resultant
Measure the resultant in cm.
What is this in N?
Now calculate the resultant using the Pythagorean theorem. How does your result compare?
17. Starting from point B, use a scale of 1cm = 10 m
A man walks 30 m East, 15 m North, and 10 m West
Draw a scaled vector diagram
Draw the resultant
Measure the resultant and convert it to meters.
Calculate his resultant displacement.
How does the calculated displacement compare to the measured displacement?
The Riverboat Problem
19. A motorboat traveling 7 m/s, East encounters a current traveling 3.0 m/s, North.
a) What is the resultant velocity of the motorboat?
b) If the width of the river is 80 meters wide, then how much time does it take the boat to travel shore to shore?
c) What distance downstream does the boat reach the opposite shore?
Speed/Velocity
Distance
20. A model airplane heads due east at 1.50 meters per second, while the wind blows due north at 0.70 meter per second.
a) What is the resultant velocity of the airplane? (make a sketch)
Distance
Velocity
b) If it’s flying East across a highway which runs North­
South which is 10 m wide, how long does it take to reach the other side? (make a sketch)
c) How far down the highway does it drift in the wind?
21. An ant is crossing a treadmill at 0.05 m/s. The treadmill moves forward at 0.10 m/s
a) What is the magnitude of the resultant velocity of the ant?
b) If the treadmill is 0.50 m wide, how much time does it take the ant to cross?
c) How far forward does the treadmill move as the ant crosses?
d) What is the resultant displacement of the ant?
22. A motorboat, which has a speed of 5 meters per second in still water, is headed east as it crosses a river flowing south at 3.3 meters per second. What is the magnitude of the boat’s resultant velocity with respect to the starting point?
It takes 22 seconds to cross the river. How wide is the river?
23. A cruise ship is headed North at 10 m/s while a jogger runs across it at 3 m/s. What is the magnitude of his resultant velocity?