1. Enduring Understanding: Modeling is widely used to represent physical and kinematic information.
Essential Question: What are the practical applications of modeling physical and kinematic information?

2. Let’s say you draw an arrow. How long is it?
60 cm

3. 1 cm = 20 miles Now, the original arrow represents something else.
How might this be useful?

6. Vectors are scale model arrow representations.
Vectors have magnitude (size) and direction.
Recall: Magnitude without direction is called a scalar.

7. Scalar Vector
Distance Displacement
Speed Velocity
Time Acceleration

8. Vector Addition
The resultant is one vector that represents the addition of two or more vectors.
What is the resultant displacement if you add 2 m east to 3 m east?

9. Add vectors head to tail
Add vectors head to tail. The arrow is the head, the other end is the tail.
2 m east m east
Resultant 5 m east

10. Head to Tail Vector Addition

12. What is the resultant of 3 m east + 4 m north?

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

14. What is the resultant magnitude of 3 m east + 4 m north?
The magnitude = 5 m
4 m
3 m

15. To find direction:
Find the angle of the resultant and name the direction. Find the angle closest to the starting point (where 2 tails meet). Use inverse tangent
θ = tan-1 (4/3)
= 53.1°
4 m θ
3 m

16. For direction, it’s always the 2nd direction ‘of’ the 1st direction.
In this case, the 1st direction traveled is
East, the 2nd direction is North.
Therefore the direction is N of E.
The resultant is:
53.1° N of E
4 m θ
3 m

18. A boat heads west across a 300 m wide river at 10 m/s
A boat heads west across a 300 m wide river at 10 m/s. The current pushes the boat south at 5 m/s. What is the resultant speed of the boat?
° S of W
How long does it take to cross the river?
30s
How far downstream does the boat travel?
150 m

19. Vector Representation of Projectile Motion

20. A vector has magnitude and direction.
For each ball, draw vectors to represent the horizontal component of the velocity.

21. For each ball, draw vectors to represent the vertical component of the velocity.

22. Even More Vector Practice
1) A person walks 5 m to the south. They stop for a rest. They then travel 20 m farther to the south. What was their resultant displacement?
25 m south

23. 2) Galileo is riding a bicycle and is pedaling at 8 m/s to the west
2) Galileo is riding a bicycle and is pedaling at 8 m/s to the west. A 2 m/s wind is pushing to the east. What is the resultant velocity of Galileo and his bicycle?
6 m/s west

24. 3) A hiker travels 300 m west, stops, checks a map and then travels 700 m north. How far is the hiker from their starting point?
762 m

25. 4) A boater drives a motorboat out in the bay
4) A boater drives a motorboat out in the bay. The boat has a speed 40 m/s directed west but a current pushes it south at m/s. What is the resultant speed of the boat?
41.2 m/s

26. 5) How long will it take the boat to travel a distance of 2500 m?

27. 6) A person walks 40 m east, 20 m south, 10 m west and 40 m north
6) A person walks 40 m east, 20 m south, 10 m west and m north. How far are they from their starting point?
36 m