Module 1.1 – Displacement and Velocity Vectors

1. Module 1.1 – Displacement and Velocity Vectors • Displacement and velocity vectors were studied in physics 11 for one dimensional motion. This module extends the study of kinematics to two dimensions, allowing us to study a much wider range of situations.

2. 2D Vectors • Vector Magnitude and Direction • Components

3. Trigonometry Review Pythagorean Theorem

4. Example Assume that the angle in the diagram below is 30.0° and the magnitude of the position vector is 7.2 cm. Calculate the components.

5. Solution

6. Vector Parts

7. Check Your Learning • Calculate the components for each of the following vectors:

8. Check Your Learning Taking the signs into consideration (to the right and down), we get

9. Check Your Learning Taking the signs into consideration (to the left and up), we get

10. Expressing Direction Assume that θ=30.0° • The angle is measured from east • Must go north to get to the vector 30.0° N of E (Could also be expressed as 60.0° E of N)

11. Check Your Learning State the direction for each of the following vectors: • 20o S of E • 80o W of S • 40o W of N • 50o N of W • S • 75o E of N

12. Vector Addition We want to find

13. Vector Addition Diagram • Must draw vectors being added head to tail • Resultant (c) goes from tail of • first vector to head of second vector

14. Check Your Learning Given the following vectors, draw a vector diagram to represent each of the following vector equations (where is an unknown vector in each case):

15. Check Your Learning

16. Adding With Components

17. Example A person walks 5.0 km east and then 8.0 km in a direction 75° N of E. What is his displacement?

18. Solution

19. Check Your Learning A person walks 4.0 km south and then 7.2 km in a direction 21o W of N. What was the person’s displacement?

20. Check Your Learning

21. Check Your Learning

22. Vector Subtraction

23. Vector Subtraction

24. Check Your Learning Given the following vectors, draw a vector diagram to represent each of the following vector equations (where is an unknown vector in each case):

25. Check Your Learning

26. Velocity Vectors

27. Relative Velocity V of boat with respect to water V of boat with respect to shore V of water with respect to shore Same inside subscripts

28. Example A boat that has a speed of 5.0 m/s in still water heads north directly across a river that is 250 m wide. The velocity of the river is 3.5 m/s east. • What is the velocity of the boat with respect to the shore? • How long does it take the boat to cross the river? • How far downstream does the boat land? • What heading (direction) would the boat need in order to land directly across from its starting point?

29. Solution

30. Solution

31. Solution is north where

32. Check Your Learning In the example just completed, how long will it take the boat to cross the river in part (d)? In other words, how long will it take to cross the river when corrective action is taken so that the boat lands directly across from its starting point? How does this answer compare with the time obtained in part (a) of the example?

33. Check Your Learning It takes more time to cross the river when correcting for the flow of the river than was calculated in part (a) since some of the boat’s velocity is being used to compensate for the river and stop the boat from moving downstream.

34. Module Summary • In this module you have learned: • To represent two dimensional vectors using two methods: • components and • magnitude and direction. • How to add and subtract displacement vectors using vector diagrams and components: • When adding vectors, they must be drawn head to tail. The resultant vector goes from the tail of the first vector to the head of the last vector. • Subtracting vectors is the same operation as adding a negative vector, where a negative vector points in the direction opposite the direction of the original vector. • How to use vector diagrams and components to calculate relative velocities.

35. Module 1.2 – Force Vectors The concept of two-dimensional vectors will be applied to free body diagrams and Newton’s Laws of Motion from Unit 3. Two-dimensional situations that will be studied include forces acting at an angle and inclined planes.

36. Pulling at an Angle Must break all force down into horizontal and vertical directions

37. Vertical Forces

38. Horizontal Forces

39. Example • A 52.0 kg sled is being pulled along a frictionless horizontal ice surface by a person pulling a rope with a force of 235 N. The rope makes an angle of 35.0o with the horizontal. What is the acceleration of the sled?

40. Solution

41. Check Your Learning A 52.0 kg sled is being pulled along a horizontal surface by a person pulling a rope with a force of 235 N. The rope makes an angle of 35.0o with the horizontal. If the coefficient of friction between the sled and the surface is 0.25, What is the acceleration of the sled?

42. Check Your Learning

43. Inclined Planes No friction Acceleration will be parallel to the plane Choose new coordinate system

44. Force of Gravity Components

45. Perpendicular Forces

46. Parallel Forces

47. Example A 1200 kg car is on an icy (frictionless) hill that is inclined at an angle of 12o with the horizontal. What is the acceleration of the car down the hill?

48. Check Your Learning A 1200 kg car is on an icy hill that is inclined at an angle of 12o with the horizontal. As the car starts sliding, the driver locks the wheels. The coefficient of friction between the locked wheels and the icy surface is 0.14. What is the acceleration of the car down the hill?

49. Check Your Learning Perpendicular Forces Parallel Forces

50. Module Summary In this module you learned that • Force vectors can be broken into components so that dynamics situations can be analyzed using free body diagrams and Newton’s Second Law. • Situations involving inclined planes (ramps) can be analyzed by rotating the coordinate system so that the x-axis is parallel to the ramp and the y-axis is perpendicular to the ramp. The components for the force of gravity can then be given by