1. Vectors and Two-Dimensional Motion
Chapter 3
Vectors and Two-Dimensional Motion

2. Vectors
Vectors – physical quantities having both magnitude and direction
Vectors are labeled either a or
Vector magnitude is labeled either |a| or a
Two (or more) vectors having the same magnitude and direction are identical

3. Vector sum (resultant vector)
Not the same as algebraic sum
Triangle method of finding the resultant:
Draw the vectors “head-to-tail”
The resultant is drawn from the tail of A to the head of B
R = A + B B A

4. Addition of more than two vectors
When you have many vectors, just keep repeating the process until all are included
The resultant is still drawn from the tail of the first vector to the head of the last vector

5. Commutative law of vector addition
A + B = B + A

6. Associative law of vector addition
(A + B) + C = A + (B + C)

7. Negative vectors
Vector (- b) has the same magnitude as b but opposite direction

8. Vector subtraction
Special case of vector addition: A - B = A + (- B)

9. Multiplying a vector by a scalar
The result of the multiplication is a vector
c A = B
Vector magnitude of the product is multiplied by the scalar
|c| |A| = |B|
If the scalar is positive (negative), the direction of the result is the same as (opposite to that) of the original vector

10. Vector components
Component of a vector is the projection of the vector on an axis
To find the projection – drop perpendicular lines to the axis from both ends of the vector – resolving the vector

11. Vector components

12. Unit vectors Unit vector: Has a magnitude of 1 (unity)
Lacks both dimension and unit
Specifies a direction
Unit vectors in a right-handed coordinate system

13. Adding vectors by components
In 2D case:

14. Chapter 3
Problem 14
A quarterback takes the ball from the line of scrimmage, runs backwards for 10.0 yards, then runs sideways parallel to the line of scrimmage for 15.0 yards. At this point, he throws a 50.0-yard forward pass straight downfield, perpendicular to the line of scrimmage. What is the magnitude of the football’s resultant displacement?

15. Position
The position of an object is described by its position vector,

16. Displacement
The displacement of the object is defined as the change in its position,

17. Velocity
Average velocity
Instantaneous velocity

18. Instantaneous velocity
Vector of instantaneous velocity is always tangential to the object’s path at the object’s position

19. Acceleration
Average acceleration
Instantaneous acceleration

20. Acceleration Acceleration – the rate of change of velocity (vector)
The magnitude of the velocity (the speed) can change – tangential acceleration
The direction of the velocity can change – radial acceleration
Both the magnitude and the direction can change

21. Projectile motion A special case of 2D motion
An object moves in the presence of Earth’s gravity
We neglect the air friction and the rotation of the Earth
As a result, the object moves in a vertical plane and follows a parabolic path
The x and y directions of motion are treated independently

22. Projectile motion – X direction
A uniform motion: ax = 0
Initial velocity is
Displacement in the x direction is described as

23. Projectile motion – Y direction
Motion with a constant acceleration: ay = – g
Initial velocity is
Therefore
Displacement in the y direction is described as

24. Projectile motion: putting X and Y together

25. Projectile motion: trajectory and range

26. Projectile motion: trajectory and range

27. Chapter 3
Problem 58
A 2.00-m-tall basketball player is standing on the floor 10.0 m from the basket. If he shoots the ball at a 40.0° angle with the horizontal, at what initial speed must he throw the basketball so that it goes through the hoop without striking the backboard? The height of the basket is 3.05 m.

28. Relative motion
Reference frame: physical object and a coordinate system attached to it
Reference frames can move relative to each other
We can measure displacements, velocities, accelerations, etc. separately in different reference frames

29. Relative motion
If reference frames A and B move relative to each other with a constant velocity
Then
Acceleration measured in both reference frames will be the same

30. Questions?

31. Answers to the even-numbered problems
Chapter 3
Problem 2:
Approximately 484 km
(b) Approximately 18.1° N of W

32. Answers to the even-numbered problems
Chapter 3
Problem 6:
Approximately 6.1 units at 113°
(b) Approximately 15 units at 23°

33. Answers to the even-numbered problems
Chapter 3
Problem 10:
1.31 km north, 2.81 km east

34. Answers to the even-numbered problems
Chapter 3
Problem 28:
x = 7.23 × 103 m, y = 1.68 × 103 m