1. Motion in Two and Three Dimensions
Chapter 3
Motion in Two and Three Dimensions

2. Types of physical quantities
In physics, quantities can be divided into such general categories as scalars, vectors, matrices, etc.
Scalars – physical quantities that can be described by their value (magnitude) only
Vectors – physical quantities that can be described by their value and direction

3. Vectors 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

4. 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

5. 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

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

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

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

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

10. 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

11. 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

12. Vector components

13. 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

14. Adding vectors by components
In 2D case:

15. Chapter 3
Problem 42
Vector A has magnitude 1.0 m and points 35° clockwise from the x-axis. Vector B has magnitude 1.8 m. Find the direction of B such that A + B is in the y-direction.

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

17. Displacement
The displacement vector is defined as the change in its position,

18. Velocity
Average velocity
Instantaneous velocity

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

20. Acceleration
Average acceleration
Instantaneous acceleration

21. 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

22. Chapter 3
Problem 23
What are (a) the average velocity and (b) the average acceleration of the tip of the 2.4-cm-long hour hand of a clock in the interval from noon to 6 PM? Use unit vector notation, with the x-axis pointing toward 3 and the y-axis toward noon.

23. 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

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

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

26. Projectile motion: putting X and Y together

27. Projectile motion: trajectory and range

28. Projectile motion: trajectory and range

29. Chapter 3
Problem 33
A carpenter tosses a shingle horizontally off an 8.8-m-high roof at 11 m/s. (a) How long does it take the shingle to reach the ground? (b) How far does it move horizontally?

30. Uniform circular motion
A special case of 2D motion
An object moves around a circle at a constant speed
Period – time to make one full revolution
The x and y directions of motion are treated independently

31. Uniform circular motion
Velocity vector is tangential to the path
From the diagram
Using
We obtain

32. Centripetal acceleration

33. Centripetal acceleration
During a uniform circular motion:
the speed is constant
the velocity is changing due to centripetal (“center seeking”) acceleration
centripetal acceleration is constant in magnitude (v2/r), is normal to the velocity vector, and points radially inward

34. Chapter 3
Problem 38
How fast would a car have to round a 75-m-radius turn for its acceleration
to be numerically equal to that of gravity?

35. 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

36. 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

37. Questions?

38. Answers to the even-numbered problems
Chapter 3
Problem 28:
196 km/h

39. Answers to the even-numbered problems
Chapter 3
Problem 30:
3.6 ˆi − 4.8 ˆj m/s2
6.0 m/s2; 53°

40. Answers to the even-numbered problems
Chapter 3
Problem 34:
1.5 m