1. Chapter 3: Vectors & Two-Dimensional Motion
Vectors and Their Properties
All the physical quantities we will learn are classified
either as a vector or scalar quantity.
A scalar is specified by its magnitude, while a vector
by its magnitude AND its direction.
Examples
Scalar : temperature, speed, mass, volume, length, etc.
Vector : displacement, velocity, acceleration, force, etc.

2. Vectors and Their Properties
Notation
Equality of two vectors
if their magnitudes and the directions are the same.
Addition of vectors (geometrical method)
Commutative law of addition:

3. Vectors and Their Properties
Negative of a vector
is defined as the vector that gives zero when added to
has the same magnitude but opposite direction of
Subtraction of vectors

4. Vectors and Their Properties
Example 3.1: Taking a trip
A car travels 20 km due north
and then 35.0 km in a direction
600 west of north.
What is the net effect of the car’s
trip?

5. Components of a Vector
Components of a vector
A vector in two-dimension can be specified by a pair of coordinates
y-component
x-component
tan-1q defined in (-900,900).
Add 1800 when the vector is in
2nd or 3rd quadrant.

6. Components of a Vector
Components of a vector in different coordinates
A vector in two-dimension can be specified by a pair of coordinates
If a different coordinate system
used, the components are
different to represent the same
vector. y
Vector addition by components: By x
But use the same coordinate
system for both vectors Bx

7. Components of a Vector Example 3.3: Take a hike
An example
Example 3.3: Take a hike
1st day : 25.0 km southeast
2nd day : 40.0 km in a direction
60.0o north of east
Determine the components
of the hiker’s displacements
in the 1st and 2nd days.

8. Displacement, Velocity, & Acceleration in Two-Dimension
Displacement in 2D
A position vector describes the
position of an object at a time.
An object’s displacement from
ti to tf is defined by:

9. Displacement, Velocity, & Acceleration in Two-Dimension
Velocity in 2D
An object’s average velocity during a time interval Dt is:
An object’s instantaneous velocity is:

10. Displacement, Velocity, & Acceleration in Two-Dimension
Acceleration in 2D
An object’s average acceleration during a time interval Dt is:
An object’s instantaneous acceleration is:

11. Motion in Two-Dimension
Motion in 2D: horizontal and vertical direction
In this chapter, we will learn movement of an object in both the
x- and y-direction simultaneously under constant acceleration.
An example: projectile motion under influence of gravity

12. Motion in Two-Dimension
Projectile motion under influence of gravity
Let’s examine the motion of an object that is tossed into air but
let’s neglect the effects of air resistance and the rotation of Earth.
It was experimentally proven that the horizontal and vertical motions
are completely independent of each other.
Motion in one direction has no effect on motion in the other
direction.
So, in general, the equations of constant acceleration we
learned in Lecture 2 follow separately for both the x-direction
and the y-direction.

13. Motion in Two-Dimension
Projectile motion under influence of gravity
Let’s assume that at t=0, the projectile leaves the origin with an
initial velocity with an angle with the horizontal.
x-direction:

14. Motion in Two-Dimension
Projectile motion under influence of gravity
Let’s assume that at t=0, the projectile leaves the origin with an
initial velocity with an angle with the horizontal.
y-direction:

15. Motion in Two-Dimension
Projectile motion under influence of gravity
Plug-in all the known quantities.
Equations that describe the motion in the x-direction:
Equations that describe the motion in the y-direction:
Velocity :

16. Motion in Two-Dimension
Projectile motion under influence of gravity
Trajectories as a function of the projection angle
Note : Given the displacement in x
there are two corresponding
projection angles. o o o o o

17. Motion in Two-Dimension
Examples
Problem 3.5: Stranded explorers
(a) What is the range of the package?
range of package

18. Motion in Two-Dimension
Examples
Problem 3.5: Stranded explorers
(b) What are the velocity components of the package at impact?
x component:
y component:
range of package

19. Motion in Two-Dimension
Examples
Problem 3.6: The long jump
(a) How long does it take for the jumper to reach the max. height?
y component:
at max.height vy =0
v0= 11.0 m/s
q=20.0o

20. Motion in Two-Dimension
Examples
Problem 3.6: The long jump
(b) Find the maximum height he reaches.
y component:
From part (a)
v0= 11.0 m/s
q=20.0o

21. Motion in Two-Dimension
Examples
Problem 3.6: The long jump
(c) Find the horizontal distance he jumps.
Displacement in x:
v0= 11.0 m/s
q=20.0o

22. Motion in Two-Dimension
Examples
Problem 3.8: The rocket
(a) Find the rocket’s velocity in y direction.
Eq.3.14c:
(b) Find the rocket’s velocity in x direction.
Eq.3.12a:
Eq.3.11a:

23. Motion in Two-Dimension
Examples
Problem 3.8: The rocket
(c) Find the magnitude and direction of the rocket’s velocity.

24. Relative Velocity
What is relative velocity?
The measured velocity of an object depends on the velocity
of the observer with respect to the object.
Relative velocity relates velocities measured by two different
observers, one moving with respect to the other.
Measurements of velocity depend on the reference frame of the
observer where the reference frame is a just coordinate system
used to measure physical quantities such as velocity, acceleration
etc. Most of time, we will use a stationary frame of reference, relative
to earth, but occasionally we will use a moving frame of reference.

25. Relative Velocity More elaborate definition:
What is relative velocity? (cont’d)
More elaborate definition:
- Let’s define E as a stationary observer with respect to Earth
A and B as two moving cars

26. Relative Velocity relative to; with respect to
What is relative velocity? (cont’d)
relative to; with respect to

27. Relative Velocity
Examples
Example 3.10: Crossing a river

28. Relative Velocity
Examples
Example 3.10: Crossing a river (cont’d)