Mechanics is the branch of physics that deals with the study of motion and forces on objects, and it is classified into statics, dynamics, and kinematics; statics concerns equilibrium, dynamics concerns forces on moving objects, and kinematics concerns motion without forces. Mechanics studies the motion of macroscopic bodies using concepts like position, displacement, distance, vectors, and methods for adding vectors like the component method using trigonometry or the graphical head-to-tail method.

2. MECHANICS
Mechanics is a branch of physics which deals
with the study of the forces which act on a
body and keep it in equilibrium or in motion. It
is one of the largest subjects in science and
technology. The bodies whose motion we
study in "Mechanics" are macroscopic bodies
i.e. bodies that we can easily see.
However, we may study motion with respect to
solids, liquids and gases, where we may be
dealing with the motion of large molecules.

3. MECHANICS
Mechanics is classified into three divisions
as:
Statics
• Statics is a branch of physics (mechanics) which
concerned with equilibrium state of bodies under the
action of forces. When a system of bodies is in static
equilibrium, the system is either at rest, or moving at
constant velocity through its center of mass. It can also
be understood as the study of the forces affecting non-
moving objects

4. MECHANICS
Dynamics
• The branch of physics (mechanics) which deals with
the effect of forces on the motion of bodies. It can also
be understood as the study of the forces affecting
moving objects.
Kinematics
• It is the branch of physics (mechanics) concerned with
the motions of objects without being concerned with
the forces that cause the motion.

5. KINEMATICS
In this unit, we study mechanics, which is the
study of the motion of objects and the related
concepts of force and energy. This section is
devoted to the study of kinematics, a
description of how objects move. We see
many examples of moving objects in our lives
from automobiles to ball motion in sports, to
the motion of the moon and the sun and
satellites.

6. SCALAR AND VECTOR
QUANTITIES
Physics is a mathematical science - that is, the
underlying concepts and principles have a
mathematical basis. Throughout the course of our
study of physics, we will encounter a variety of
concepts which have a mathematical basis
associated with them.
The motion of objects can be described by words -
words such as distance, displacement, position,
speed, velocity, and acceleration. These
mathematical quantities which are used to
describe the motion of objects can be divided into
two categories.

7. Scalar and Vector
Quantities
The quantity is either a vector or a scalar.
These two categories can be distinguished
from one another by their distinct definitions:
• Scalars are quantities that are fully
described by a magnitude alone.
• Vectors are quantities that are fully
described by both a magnitude and a
direction.

8. Scalar and Vector
Video

9. Scalar and Vector
Quantities
Check Your Understanding
Categorize each quantity as being either a vector or a scalar.
a. 5 m b. 30 m/sec, East
c. 5 mi., 5o d. 20 degrees Celsius
e. 256 bytes f. 4000 Calorie

10. POSITION, DISPLACEMENT, &
DISTANCE
Distance and displacement are two
quantities which may seem to mean the
same thing, yet they have distinctly
different meanings and definitions.
Distance is a scalar quantity which refers to
"how much ground an object has covered"
during its motion.
Displacement is a vector quantity which
refers to "how far out of place an object is";
it is the object's change in position.

11. POSITION, DISPLACEMENT, &
DISTANCE
The position of an object is a description of its
location. In order to describe the position of an
object, one must know the distance and direction
from a known origin.
Example
• A physics teacher walks 4 meters East, 2 meters South, 4 meters
West, and finally 2 meters North

12. POSITION, DISPLACEMENT, &
DISTANCE
Example
• Use the diagram to determine the distance traveled by the skier and
the resulting displacement during these three minutes.

13. POSITION, DISPLACEMENT, &
DISTANCE
Question:
What are the position, distance and the
displacement of the racecar drivers in the Indy
500?

14. VECTORS & DIRECTION
• Examples of vector include displacement,
velocity, acceleration, and force. Each of these
quantities are unique in that a full description of
the quantity demands that both a magnitude and
a direction are listed.
• Vector quantities are not fully described unless
both magnitude and direction are described
• Vector quantities are often represented by scaled
diagrams. Vector diagrams depict a vector by use
of an arrow drawn to scale in a specific direction.

15. VECTORS & DIRECTION
Observe that there are several
characteristics of this diagram which
make it an appropriately drawn vector
diagram.
• a scale is clearly listed
• an arrow (with arrowhead) is drawn
in a specified direction; thus, the
vector has a head and a tail.
• the magnitude and direction of the
vector is clearly labeled; in this
case, the diagram shows
magnitude is 20m and the direction
is (30 degrees West of North).

16. VECTORS & DIRECTION
The direction of a vector is often expressed as an
counterclockwise angle of rotation of the vector
about its "tail" from due East. Using this convention,
a vector with a direction of 30 degrees is a vector
that has been rotated 30 degrees in a
counterclockwise direction relative to due east. A
vector with a direction of 160 degrees is a vector
that has been rotated degrees in a
counterclockwise direction relative to due east. A
vector with a direction of 270 degrees is a vector
that has been rotated 270 degrees in a
counterclockwise direction relative to due east.

17. VECTORS & DIRECTION

18. VECTORS & DIRECTION
The magnitude of a vector in a scaled vector
diagram is depicted by the length of the arrow. The
arrow is drawn a precise length in accordance with
a chosen scale. For example, if a diagram shows a
vector with a magnitude of 15 km and the scale
used constructing the diagram is 1 cm = 5 km, the
vector arrow is drawn with a length of 3 cm.

19. COMPOSITION (ADDING) OF
VECTORS
• The composition of vectors is the process of
combining vectors to find a single vector that has
the same effect as the combination of single
vectors.
• Each separate vector is called a component
and the single vector that produces the same
result as the combined components is called the
resultant.
• The resultant vector is a vector that replaces two
or more vectors that have been added together.

20. COMPOSITION OF VECTORS
There are a variety of methods for
determining the magnitude and direction
of the result of adding two or more
vectors. These methods include:
• the Pythagorean theorem and
trigonometric methods (Component
Method)
• the head-to-tail method using a scaled
vector diagram (Graphical Method)

21. ADDING LINEAR VECTORS
• Two vectors, going in the same direction can be
added together by simply placing them “head-to-
tail”. The resultant will have a new magnitude
that is equal to the sum of the two individual
vector magnitudes.
• Vectors can also be subtracted. The direction of
the resultant vector is the same as the direction
of the largest vector.
• To determine the magnitude of the resultant
vector you place them “tail-to-tail”. The longer
one then cancels out the shorter one, leaving the
resultant.

22. ADDING LINEAR VECTORS

23. ADDING LINEAR
VECTORS
Practice Problems:
1. A jet fighter plane traveling south at
490 km/h fires a missile forward at 150
km/h. What is the velocity of the
missile?
2. A child pulls east on a rope with a force
of 30 newtons. Another child pulls with a
force of 25 newtons in the opposite
direction. What is the resultant force on the
rope?

24. ADDING PERPENDICULAR
VECTORS
The Pythagorean theorem is a useful method for
determining the result of adding two (and only two)
vectors that make a right angle to each other. The
method is not applicable for adding more than two
vectors or for adding vectors which are not at 90-
degrees to each other.

25. Adding Perpendicular
Vectors
Example
• A hiker leaves camp and hikes 11 km, north and
then hikes 11 km east. Determine the resulting
displacement of the hiker.

26. Adding Perpendicular
Vectors
The direction of vector R in the diagram above can be
determined by use of trigonometric functions. Recall
the meaning of the useful mnemonic - SOH CAH
TOA.

27. Adding Perpendicular
Vectors
The three equations below summarize these three
functions in equation form.

28. Adding Perpendicular
Vectors
Example
• A hiker leaves camp and hikes 11 km, north and then hikes 11 km
east. Determine the resulting displacement of the hiker.

29. Adding Perpendicular
Vectors
Practice Problems:

30. GRAPHICAL METHOD
• The magnitude and direction of the sum of two
or more vectors can also be determined by use
of an accurately drawn scaled vector diagram.
Using a scaled diagram, the head-to-tail method
is to determine the resultant.
• The head-to-tail method involves drawing a
vector to scale on a sheet of paper beginning at
a designated starting position; where the head of
this vector ends the tail of the next vector begins
(thus, head-to-tail method).

31. GRAPHICAL METHOD
• The process is repeated for all vectors that are
added. Once all vectors have been added head-
to-tail, the resultant is drawn from the tail of the
first vector to the head of the last vector; i.e.,
from start to finish. Once the resultant is drawn,
its length can be measured and converted to real
units using the given scale.
• The direction of the resultant can be determined
by using a protractor and measuring its
counterclockwise angle of rotation from due
East.

32. A VECTOR JOURNEY
Your assignment is to find the displacement
of the following journey. You must draw a
scaled vector Diagram, showing the
displacement.
Starting Point – X marks the spot (Mr.
Coulter’s Room)
End Point – X marks the spot (in front of the
water fountain in the north hallway)

33. VECTOR COMPONENTS
• Any vector directed in two dimensions can be
thought of as having an influence in two different
directions. That is, it can be thought of as having two
parts.
• Each part of a two-dimensional vector is known as a
component. The components of a vector depict the
influence of that vector in a given direction.
• The combined influence of the two components is
equivalent to the influence of the single two-
dimensional vector. The single two-dimensional
vector could be replaced by the two components.

34. VECTOR COMPONENTS

35. VECTOR RESOLUTION
There are two basic methods for determining the
magnitudes of the components of a vector directed
in two dimensions. The process of determining the
magnitude of a vector is known as vector
resolution. The two methods of vector resolution
that we will examine are:
• The Parallelogram Method
• Trigonometric Method

36. THE PARALLELOGRAM
METHOD
The parallelogram method of vector resolution involves
using an accurately drawn, scaled vector diagram to
determine the components of the vector. The method
involves drawing the vector to scale in the indicated
direction, sketching a parallelogram around the vector.

37. THE TRIGONOMETRIC
METHOD
The trigonometric method of vector resolution involves using
trigonometric functions to determine components of the
vector.

38. Adding Non-
Perpendicular or Non-
Linear Vectors
Using the components of vectors, it is now easy to
add vectors that are not linear or perpendicular to
one another using the component method.

39. Adding Non-
Perpendicular or Non-
Linear Vectors
Example:
Find the resultant vector of A and B given in the
graph below.

40. Adding Non-
Perpendicular or Non-
Linear Vectors
We use trigonometric
equations first and
find the components
of the vectors then,
make addition and
subtraction between
the vectors sharing
same direction.

41. Adding Non-
Perpendicular or Non-
Linear Vectors
Vector X Component Y Component
A
B
Result-
ant

42. INSTANT & INTERVAL OF
TIME
• The notion of time is important in physics. A
common misconception is that an instant in time
is a very short interval of time. However, an
instant is considered to be a single clock reading
(t). If time were plotted on an axis, an instant is
just a single coordinate along that axis.
• An interval is a duration in time, i.e. the interval
separating two instants on the time axis (∆t). The
combination of a clock reading and
instantaneous position is called an event and
later becomes useful introducing relativity.

43. INSTANT & INTERVAL OF
TIME
Note: ∆, pronounced “delta”, is used to represent the phrase: “
change in”, and is calculated as “final – initial”.
Ex. ∆ t is pronounced as: the change in time
(∆ t = tfinal – tinitial )
∆ v is pronounced as: the change in velocity
(∆ v = vfinal - vinitial )

44. INSTANT & INTERVAL OF
TIME
Note: ∆, pronounced “delta”, is used to represent the phrase: “
change in”, and is calculated as “final – initial”.
Ex. ∆ t is pronounced as: the change in time
(∆ t = tfinal – tinitial )
∆ v is pronounced as: the change in velocity
(∆ v = vfinal - vinitial )

45. Describing Motion
15
10
Position
5
0
0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34
Time

46. POSITION VS.
TIME
GRAPHS

47. Position vs. Time Graphs
The specific features of the motion of objects are demonstrated
by the shape and the slope of the lines on a position vs. time
graph.
The first part involves a study of the relationship between the
shape of a p-t graph and the motion of the object.

48. Position vs. Time Graphs
Consider a car moving with a constant,
rightward (+) velocity - say of +10 m/s.

49. Position vs. Time Graphs
Consider a car moving with a constant,
rightward (+) velocity - say of +10 m/s.

50. Position vs. Time Graphs
Consider a car moving with a rightward (+),
changing velocity - that is, a car that is moving
rightward but speeding up or accelerating.

51. Position vs. Time Graphs
Consider a car moving with a rightward (+),
changing velocity - that is, a car that is moving
rightward but speeding up or accelerating.

52. Position vs. Time Graphs
The slope of the line on a position-time graph
reveals useful information about the velocity of
the object.
If the velocity is constant, then the slope is
constant (i.e., a straight line).
If the velocity is changing, then the slope is
changing (i.e., a curved line).
If the velocity is positive, then the slope is
positive (i.e., moving upwards and to the right).

53. Position vs. Time Graphs
Slow, Rightward (+), Fast, Rightward (+),
Constant Velocity Constant Velocity

54. THE MEANING OF SLOPE FOR A
P-T GRAPH
The slope of a position vs. time graph reveals
pertinent information about an object's velocity.
• a small slope means a small velocity;
• a negative slope means a negative velocity;
• a constant slope (straight line) means a constant velocity;
• a changing slope (curved line) means a changing velocity.

55. The Meaning of Slope for a
p-t Graph
Consider a car moving at a constant velocity
of +5 m/s for 5 seconds, abruptly stopping,
and then remaining at rest (v = 0 m/s) for 5
seconds.

56. The Meaning of Slope for a
p-t Graph
Consider a car moving at a constant velocity of
+5 m/s for 5 seconds, abruptly stopping, and
then remaining at rest (v = 0 m/s) for 5
seconds.

57. Determining the Slope for a
p-t Graph
The slope of the line on a position vs. time
graph is equal to the velocity of the object.

58. The Meaning of Slope for a
p-t Graph
Determine the velocity (i.e., slope) of the object
as portrayed by the graph below.

59. VELOCITY VS
TIME
GRAPHS

60. Velocity vs. Time Graphs
Velocity is a vector quantity that refers to "the rate at which an
object changes its position." As such, velocity is "direction-
aware." When evaluating the velocity of an object, one must
keep track of direction.
To describe the motion of an object, velocity vs. time graphs can
be used.

61. Velocity vs. Time Graphs
Consider a car moving with a constant, rightward
(+) velocity - say of +10 m/s. As learned earlier, a
car moving with a constant velocity is a car with
zero acceleration.

62. Velocity vs. Time Graphs
Note that a motion described as a constant,
positive velocity results in a line of zero slope (a
horizontal line has zero slope) when plotted as
a velocity-time graph.

63. Velocity vs. Time Graphs
Consider a car moving with a rightward (+),
changing velocity - that is, a car that is moving
rightward but speeding up or accelerating. Since
the car is moving in the positive direction and
speeding up, the car is said to have a positive
acceleration.

64. Velocity vs. Time Graphs
Note that a motion described as a changing,
positive velocity results in a sloped line when
plotted as a velocity-time graph. The slope of the
line is positive, corresponding to the positive
acceleration.

65. Velocity vs. Time Graphs
Positive Velocity Positive Velocity
Positive Acceleration Zero Acceleration

66. Velocity vs. Time Graphs

67. KINEMATIC
EQUATIONS
These are
the basic
equations
that you
will be
working
with. Keep
in mind
that these
equations
can be
rearranged
or
combined
to make
different
equations

68. KINEMATIC
EQUATIONS
Illustrative Example #1
An object moving at 3.0 m/s accelerates for 4.0 s
with a uniform acceleration of 2.0 m/s2. Find the
displacement of the object.

69. KINEMATIC
EQUATIONS
Illustrative Example #2
A car traveling at 10.0 m/s accelerates at a rate of
3.0 m/s2 to a speed of 25.0 m/s. What is the
displacement of the car during the acceleration?

70. KINEMATIC
EQUATIONS
Illustrative Example #3
If an object is accelerated at 5.00 m/s2 and starts from rest, what is
its velocity after 20.0 m?