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College and Engineering Physics
Velocity and Speed
The Particle Model
Velocity
Motion in One Dimension
Uniform Motion
Instantaneous Velocity
Finding Position from Velocity
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College and Engineering Physics
Velocity and Speed
The Particle Model
A particle is an object that can be represented as a mass at a single point
in space.
This point is called the center of mass of the object. We will learn about
this later.
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The relationship between the force on the particle and the particle’s
acceleration determine the path (trajectory) that the particle will follow.
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Velocity
College and Engineering Physics
Velocity and Speed
G
v
Velocity measures the motion of an object
It consists of two parts …
1. The magnitude (speed) which is larger when the object is moving fast
and smaller when the object is moving slow.
The turtle has a smaller magnitude of velocity than the rabbit.
2. The direction tells you where the object is going.
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The velocity of the airplanes has the same magnitude,
but a different direction.
College and Engineering Physics
Velocity and Speed
Horizontal Motion
The x-component of most vectors (though not all) will be the component
which is horizontal (lies along the plane of the earth).
Variable
Symbol
(> 0)
(< 0)
Positive
Negative
y
y
Meaning
of the
Sign
Determines a
particle’s
location
Position
x
Velocity
vx
Determines the
direction of the
particle’s motion
Acceleration
ax
Determines the
direction of the
change in a
particle’s motion
x
x
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College and Engineering Physics
Velocity and Speed
Vertical Motion
The y-component of most vectors (though not all) will be the component
which is vertical (points perpendicular to the plane of the earth).
Variable
Symbol
(> 0)
(< 0)
Positive
Negative
y
y
Meaning
of the
Sign
Determines a
particle’s
location
Position
y
Velocity
vy
Determine the
direction of the
particle’s motion
Acceleration
ay
Determines the
direction of the
change in a
particle’s motion
x
x
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College and Engineering Physics
Velocity and Speed
Average Velocity
The direction and quantity of “motion” of a particle is given by its velocity.
x
The slope of this line is
the x-component of the
average velocity.
∆x
∆t
t
The average velocity of a particle (whether or not the motion is uniform) is
given be a simple equation …
G
G
∆r
vavg =
∆t
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In words, we would say this equation tells us that “the average velocity of a
particle is its displacement over the interval during which it moved”.
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College and Engineering Physics
Velocity and Speed
Motion with Constant Velocity (Uniform Motion)
If the position vs. time graph of a
particle’s motion is a straight line,
then the average velocity is the
slope of this line and this
component of the motion is uniform.
x
∆x
∆t
t
Velocity is constant if it does not change.
This means that the quantity (known as the magnitude) and the direction
stay the same.
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College and Engineering Physics
Velocity and Speed
G
r vs. t
Position vs. Time Graph
To model the way a particle moves, we can use a graph of position vs. time.
x
y
t
y
x
8
t
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College and Engineering Physics
Velocity and Speed
G
v vs. t
Velocity vs. Time Graph
Another way to model the motion of a particle is to graph velocity vs. time.
Velocity is the derivative (slope) of the position over time.
vx
x
t
t
0s 1s 2s 3s 4s 5s 6s 7s 8s
0s 1s 2s 3s 4s 5s 6s 7s 8s
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College and Engineering Physics
Velocity and Speed
G
v vs. t
Velocity vs. Time
To determine where an object will be at a later time, we can use velocity vs.
time curve.
Time
Vx (m/s)
Velocity
Position
4
3
2
1
t (s)
0
-1
-2
-3
-4
0
1
2
3
4
5
6
7
8
10
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College and Engineering Physics
Velocity and Speed
G
x vs. t
Position vs. Time
To determine where an object will be at a later time, we can use velocity vs.
time curve.
Time
x (m)
Velocity
Position
8
6
4
2
t (s)
0
-2
-4
-6
-8
0
1
2
3
4
5
6
7
8
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College and Engineering Physics
Velocity and Speed
Comparison
To determine where an object will be at a later time, we can use velocity vs.
time curve.
Vx (m/s)
x (m)
4
8
3
6
2
4
1
2
0
0
-1
-2
-2
-4
-3
-6
-4
12
-8
0
1
2
3
4
5
6
7
8
t (s)
0
1
2
3
4
5
6
7
8
x0 = 2 m
x0 = −3 m
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