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Lecture 5
(Speed,Velocity, Acceleration,
and Motion in 1-D)
Physics 160
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Introduction
•  Kinematics is the study of motion.
•  Velocity and acceleration are important physical
quantities.
•  A bungee jumper speeds up during the first part of his
fall and then slows to a halt.
Copyright © 2012 Pearson Education Inc.
Motion in One Dimension
• We need to define some terms:
–
–
–
–
–
–
Distance
Displacement
Speed
Velocity
Acceleration
Time
(scalar)
(vector)
(scalar)
(vector)
(vector)
(scalar)
[m]
[m]
[m/s]
[m/s]
[m/s2]
[s]
Average Speed and Velocity
• When we talk about speed and velocity, we are
referring to changes in distance and displacement
over a change in time.
• Important to be specific about these changes:
savg
vavg
total distance
total time
total displacement
total time
x
t
xf
xi
tf
ti
Displacement, time, and average velocity—Figure 2.1
•  A particle moving along the x-axis has a coordinate x.
•  The change in the particle s coordinate is Δx = x2 - x1.
•  The average x-velocity of the particle is vav-x = Δx/Δt.
•  Figure 2.1 illustrates how these quantities are related.
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Motion and Graphs
y [cm]
10s
0s
9s
1s
8s
7s
6s
2s
3s
4s
5s
x [cm]
x [cm]
18
16
14
12
10
8
6
4
2
1
2
1
3
4
2
5
6
3
7
8
4
9 10 11 12 13 14 15 16 17 18
5
6
7
8
9
10
t [s]
Negative velocity
•  The average x-velocity is negative during a time interval if
the particle moves in the negative x-direction for that time
interval. Figure 2.2 illustrates this situation.
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A position-time graph—Figure 2.3
•  A position-time graph (an x-t graph) shows the particle s
position x as a function of time t.
•  Figure 2.3 shows how the average x-velocity is related to the
slope of an x-t graph.
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x [cm]
Average Speed and Velocity
18
16
14
12
10
8
6
4
2
1
2
3
4
5
6
7
8
9
10
t [s]
• What is the average speed from 0 – 5s? From 0 – 10s?
• What is the average velocity from 0 – 5s? From 0 – 10s?
Instantaneous velocity—Figure 2.4
•  The instantaneous velocity is the velocity at a
specific instant of time or specific point along the
path and is given by vx = dx/dt.
•  The average speed is not the magnitude of the
average velocity!
Average and instantaneous velocities
•  In Example 2.1, the cheetah s instantaneous velocity
increases with time. (Follow Example 2.1)
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Finding velocity on an x-t graph
•  At any point on an x-t graph, the instantaneous xvelocity is equal to the slope of the tangent to the
curve at that point.
Copyright © 2012 Pearson Education Inc.
Instantaneous Velocity
x [cm]
18
16
14
12
10
8
6
4
2
x
t
1
2
3
4
5
6
7
8
9
10
t [s]
• What is the average velocity from 1 – 2s?
vavg
x
t
xf
xi
tf
ti
6cm iˆ
2cm iˆ
2s 1s
4
cm ˆ
i
s
4
cm
s
1m ˆ
i
100cm
0.04
m ˆ
i
s
• Note that this is the slope of the line connecting these two points.
Instantaneous Velocity
x [cm]
18
16
14
12
10
8
6
4
2
1
2
3
4
5
6
7
8
x
9
dx
10
t [s]
v t
lim vavg lim
• What is the velocity at 2s?
t 0
t 0
t dt
• This is the definition of the derivative of the function that
describes the position as a function of time.
• It gives a the slope of the tangent line to the function at any point.
Motion diagrams
•  A motion diagram shows the position of a particle at
various instants, and arrows represent its velocity at each
instant.
•  Figure 2.8 shows the x-t graph and the motion diagram for a
moving particle.
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Average acceleration
•  Acceleration describes the rate of change of velocity with time.
•  The average x-acceleration is aav-x = Δvx/Δt.
•  Follow Example 2.2 for an astronaut.
Copyright © 2012 Pearson Education Inc.
Instantaneous acceleration
•  The instantaneous acceleration is ax = dvx/dt.
•  Follow Example 2.3, which illustrates an accelerating racing
car.
Finding acceleration on a vx-t graph
•  As shown in Figure 2.12, the υx-t graph may be used to
find the instantaneous acceleration and the average
acceleration.
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An x-t graph and a motion diagram
•  Figure 2.14 shows the x-t graph and the motion diagram
for a particle.
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for a particle.
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A vx-t graph and a motion diagram
•  Figure 2.13 shows the vx-t graph and the motion diagram
for a particle.
Copyright © 2012 Pearson Education Inc.