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Chapter 2
Motion in a Straight Line
Linear motion
In this chapter we will consider moving objects:
• Along a straight line
• With every portion of an object moving in the same direction and at the same rate (particle-like motion)
Distance, position, and displacement
• Distance a total length of the path traveled regardless of direction (SI unit: m)
• In each instance we choose an origin – a reference point, convenient for further calculations
• Position of an object is described by the shortest distance from the origin and direction relative to the origin
• Displacement – a change from position x1 to position x2
12 xxx
Velocity and speed
• Average speed - a ratio of distance traveled (over a time interval) to that time interval (SI unit: m/s)
• Average velocity - a ratio of displacement (over a time interval) to that time interval
• Instantaneous velocity – velocity at a given instant
• Instantaneous speed – a magnitude of an instantaneous velocity
t
xv
t
xv
t
0
lim
12
12
tt
xx
dt
dx
Velocity and speed
Velocity and speed
Instantaneous velocity
• The instantaneous velocity is the slope of the line tangent to the x vs. t curve
• This would be the green line
• The light blue lines show that as Δt gets smaller, they approach the green line
t
xv
t
0
lim
Acceleration
• Average acceleration - a ratio of change of velocity (over a time interval) to that time interval (SI unit = (m/s)/s = m/s2)
• Instantaneous acceleration – a rate of change of velocity at a given instant
2
2
dt
xd
t
va
12
12
tt
vv
t
va
t
0
limdt
dv
dt
dx
dt
d
Acceleration
• The blue line is the average acceleration
• The slope (green line) of the velocity-time graph is the acceleration
t
va
t
0
lim
Chapter 2Problem 43
You allow 40 min to drive 25 mi to the airport, but you’re caught in heavy traffic and average only 20 mi/h for the first 15 min. What must your average speed be on the rest of the trip if you’re to make your flight?
Case of constant acceleration
• Average and instantaneous accelerations are the same
• Conventionally
• Then
0it
t
vaa
if
if
tt
vv
0
t
vv if
atvv if
tt f
Case of constant acceleration
• Average and instantaneous accelerations are the same
• Conventionally
• Then
0it tt f
t
xv
0
t
xx if tvxx if
2221 fi vvvv
v
2
atvi
if
if
tt
xx
2
)( atvv ii
2
2attvxx iif
Case of constant acceleration
dt
dxv
dt
dva
atvv if
2
2attvxx iif
Case of constant acceleration
Case of constant acceleration
To help you solve problems
Equations Missing variables
2
2attvxx iif
atvv if
)(222ifif xxavv
2
)( tvvxx fiif
2
2attvxx fif
fv
if xx
t
a
iv
Chapter 2Problem 36
You’re driving at speed v0 when you spot a stationary moose on the road, a distance d ahead. Find an expression for the magnitude of the acceleration you need if you’re to stop before hitting the moose.
Case of free-fall acceleration
• At sea level of Earth’s mid-latitudes all objects fall (in vacuum) with constant (downward) acceleration of
a = - g ≈ - 9.8 m/s2 ≈ - 32 ft/s2
• Conventionally, free fall is along a vertical (upward) y-axis
gtvv if
2
2gttvyy iif
Chapter 2Problem 37
You drop a rock into a deep well and 4.4 s later hear a splash. How far down is the water? Neglect the travel time of sound.
Graphical representation
Graphical representation
Graphical representation
Graphical representation
Graphical representation
Graphical representation
Graphical representation
Graphical representation
Graphical representation
Questions?
Answers to the even-numbered problems
Chapter 2
Problem 14:
(a) 24 km (b) 9.6 km/h(c) - 16 km/h(d) 0 km(e) 0 km/h
Answers to the even-numbered problems
Chapter 2
Problem 24:
15 m/s2
Answers to the even-numbered problems
Chapter 2
Problem 26:
31 s
Answers to the even-numbered problems
Chapter 2
Problem 28:
100 m
Answers to the even-numbered problems
Chapter 2
Problem 38:
11 m/s
Answers to the even-numbered problems
Chapter 2
Problem 42:
3.8 m/s2
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