Chapter 2 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
x  x2  x1
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
x x2  x1
v

t
t 2  t1
• Instantaneous velocity – velocity at a given instant
x dx
v  lim

t 0 t
dt
• Instantaneous speed – a magnitude of an
instantaneous velocity
Velocity and speed
Velocity and speed
Instantaneous velocity
x
v  lim
t 0 t
• 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
Acceleration
• Average acceleration - a ratio of change of velocity
(over a time interval) to that time interval (SI unit =
(m/s)/s = m/s2)
v v2  v1
a

t
t 2  t1
• Instantaneous acceleration – a rate of change of
velocity at a given instant
2
d  dx  d x
v dv

 
a  lim

2
t  0  t
dt  dt  dt
dt
Acceleration
• The blue line is the average acceleration
• The slope (green line) of the velocity-time graph is
the acceleration
v
a  lim
t  0  t
Chapter 2
Problem 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
ti  0 t f  t
• Then
v v f  vi v f  vi
aa 


t
t 0
t f  ti
 v f  vi  at
Case of constant acceleration
• Average and instantaneous accelerations are the
same
• Conventionally
• Then
ti  0 t f  t
x x f  xi x f  xi
 x f  xi  v t
v


t
t f  ti
t 0
vi  (vi  at )
v1  v2 vi  v f
at

 vi 
v

2
2
2
2
2
at
 x f  xi  vi t 
2
Case of constant acceleration
2
at
x f  xi  vi t 
2
dx
v
dt
v f  vi  at
dv
a
dt
Case of constant acceleration
Case of constant acceleration
Equations
Missing variables
v f  vi  at
x f  xi
at 2
x f  xi  vi t 
2
vf
v f  vi  2a( x f  xi )
2
2
x f  xi 
(vi  v f )t
2
at 2
x f  xi  v f t 
2
t
a
vi
Chapter 2
Problem 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
v f  vi  gt
gt
y f  yi  vi t 
2
2
Chapter 2
Problem 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?
Chapter 2
Problem 14:
(a) 24 km
(b) 9.6 km/h
(c) - 16 km/h
(d) 0 km
(e) 0 km/h
Chapter 2
Problem 24:
15 m/s2
Chapter 2
Problem 26:
31 s