Displacement vs. position
Chapter 2: One-Dimensional Motion
Position: x (relative to origin)
Displacement: !x = xf-xi
•Motion at fixed velocity
•Definition of average velocity
•Motion with fixed acceleration
•Graphical representations
1
2
Average velocity
Instantaneous velocity
basic formula
basic formula
!x x f " xi
v=
=
!t
t
v=
Average velocity
!x x f " xi
=
!t
t
Let time interval approach zero
•Can be positive or negative
•Depends only on initial/final positions
•e.g., if you return to original position,
average velocity is zero
•Defined for every instance in time
•Equals average velocity if v = constant
•SPEED is absolute value of velocity
3
4
Graphical Representation of
Instantaneous Velocity
Graphical Representation of Average Velocity
v(t=3.0) is slope of tangent (green line)
Between A and D , v is slope of blue line
5
6
Example 2.1
Example 2.2
Carol starts at a position x(t=0) = 1.5 m.
At t=2.0 s, Carol’s position is x(t=2 s)=4.5 m
At t=4.0 s, Carol’s position is x(t=4 s)=-2.5 m
a) What is Carol’s average velocity between t=0 and t=2 s?
b) What is Carol’s average velocity between t=2 and t=4 s?
c) What is Carol’s average velocity between t=0 and t=4 s?
On a mission to rid Spartan Stadium of
vermin, an archer shoots an arrow across
the stadium at an unlucky rat 200 meters
away. The archer hears the squeal 2.2
seconds later. What was the velocity of
the arrow? The speed of sound is 330 m/s.
a) 1.5 m/s
b) -3.5 m/s
c) -1.0 m/s
7
8
Example 2.2
Example 2.2:
Visualize the problem!
On a mission to rid Spartan Stadium of
vermin, an archer shoots an arrow across
the stadium at an unlucky rat 200 meters
away. The archer hears the squeal 2.2
seconds later. What was the velocity of
the arrow? The speed of sound is 330 m/s.
V = 125 m/s
9
Example 2.3a
The instantaneous velocity
is zero at ___
10
Example 2.3b
A) a
B) b & d
C) c & e
The instantaneous velocity is
negative at _____
11
A)
B)
C)
D)
E)
a
b
c
d
e
12
Example 2.3c
Example 2.3d
The average velocity is zero in
the interval _____
A)
B)
C)
D)
E)
a-c
b-d
c-d
c-e
d-e
A)
B)
C)
D)
The average velocity is
negative in the interval(s)
_________
a-b
a-c
c-e
d-e
13
14
Example 2.4
SPEED
x (m)
• Speed is |v| and is always positive
• Average speed is sum over |!x| elements
divided by elapsed time
a) What is the average
velocity between B and E?
8
b) What is the average
speed between B and E?
4
D
6
A
0
E
B
2
C
0
2
4
6
8
10
12
t (s)
a) 0.2 m/s
b) 1.2 m/s
15
16
Acceleration
The rate of change of the velocity
a=
Accelerometer Demo
v f ! vi
t
Average acceleration:
measured over finite time interval
Instantaneous acceleration:
measured over infinitesimal interval, !t -> 0
17
18
a < 0
Graphical
Description of
Acceleration
Graphical
Description of
Acceleration
a > 0
a is positive/negative
when v vs. t is
rising/falling
or when x vs t curves
upwards/downwards
Acceleration is slope of
tangent line in v vs. t
graph
19
Example
20
Example
e
e
b
b
c
a
c
d
a
At which point(s) does the position equal zero?
A)
B)
C)
D)
a
a
b
b
d
At which point(s) does the velocity equal zero?
only
and d
only
& d
A)
B)
C)
D)
E)
21
Example 2.5c
22
Example
e
e
b
b
c
a
a
b only
c only
b & d
a & d
c
d
At which point is the velocity negative?
a
A)
B)
C)
D)
E)
a
b
c
d
e
d
At which segment(s) is the
acceleration negative?
23
A)
B)
C)
D)
a-c
c-d
c-e
d-e
24
Constant Acceleration
Example 2.5e
• a vs. t is a constant
• v vs t is a straight line
• x vs t is a parabola
e
b
c
Eq.s of Motion
d
a
At which point(s) does the
acceleration equal zero?
v f = v0 + at
A)
B)
C)
D)
E)
!x =
none of the below
b
c
d
e
1
(v0 + v f )t
2
25
26
Example 2.6
Solving Problems with Eq.s of Motion
5 variables: !x, t, v0, vf, a
2 equations:
v f = v0 + at
!x =
Crash Houlihan speeds down the intersate, when she
slams on the brakes and slides into a concrete
barrier. The police measure skid marks to be 60 m
long, and from a tape recording, know that she was
breaking from 3.5 seconds. Furthermore, they know
that her Mercedes would decelerate at 5.5 m/s2
while skidding. What was Crash’s speed when she hit
the barrier?
7.52 m/s
1
(v0 + v f )t
2
3 variables must be given so that
2 equations can solve for 2 unknowns
27
Other Forms of Eq.s of Motion
28
Other Forms of Eq.s of Motion
v f = v0 + at
v f = v0 + at
1
!x = (v0 + v f )t
2
!x =
Substitute to eliminate vf
1
(v0 + v f )t
2
Substitute to eliminate v0
v0 + (v0 + at)
t
2
1
!x = v0 t + at 2
2
!x =
!x =
(v f " at) + v f
2
1
!x = v f t " at 2
2
29
t
30
Final List of 1-d Equations
basic equations:
Other Forms of Eq.s of Motion
1) v = v0 + at
1
2) !x = (v0 + v)t
2
1
3) !x = v0 t + at 2
2
1
4) !x = v f t " at 2
2
2
v f v 20
5) a!x =
"
2
2
v f = v0 + at
!x =
1
(v0 + v f )t
2
Substitute to eliminate t
!x =
(v0 + v f ) (v f " v0 )
2
a!x =
a
v 2f
2
"
v02
2
Which one should I use?
Each Eq. has 4 of the 5 variables:
!x, t, v0, v & a
Ask yourself
“Which variable am I not given
and not interested in?”
If that variable is t, use Eq. (5).
31
Example 2.6
(Revisited)
32
A drag racer starts her
car from rest and
accelerates at 10.0 m/s2
for the entire distance of
a 400 m (1/4 mi) race.
How much time was
required to finish the race?
Crash Houlihan speeds down the intersate, when she
slams on the brakes and slides into a concrete
barrier. The police measure skid marks to be 60 m
long, and from a tape recording, know that she was
breaking from 3.5 seconds. Furthermore, they know
that her Mercedes would decelerate at 5.5 m/s2
while skidding. What was Crash’s speed when she hit
the barrier?
7.52 m/s
Example 2.7a
a) v = v0 + at
1
b) !x = (v0 + v)t
2
1
c) !x = v0 t + at 2
2
1
d) !x = v f t " at 2
2
2
v f v 20
e) a!x =
"
2
2
33
A drag racer starts her
car from rest and
accelerates at 10.0 m/s2
for the entire distance of
a 400 m (1/4 mi) race.
What was her final speed?
Example 2.7b
a) v = v0 + at
1
b) !x = (v0 + v)t
2
1
c) !x = v0 t + at 2
2
1
d) !x = v f t " at 2
2
2
v f v 20
e) a!x =
"
2
2
35
34
A drag racer starts her
car from rest and finishes
a race in 3.5 seconds with
a constant acceleration for
the entire distance of a
400 m (1/4 mi) race.
What was her final speed?
Example 2.7c
a) v = v0 + at
1
b) !x = (v0 + v)t
2
1
c) !x = v0 t + at 2
2
1
d) !x = v f t " at 2
2
2
v f v 20
e) a!x =
"
2
2
36
Free Fall
Galileo
• Objects under the influence of gravity (no
resistance) fall with constant downward
acceleration (if near Earth’s surface).
g = 9.81 m/s2
• Use the usual equations with a --> -g
37
Example 2.8a
A man drops a brick off the top
of a 50-m building. The brick
has zero initial velocity.
A
A man throws a brick upward from the top
of a 50 m building. The brick has an initial
upward velocity of 20 m/s.
B
A
B
a) How high above the building does the
brick get before it falls?
b) How much time does the brick spend
going upwards?
c
c
c) What is the velocity of the brick when
it passes the man going downwards?
b) What is the velocity of the
brick when it hits the ground?
a) 3.19 s
b) -31.3 m/s
d) What is the velocity of the brick when
it hits the ground?
39
Example 2.8b
e) At what time does the brick hit the
ground?
Example 2.9a
A man throws a brick upward from the top
of a 50 m building. The brick has an initial
upward velocity of 20 m/s.
A man throws a brick upward from the
top of a building. (Assume the coordinate
system is defined with positve defined as
upward)
a) How high above the building does the
brick get before it falls?
c) What is the velocity of the brick when
it passes the man going downwards?
38
Example 2.8b
a) How much time is required
for the brick to hit the ground?
b) How much time does the brick spend
going upwards?
•Father was a musician, experimented with music
•Initially was a professor teaching pre-meds
•Developed telescope ~ 1610:
Milky Way = stars
Moons of Jupiter
Phases of Venus…
•Measured g
•Quantified mechanics
•In 1632, published Dialogue
concerning the two greatest
world systems
•Was found guilty of heresy
a)
b)
c)
d)
e)
At ‘A’ the acceleration is positive
a) True
b) False
20.4 m
2.04 s
-20 m/s
-37.2 m/s
5.83 s
40
B
AA
C
C
D
D
d) What is the velocity of the brick when
it hits the ground?
e) At what time does the brick hit the
ground?
E
41
42
Example 2.9b
A man throws a brick upward from the
top of a building. (Assume the coordinate
system is defined with positve defined as
upward)
B
AA
At ‘B’ the velocity is zero
a) True
b) False
Example 2.9c
A man throws a brick upward from the
top of a building. (Assume the coordinate
system is defined with positve defined as
upward)
C
C
B
AA
At ‘B’ the acceleration is zero
a) True
b) False
D
D
C
C
D
D
E
E
43
Example 2.9d
A man throws a brick upward from the
top of a building. (Assume the coordinate
system is defined with positve defined as
upward)
B
AA
At ‘C’ the velocity is negative
a) True
b) False
44
Example 2.9e
A man throws a brick upward from the
top of a building. (Assume the coordinate
system is defined with positve defined as
upward)
C
C
B
AA
At ‘C’ the acceleration is negative
a) True
b) False
D
D
C
C
D
D
E
E
45
Example 2.9f
A man throws a brick upward from the
top of a building. (Assume the coordinate
system is defined with positve defined as
upward)
The speed at ‘C’ and at ‘A’ are equal
a) True
b) False
B
AA
46
Example 2.9g
A man throws a brick upward from the
top of a building. TRUE OR FALSE.
(Assume the coordinate system is
defined with positve defined as upward)
C
C
The velocity at ‘C’ and at ‘A’ are equal
a) True
b) False
D
D
E
B
AA
C
C
D
D
E
47
48
Example 2.9h
A man throws a brick upward from the
top of a building. (Assume the coordinate
system is defined with positve defined as
upward)
The speed is greatest at ‘E’
a) True
b) False
B
AA
C
C
D
D
E
49