PHYSICS 231
INTRODUCTORY PHYSICS I
Lecture 2
PHYSICS 231
INTRODUCTORY PHYSICS I
• Lecturer: Carl Schmidt (Sec. 001)
•schmidt@pa.msu.edu
•(517) 355-9200, ext. 2128
•Office Hours:
Friday 1-2:30 pm in 1248 BPS
or by appointment
Main points of last lecture
•
SI units:
•
•
•
Mass: kilograms (kg)
Length: meters (m)
Time: seconds (s)
•
•
•
Unit conversion
Dimensional analysis / Unit consistency
Scientific Notation and Significant figures
•
Displacement: Dx = xf-xi
•
Average Velocity:
Dx
v
Dt
Graphical Representation of Average Velocity
Between A and D , v is slope of blue line
40m
v
13.3m/s
3.0s
Instantaneous velocity
basic formula
Dx x f  xi
v

Dt
t
Let time interval approach zero
•Defined for every instance in time
•Equals average velocity if v = constant
•SPEED is absolute value of velocity
Graphical Representation of Average Velocity
Between A and D , v is slope of blue line
Graphical Representation of Instantaneous
Velocity
Dx
Dt
Dx
v  lim
Dt 0 Dt
= Slope of tangent at that point
Graphical Representation of Instantaneous
Velocity
v(t=3.0) is slope of tangent (green line)
Example 2.2a
The instantaneous velocity
is zero at ___
A) a
B) b & d
C) c & e
Example 2.2b
The instantaneous velocity is
negative at _____
A)
B)
C)
D)
E)
a
b
c
d
e
Example 2.2c
The average velocity is zero in
the interval _____
A)
B)
C)
D)
E)
a-c
b-d
c-d
c-e
d-e
Example 2.2d
The average velocity is
negative in the interval(s)
_________
A)
B)
C)
D)
a-b
a-c
c-e
d-e
SPEED
• Speed is |v| and is always positive
• Average speed is sum over |Dx| elements
divided by elapsed time
Example 2.3
x (m)
a) What is the average
velocity between B and E?
8
b) What is the average
speed between B and E?
4
6
E
B
2
A
0
a) 0.2 m/s
b) 1.2 m/s
D
C
0
2
4
6
8
10
12
t (s)
Acceleration
The rate of change of the velocity
a
v f  vi
t
Average acceleration:
measured over finite time interval
Instantaneous acceleration:
measured over infinitesimal interval, Dt -> 0
Accelerometer Demo
Example 2.4
A speed boat starts from rest and reaches 3.2 m/s
in 2 s.
1. What is its average acceleration?
a = 16 m/s2
2. Assuming acceleration is constant, what is its
velocity after 5 s?
vf = v0+at = 8 m/s
Graphical
Description of
Acceleration
Acceleration is slope of
tangent line in v vs. t
graph
Graphical
Description of
Acceleration
a is positive/negative
when v vs. t is
rising/falling
or when x vs t curves
upwards/downwards
a < 0
a > 0
Example 2.5a
e
b
c
a
d
At which segment(s) is the
acceleration negative?
A)
B)
C)
D)
a-c
c-d
c-e
d-e
Example 2.5b
e
b
c
a
At which point(s) does the
acceleration equal zero?
d
A)
B)
C)
D)
E)
none of the below
b
c
d
e
Constant Acceleration
• Recall v f  v 0  at
v
• v vs. t is a straight line
1

v


2 (v 0  v f )

Dx  v t
• Also
(t,v f )

v  12 (v0  v f )

(0,v0 )

 Dx  12 (v0  v f )t
• Combining:
Dx 
1
2
v 0  (v
0  at)t
 Dx  v0t  12 at 2

• x vs. t is a parabola

t
Example 2.6
A speed boat starts from rest and accelerates at a
rate of 1.6 m/s2. How far does it go after 5 s?
Dx = 20 m
Solving Problems with Eq.s of Motion
5 variables: Dx, t, v0, vf, a
v f  v 0  at
3 equations (so far):
Dx  12 (v 0  v f )t
Dx  v 0 t  12 at 2
Must be 2 more equations

Other Forms of Eq.s of Motion
v f  v0  at
1
Dx  (v0  v f )t
2
Substitute to eliminate v0
Dx 
(v f  at)  v f
2
1 2
Dx  v f t  at
2
t
Other Forms of Eq.s of Motion
v f  v0  at
1
Dx  (v0  v f )t
2
Substitute to eliminate t
Dx 
(v0  v f ) (v f  v0 )
2
v 2f
v02
aDx 

2
2
a
Final List of 1-d Equations
basic equations:
1) v  v0  at
1
2) Dx  (v0  v)t
2
1 2
3) Dx  v0 t  at
2
1 2
4) Dx  v f t  at
2
v 2f v 20
5) aDx 

2
2
Which equation to use?
Each has 4 of the 5 variables:
Dx, t, v0, v & a
Ask yourself
“Which variable am I not given
and not interested in?”
If that variable is t, use Eq. (5).
Example 2.7
Crash Houlihan speeds down the interstate at 44
m/s (100 mph), when she slams on the brakes and
slides into a concrete barrier. The police measure
skid marks to be 60 m long, and estimate that her
Mercedes would decelerate at 11 m/s2 while
skidding. What was Crash’s speed when she hit the
barrier?
25 m/s
Free Fall
• Special case of constant acceleration:
• Objects near Earth’s surface falling under
the influence of gravity (neglecting air
resistance)
• Acceleration g = 9.81 m/s2
• Use the usual equations with
(with convention up is +)

a  (g)
Galileo
•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
Example 2.8a
A man drops a brick off the top
of a 50-m building. The brick
has zero initial velocity.
A
B
a) How much time is required
for the brick to hit the ground?
a) 3.19 s
b) What is the velocity of the
brick when it hits the ground?
b) -31.3 m/s
c
Example 2.8b
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
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) What is the velocity of the brick when
it passes the man going downwards?
d) What is the velocity of the brick when
it hits the ground?
e) At what time does the brick hit the
ground?
c
Example 2.8b
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) How high above the building does the
brick get before it falls?
b) How much time does the brick spend
going upwards?
c) What is the velocity of the brick when
it passes the man going downwards?
d) What is the velocity of the brick when
it hits the ground?
e) At what time does the brick hit the
ground?
a)
b)
c)
d)
e)
20.4 m
2.04 s
-20 m/s
-37.2 m/s
5.83 s
Example 2.9a
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
C
C
At ‘A’ the acceleration is positive
a) True
b) False
D
D
E
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
C
C
At ‘B’ the velocity is zero
a) True
b) False
D
D
E
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)
B
AA
C
C
At ‘B’ the acceleration is zero
a) True
b) False
D
D
E
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
C
C
At ‘C’ the velocity is negative
a) True
b) False
D
D
E
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)
B
AA
C
C
At ‘C’ the acceleration is negative
a) True
b) False
D
D
E
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)
B
AA
C
C
The speed at ‘C’ and at ‘A’ are equal
a) True
b) False
D
D
E
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)
B
AA
C
C
The velocity at ‘C’ and at ‘A’ are equal
a) True
b) False
D
D
E
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)
B
AA
C
C
h) The speed is greatest at ‘E’
a) True
b) False
D
D
E