PHYS 1110
Lecture 2
Professor Stephen Thornton
August 30, 2012
An Issue
Other classes are also using iClickers
nearby. Therefore, we will need to change
frequencies.
iClicker 1 (old one): Press and hold the On/Off
power button on the remote until the blue Power
light begins flashing. Then press BB. A green
Vote Status light on your remote will indicate that
you have successfully reset the remote frequency.
iClicker 2: Press and hold the On/Off power
button on the remote until the BB on the LCD
begins flashing.
Reading Quiz:
What is the total displacement from
start to finish?
A) - 2 m
B)
C)
D)
E)
finish
start
+2 m
+3 m
+7 m
+10 m
Reading Quiz:
What is the total displacement from
start to finish?
A) - 2 m
B)
C)
D)
E)
finish
start
+2 m
+3 m
+7 m
+10 m
How can we change things?
• Be more energy efficient. Could reduce
electricity need by 15% by 2020, 30% by 2030.
• Develop more renewable energy.
• Energy policy like tax credits, policy changes.
• Carbon capture and storage in order to use fossil
fuels.
• Revolutionary nuclear reactors that are simpler
and safer. They probably are already imagined.
Hydraulic fracturing video:
http://www.oerb.com/Default.a
spx?tabid=242 - man talking
http://www.youtube.com/watch
?v=kv3cQngRPmw – watered
down, woman talking
Solar and wind energy did not even show up in 2008, < 1% in US.
By mid-2012 wind energy had grown to 50 GW (4.5%).
Solar is still way behind (~0.2%), but growing by 30% a year. By
some estimates it is as much as 30 GW or 3%.
Wind and solar energy represent the greatest potential increase
of renewable energy.
There are 104 nuclear reactor power plants operating in the
United States, 4 in Virginia, which generates 38% of its power.
Growth of Fuel Inputs to World Power Generation
Estimates of Levelized Cost of Electricity for New Baseload and Intermittent
Sources for 2020. Dashed is actual 2007 price; shaded is range in 2007.
And then there is the transportation energy
problem.
The United States is committed to ethanol. It
has been growing for 25 years and is now a
political issue. US law requires us to use 10%
ethanol in our gas – 15% in some places.
Ethanol use is required to increase every year.
Biofuel generation has not worked, but there
is progress.
The electrical distribution system is a huge
problem. At least 10% of our electricity is
lost. It is a patchwork and archaic system.
Quiz:
Which of the following energies had
the greatest increase in the last few
years in the US?
A)
B)
C)
D)
E)
Concentrated solar power
Wind
Biomass
Geothermal
Hydropower
Quiz:
Which of the following had the
greatest increase in the last few years
in the US?
A)
B)
C)
D)
E)
Concentrated solar power
Wind
Biomass
Geothermal
Hydropower
One-Dimensional Coordinate System
Distance = total length traveled
Example: You can run 2 m/s.
How far can you run in 4 s?
Answer: 2 m/s x 4 s = 8 m
Displacement
Definition:
displacement = change in position
= final position – initial position
Δx = xf - xi
One-Dimensional Motion Along the
x-axis – do quiz
finish
start
What is the total distance traveled (0-4s)?
What is the displacement? – Reading Quiz
What is the total distance traveled from
start to finish?
A)
B)
C)
D)
E(
-4m
+4 m
+3 m
+8 m
+10 m
finish
start
What is the total distance traveled from
start to finish?
A)
B)
C)
D)
E(
-4m
+4 m
+3 m
+8 m
+10 m
finish
start
Average Speed
distance
average speed =
elapsed time
Note that this is always a positive
number.
Average Velocity
Velocity is different than speed,
because velocity is a vector.
displacement
average velocity =
elapsed time
x x f  xi
vav 

t t f  ti
One-Dimensional Motion Along the x Axis
finish
start
What is average velocity (0 – 4 s)?
(1m) - (1m) 2m
vav 

 0.5 m/s
4s-0s
4s
Motion Along the x axis represented
with an x-versus-t graph
Average Velocity on an x-Versus-t Graph
trajectory
Instantaneous Velocity
x dx
v  lim

t 0 t
dt
In this way we can find the
velocity at any particular
instant of time.
What is the instantaneous velocity
at t = 1 s?
Graphical Interpretation of Average
and Instantaneous Velocity
Acceleration
Just like velocity is given by the rate
of change of position with respect to
time, the acceleration is given by the
rate of change of velocity with respect
to time.
We are still dealing with onedimensional motion, so vector
direction is simple.
Average Acceleration
v v f  vi
aav 

t t f  ti
We must be very careful with
units. What are they?
m/s2
Instantaneous Acceleration
x dx
v  lim

t 0 t
dt
v dv
a  lim

t 0 t
dt
Similarity between velocity and
acceleration is clear.
Graphical Interpretation of Average
and Instantaneous Acceleration
If you don’t
remember
about tangents,
please review!
We use signs to denote the directions of both velocity and
acceleration along a particular axis.
x
 When v is +, motion is to right.
 When v is -, motion is to left.
 When motion is to the right, and a is +, then object
speeds up (accelerates) to the right.
 When motion is to the left,
and a is +, then object is
slowing down and will
eventually turn to the right.
(not shown here)
Cars Accelerating or Decelerating
speeding up
slowing down
slowing down
speeding up
Much easier to see what is happening,
when we draw a picture.
Conceptual Quiz: The graph shows position as a
function of time for two trains running on parallel
tracks. Which of the following is true?
A) At time tB, both trains have the same velocity.
B) Both trains speed up all the time.
C) Both trains have the
same velocity at some
time before tB.
D) Somewhere on the
graph, both trains
have the same
acceleration.
time
Conceptual Quiz: The graph shows position as a
function of time for two trains running on parallel
tracks. Which of the following is true?
A) At time tB, both trains have the same velocity.
B) Both trains speed up all the time.
C) Both trains have the
same velocity at some
time before tB.
D) Somewhere on the
graph, both trains
have the same
acceleration.
time
Motion with Constant Acceleration
If the acceleration is constant, then we have
v  v0  at
where v = v0 at t = 0. This result is easy to show
from our definition of a.
The Average Velocity
v
f
vi
constant acceleration
The Average Velocity
nonconstant
acceleration
Constant acceleration
Another important result:
1
1
vav   v0  v   (vi  v f )
2
2
Note that I have used vi and vf , which is
more general than using v0 and v, because
we may want to find the average between
some initial and final position other than v0
and a general v.
Let’s determine some important equations.
x x f  xi x  x0
vav 


t t f  ti
t 0
x  x0
vav 
t
or vavt  x  x0
solve for x : x  x0  vavt ****
Insert our previous result for vav
1
x  x0  (v0  v)t
2
Only for constant acceleration!!!!!
1
x  x0  (v0  v)t
2
This is a very important equation.
It relates the position x to the
velocity v as a function of time t.
But we also know the relationship
between velocity v and
acceleration a. It was
v  v0  aconstant t
1
x  x0  (v0  v)t
2
1
x  x0  v0  (v0  aconstant t ) t
2
1
2
x  x0  v0t  aconstant t
2
v  v0  at
solve for t , a is constant
v  v0
t
aconstant
remember a  aconstant
1
We had x  x0  (v0  v)t
2
Substitute in for t from above
 v  v0 
1
x  x0  (v  v0 ) 

2
 aconstant 
v v
x  x0 
2aconstant
2
2
0
Four important equations
v = v0 + at
1
vav = (v0 + v )
2
1 2
x = x0 + v0t + at
2
2
2
v = v0 + 2a ( x - x0 )
Four important equations
with initial time t0
v = v0 + a (t - t0 )
1
vav = (v0 + v)
2
1
2
x = x0 + v0 (t - t0 ) + a(t - t0 )
2
2
2
v = v0 + 2a( x - x0 )
Freely falling objects
Most important example of constant
acceleration; a = ± g = 9.81 m/s2
Do demo: Paper and racquetball
Nickel and feather
Galileo – father of physics
We let x be downward. Look at our
previous equation:
1
2
x  x0  v0t  aconstant t
2
Let’s release an object at x0 = 0 at t = 0.
We then also have v0 = 0. a = g
x
1 2
x  gt
2
Note that g is always positive. Here
x is down.
Our previous equations for v become
v  v0  at  0  gt  gt
v v
v
x  x0 
0
2a
2g
which can be rewritten as
2
v  2 gx or
2
2
0
2
v  2 gx
These are useful equations. Drop
from rest.
Now use acceleration of gravity,
with a = - g . Note y is up.
v = v0 - g (t - t0 )
1
vav = (v0 + v)
2
y
1
2
y = y0 + v0 (t - t0 ) - g (t - t0 )
2
2
2
v = v0 - 2 g ( y - y0 )
x = 4.9 m
Free fall from rest
x = 14.7 m9
x =x=4.91
24.5 m m
x = 34.4 m
v  gt
1 2
x  gt
2
So what would happen if we dropped a
rope that had masses at equal intervals?
Do demo. Free fall
What would happen if we dropped a
rope that had masses spaced out as t2?
Do demo. Free fall
What happens if we throw a ball up?
We throw a ball up at x = 0 with speed v0.
What is its speed when it returns? v0
How long does it take to return? 2v0/g
How can we determine these numbers?
The equations we have determined must
tell us these answers!
x=0
Conceptual Quiz: Throw ball up.
Initial speed = v0. Round trip time
is 2v0/g. What is minimum speed?
A)
B)
C)
D)
E)
0
-v0
v0
-2v0
2v0
Conceptual Quiz: Throw ball up.
Initial speed = v0. Round trip time
is 2v0/g. What is minimum speed?
A) 0
B) -v0
C) v0
D) -2v0
E) 2v0
Conceptual Quiz: Throw ball up.
Initial speed = v0. Round trip time
is 2v0/g. What is time when
minimum speed is reached?
A)
B)
C)
D)
0
v0/g
2v0/g
Can not be determined
Conceptual Quiz: Throw ball up.
Initial speed = v0. Round trip time
is 2v0/g. What is time when
minimum speed is reached?
A)
B)
C)
D)
0
v0/g
2v0/g
Can not be determined
Review of Vectors
A scalar is a number with units. It
can be positive, negative, or zero.
A vector has both a magnitude and
direction.
We will put an arrow over a quantity
that is a vector. Sometimes a vector
is in boldface.
Directions to the library
3 blocks west, 3
blocks north.
Start
Ax  A cos
Ay  A sin 
A A  A
2
x
2
y
 Ay 
  tan  
 Ax 
1
The Sum of Two Vectors
We can add
vectors.
C=A+B
Component Addition of Vectors
Unit Vectors
More common to use
ˆi and ˆj or just i, j or i, j
ˆ y.
ˆ
than x,
Multiplying a Vector by a Scalar
We can
multiply a
vector by a
scalar.
Vector Component Use
A  3iˆ + 4jˆ
B  2iˆ -2jˆ
ˆ ˆ  (4jˆ  2j)
ˆ = 5iˆ  2ˆj
A  B  (3i+2i)
ˆ ˆ  (2iˆ  2j)
ˆ  ˆi  6ˆj
A - B  (3i+4j)
Unit vectors make vector
addition and subtraction
reasonably easy.
Good review of vector use:
http://www.physics.uoguelph.ca/tutorials/
vectors/vectors.html