PHYS 1110

Lecture 2

Professor Stephen Thornton

August 30, 2012

An IssueOther 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) +2 mC) +3 mD) +7 mE) +10 m

finish

start

Reading Quiz:What is the total displacement from start to finish? A) - 2 m

B) +2 mC) +3 mD) +7 mE) +10 m

finish

start

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.aspx?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) Concentrated solar powerB) WindC) BiomassD) GeothermalE) Hydropower

Quiz:Which of the following had the greatest increase in the last few years in the US?

A) Concentrated solar powerB) WindC) BiomassD) GeothermalE) 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

What is the total distance traveled (0-4s)?

What is the displacement? – Reading Quiz

start

finish

What is the total distance traveled from start to finish?

A) - 4 mB) +4 mC) +3 mD) +8 mE( +10 m

start

finish

What is the total distance traveled from start to finish?

A) - 4 mB) +4 mC) +3 mD) +8 mE( +10 m

start

finish

Average Speed

distanceaverage speed =

elapsed time

Note that this is always a positive number.

Average Velocity

Velocity is different than speed, because velocity is a vector.

displacementaverage velocity =

elapsed time

f iav

f i

x xxv

t t t

One-Dimensional Motion Along the x Axis

What is average velocity (0 – 4 s)?

( 1m) - (1m) 2m0.5 m/s

4s-0s 4savv

start

finish

Motion Along the x axis represented with an x-versus-t graph

Average Velocity on an x-Versus-t Graph

trajectory

Instantaneous Velocity

0

dlim

dt

x xv

t t

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

AccelerationJust 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 one-dimensional motion, so vector direction is simple.

Average Acceleration

f iav

f i

v vva

t t t

We must be very careful with units. What are they?

m/s2

Instantaneous Acceleration

0

dlim

dt

v va

t t

0

dlim

dt

x xv

t t

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

speeding upslowing down

slowing down

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 AccelerationIf the acceleration is constant, then we have

where v = v0 at t = 0. This result is easy to show from our definition of a.

0v v at

The Average Velocity

constant accelerationiv

fv

The Average Velocity

nonconstant acceleration

Constant accelerationAnother important result:

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.

av 0

1 1( )

2 2 i fv v v v v

0av

0av

av 0

0 av

0

or

solve for : ****

f i

f i

x xx x xv

t t t t

x xv

tv t x x

x x x v t

Insert our previous result for vav

0 0

1( )

2x x v v t

Only for constant acceleration!!!!!

0 0

1( )

2x x v v t

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

0 constantv v a t

0 0

0 0 0 constant

20 0 constant

1( )

21

( )2

1

2

x x v v t

x x v v a t t

x x v t a t

0

0constant

constant

0 0

00 0

constant

2 20

0constant

solve for , is constant

remember

1We had ( )

2Substitute in for from above

1( )

2

2

v v at t a

v vt a a

a

x x v v t

t

v vx x v v

a

v vx x

a

0

av 0

20 0

2 20 0

1( )

21

2

2 ( )

v v at

v v v

x x v t at

v v a x x

= +

= +

= + +

= + -

Four important equations

0 0

av 0

20 0 0 0

2 20 0

( )

1( )

21

( ) ( )2

2 ( )

v v a t t

v v v

x x v t t a t t

v v a x x

= + -

= +

= + - + -

= + -

Four important equations with initial time t0

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:

x

20 0 constant

1

2x x v t a t

Let’s release an object at x0 = 0 at t = 0. We then also have v0 = 0. a = g

21

2x gt

Note that g is always positive. Here x is down.

Our previous equations for v become

0 0v v at gt gt 2 2 2

00

2

02 2

which can be rewritten as

2 or 2

v v vx x

a g

v gx v gx

These are useful equations. Drop from rest.

0 0

av 0

20 0 0 0

2 20 0

( )

1( )

21

( ) ( )2

2 ( )

v v g t t

v v v

y y v t t g t t

v v g y y

= - -

= +

= + - - -

= - -

Now use acceleration of gravity, with . Note y is up.

y

a g=-

Free fall from restx = 14.7 m9

x=4.91 mx = 24.5 m

x = 34.4 m 21

2

v gt

x gt

x = 4.9 m

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) 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 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) 0

B) v0/g

C) 2v0/g

D) 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) 0

B) v0/g

C) 2v0/g

D) 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

2 2 1

cos sin

tan

x y

yx y

x

A A A A

AA A A

A

The Sum of Two Vectors

We can add vectors.

C = A + B

Component Addition of Vectors

Unit Vectors

More common to use ˆ ˆi and j or just i, j or ,

ˆ ˆthan x, y.i j

Multiplying a Vector by a Scalar

We can multiply a vector by a scalar.

Vector Component Use

Unit vectors make vector addition and subtraction reasonably easy.

ˆ ˆ ˆ ˆA 3i + 4j B 2i -2j

ˆ ˆ ˆ ˆ ˆ ˆA B (3i+2i) (4j 2j) = 5i 2 j

ˆ ˆ ˆ ˆ ˆ ˆA - B (3i+4j) (2i 2j) i 6 j

Good review of vector use:

http://www.physics.uoguelph.ca/tutorials/vectors/vectors.html