© 2010 pearson education, inc. lecture chapter 2 college physics, 7 th edition wilson / buffa / lou

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Lecture

Chapter 2

College Physics, 7th Edition

Wilson / Buffa / Lou

Chapter 2Kinematics: Description of

Motion

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Units of Chapter 2

Distance and Speed: Scalar Quantities

One-Dimensional Displacement and Velocity: Vector Quantities

Acceleration

Kinematic Equations (Constant Acceleration)

Free Fall

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Defining the important variablesKinematics is a way of describing the motion of objects

without describing the causes. You can describe an object’s motion:

In wordsMathematically Pictorially Graphically

Symbol Variable Units

t Time s

a Acceleration m/s/s

x or y Displacement m

vo Initial velocity m/s

v Final velocity m/s

g or ag Acceleration due to gravity

m/s/s

No matter HOW we describe the motion, there are several KEY VARIABLES that we use.

2.1 Distance and Speed: Scalar Quantities

Distance is the path length traveled from one location to another. It will vary depending on the path.

Distance is a scalar quantity—it is described only by a magnitude.

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2.1 Distance and Speed: Scalar Quantities

Average speed is the distance traveled divided by the elapsed time:

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2.1 Distance and Speed: Scalar Quantities

Since distance is a scalar, speed is also a scalar (as is time).

Instantaneous speed is the speed measured over a very short time span. This is what a speedometer reads.

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Scalar Example

Magnitude

Speed 20 m/s

Distance 10 m

Age 15 years

Heat 1000 calories

2.2 One-Dimensional Displacement and Velocity: Vector Quantities

A vector has both magnitude and direction. Manipulating vectors means defining a coordinate system, as shown in the diagrams to the left.

Right-hand Rule… first of many

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2.2 One-Dimensional Displacement and Velocity: Vector Quantities

Displacement is a vector that points from the initial position to the final position of an object. Different from distance.

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2.2 One-Dimensional Displacement and Velocity: Vector Quantities

Note that an object’s position coordinate may be negative, while its velocity may be positive; the two are independent.

This can be VERY tricky for some students.

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2.2 One-Dimensional Displacement and Velocity: Vector Quantities

For motion in a straight line with no reversals, the average speed and the average velocity are the same.

Otherwise, they are not.

The average velocity of a round trip is zero, as the total displacement is zero!

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2.2 One-Dimensional Displacement and Velocity: Vector Quantities

Different ways of visualizing uniform velocity:

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2.2 One-Dimensional Displacement and Velocity: Vector Quantities

This object’s velocity is not uniform. Does it ever change direction, or is it just slowing down and speeding up?

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2.2 One-Dimensional Displacement and Velocity: Vector Quantities

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Vector Magnitude & Direction

Velocity 20 m/s, N

Acceleration 10 m/s/s, E

Force 5 N, West

2.3 Acceleration

Acceleration is the rate at which velocity changes. Since velocity is a vector, so too is acceleration. Notice the bar over the a.

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2.3 AccelerationAcceleration means that the speed of an object is changing, or its direction is, or both. Ex B can be tricky.

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2.3 Acceleration

Acceleration may result in an object either speeding up or slowing down (or simply changing its direction).

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2.3 AccelerationIf the acceleration is constant, we can find the velocity as a function of time:

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2.4 Kinematic Equations (Constant Acceleration)

From previous sections:

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2.4 Kinematic Equations (Constant Acceleration)

Substitution gives:

and:

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2.4 Kinematic Equations (Constant Acceleration)

These are all the equations we have derived for constant accelerationconstant acceleration. The correct equation for a problem should be selected considering the information given and the desired result.

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2.4 Kinematic Equations (Constant Acceleration)

These are all the equations we have derived for constant accelerationconstant acceleration. The correct equation for a problem should be selected considering the information given and the desired result.

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Kinematic Example #1Example: A boat moves slowly out of a marina (so as to not

leave a wake) with a speed of 1.50 m/s. As soon as it passes the breakwater, leaving the marina, it throttles up and accelerates at 2.40 m/s/s.

What do I know?

What do I want?

vo= 1.50 m/s v = ?

a = 2.40 m/s/s

t = 5 s

v

v

atvv o

)5)(40.2()50.1(

a) How fast is the boat moving after accelerating for 5 seconds?

13.5 m/s

Kinematic Example #22

21 attvxx oxo

x

x

attvxx oxo

)5)(40.2(21)5)(5.1(0

21

2

2

b) How far did the boat travel during that time?

37.5 m

Does all this make sense?

mA

AbhA

50.7

)5.1)(5(

mA

bhA

30

)12)(5(2

1

2

1

mA

AbhA

50.7

)5.1)(5(

1.5 m/s

13.5 m/s

Total displacement = 7.50 + 30 = 37.5 m = Total AREA under the line.Total AREA under the line.

Interesting to Note

vttvxx

atttvxx

attvxx

oxo

oxo

oxo

2121

21 2

A = HB

A=1/2HBMost of the time, xo=0, but if it is not don’t forget to ADD in the initial position of the object.

Kinematic Example #3)(222oo xxavv

What do I know?

What do I want?

vo= 12 m/s x = ?

a = -3.5 m/s/s

V = 0 m/s

x

x

x

xxavv oo

7144

)0)(5.3(2120

)(22

22

Example: You are driving through town at 12 m/s when suddenly a ball rolls out in front of your car. You apply the brakes and begin decelerating at 3.5 m/s/s.

How far do you travel before coming to a complete stop?

20.57 m

2.5 Free Fall

An object in free fall has a constant acceleration (in the absence of air resistance) due to the Earth’s gravity.

This acceleration is directed downward.

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2.5 Free Fall

The effects of air resistance are particularly obvious when dropping a small, heavy object such as a rock, as well as a larger light one such as a feather or a piece of paper.

However, if the same objects are dropped in a vacuum, they fall with the same acceleration.

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2.5 Free Fall

Here are the constant-acceleration equations for free fall:

**The positive y-direction has been chosen to be upwards. If it is chosen to be downwards, the sign of g would need to be changed.

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ExamplesA stone is dropped at rest from the top of a cliff. It is

observed to hit the ground 5.78 s later. How high is the cliff?

What do I know?

What do I want?

voy= 0 m/s y = ?

g = -9.8 m/s2

yo=0 m

t = 5.78 s

2

21 gttvyy oyo

Which variable is NOT given andNOT asked for?

Final Velocity!

y

y 2)78.5(9.4)78.5)(0(

-163.7 mH =163.7m

ExamplesA pitcher throws a fastball with a velocity of 43.5 m/s. It is

determined that during the windup and delivery the ball covers a displacement of 2.5 meters. This is from the point behind the body when the ball is at rest to the point of release. Calculate the acceleration during his throwing motion.

What do I know?

What do I want?

vo= 0 m/s a = ?

x = 2.5 m

v = 43.5 m/s

)(222oo xxavv

Which variable is NOT given andNOT asked for? TIME

a

a )05.2(205.43 22

378.5 m/s/s

ExamplesHow long does it take a car at rest to cross a 35.0 m

intersection after the light turns green, if the acceleration of the car is a constant 2.00 m/s/s?

What do I know?

What do I want?

vo= 0 m/s t = ?

x = 35 m

a = 2.00 m/s/s

t

t 2)2(21)0(035

Which variable is NOT given andNOT asked for? Final Velocity

2

21 attvxx oxo

5.92 s

ExamplesA car accelerates from 12.5 m/s to 25 m/s in 6.0

seconds. What was the acceleration?

What do I know?

What do I want?

vo= 12.5 m/s a = ?

v = 25 m/s

t = 6satvv o

Which variable is NOT given andNOT asked for?

DISPLACEMENT

a

a )6(5.1225

2.08 m/s/s

Graphical Analysis of Motion

Very small section in your text. I am not happy with it so I have added the following…

Slope – A basic graph modelA basic model for understanding graphs in physics is SLOPE.

Using the model - Look at the formula for velocity.

Who gets to play the role of the slope?

Who gets to play the role of the y-axis or the rise?

Who get to play the role of the x-axis or the run? 

What does all the mean? It means that if your are given a graph, to find the velocity of an object during specific time intervals simply find the slope.

t

xv

Run

Riseslope

Velocity

Displacement

Time

Displacement vs. Time graphWhat is the velocity of the object from 0 seconds to 3 seconds?

The velocity is the slope!

Displacement vs. Time graphWhat is the velocity of the object from 7 seconds to 8

seconds?  Once again...find the slope!

A velocity of 0 m/s. What does this mean? It is simple....the object has simply stopped moving for 1 second.

Displacement vs. Time graphWhat is the velocity from 8-10 seconds? You must remember to

find the change it is final - initial.

The answer is negative! It is no surprise, because the slope is considered to be negative.

This value could mean several things: The object could be traveling WEST or SOUTH. The object is going backwards - this being the more likely choice!

You should also understand that the slope does NOT change from 0-3s , 5 to 7s and 8- 10s.

This means that the object has a CONSTANT VELOCITY or IT IS NOT ACCELERATING.

Example It is very important that you are able to look at a graph and explain it's motion in greatgreat detail. These graphs can be very conceptual.

Look at the time interval t = 0 to t = 9 seconds. What does the slope do?

It increases, the velocity is increasingLook at the time interval t = 9 to t = 11 seconds. What does the slope do?

No slope. The velocity is ZERO.Look at the time interval t = 11 to t = 15 seconds. What does the slope do?

The slope is constant and positive. The object is moving forwards at a constant velocity.Look at the time interval t = 15 to t = 17 seconds. What does the slope do? The slope is constant and negative. The object is moving backwards at a constant velocity.

Slope – A basic graph modelLet’s look at another model

Who gets to play the role of the slope for v-t graph?

Who gets to play the role of the y-axis or the rise?

Who get to play the role of the x-axis or the run? 

What does all the mean? It means that if your are given a Velocity vs. Time graph. To find the acceleration of an object during specific time intervals simply find the slope.

Acceleration

Velocity

Time

Velocity vs. Time GraphWhat is the acceleration from 0 to 6s?

What is the acceleration from 6 to 9s?

You could say one of two things here: The object has a ZERO accelerationThe object has a CONSTANT velocityWhat is the acceleration from 14 to 15s?

A negative acceleration is sometimes called DECELERATION. In other words, the object is slowing down. An object can also have a negative acceleration if it is falling. In that case the object is speeding up. CONFUSING? Be careful and make sure you understand WHY the negative sign is there.

Velocity vs. Time GraphConceptually speaking, what is the object doing during the time interval t = 9 to t = 13 seconds?

Does the steepness or slope increase or decrease?

The slope INCREASES!According to the graph the slope gets steeper or increases, but in a negative direction.

What this means is that the velocity slows down with a greater change each second. The deceleration, in this case, get larger even though the velocity decreases.

The velocity goes from 60 to 55 ( a change of 5), then from 55 to 45 ( a change of 10), then from 45 to 30 ( a change of 15), then from 30 to 10 ( a change of 20). Do you see how the change gets LARGER as the velocity gets SMALLER?

Area – the “other” basic graph modelAnother basic model for understanding graphs in

physics is AREA.

Let's try to algebraically make our formulas look like the one above. We'll start with our formula for velocity.

Who gets to play the role of the base? 

Who gets to play the role of the height?

What kind of graph is this? Who gets to play the role of the Area? A Velocity vs. Time graph  ( velocity = y-

axis & time = x-axis)

Time

Velocity

Displacement

Example What is the displacement during the time interval t = 0 to t = 5 seconds?

That happens to be the area!

What is the displacement during the time interval t = 8 to t = 12 seconds?

Once again...we have to find the area.

During this time period we have a triangle AND a square. We must find the area of each section then ADD them together.

mBHAsquare 140)35(4

mntDisplaceme

mmA

mBHA

total

triangle

260

140120

120)60)(4(2

1

2

1

Area – the “other” basic graph modelLet's use our  new model again, but for our equation for

acceleration.

What does this mean? Who gets to play the role of the base?

Who gets to play the role of the height? What kind of graph is this?

Who gets to play the role of the Area?

Time

Acceleration

An Acceleration vs. Time graph  ( acceleration = y-axis & time = x-axis)

The Velocity

Acceleration vs. Time GraphWhat is the velocity during the time interval t = 3 and t = 6 seconds? Find the Area!

smv

tavBhA

/18)6)(3(

Graphing SummaryThere are 3 types of MOTION graphs

• Displacement (position) vs. Time

• Velocity vs. Time

• Acceleration vs. Time

There are 2 basic graph models

• Slope (derivative)

• Area (integral)

Summary

There are 2 directions:

L to R = Slope

R to L = Area

• X-t -> slope -> V-t -> slope -> A-t

• A-t <- area <- V-t <- area <- X-t

t (s) t (s) t (s)

x (m)v (m/s) a (m/s/s)

slope

= v

slope

= a

area = xarea = v

Comparing and Sketching graphsOne of the more difficult applications of graphs in physics is when given a certain type of graph and asked to draw a different type of graph

t (s)

x (m)

slope

= v

t (s)

v (m/s)

List 2 adjectives to describe the SLOPE or VELOCITY1. 2.

The slope is CONSTANTThe slope is POSITIVE

How could you translate what the SLOPE is doing on the graph ABOVE to the Y axis on the graph to the right?

Example

t (s)

x (m)

t (s)

v (m/s)

1st line• •

2nd line•

3rd line• •

The slope is constant

The slope is constantThe slope is “+”

The slope is “-”The slope is “0”

Example – Graph Matching

t (s)

v (m/s)

t (s)

a (m/s/s)

t (s)

a (m/s/s)

t (s)

a (m/s/s)

What is the SLOPE of the v-t graph doing?

The slope is increasing

Which a-t graph is correct for the v-t shown?

Summary of Chapter 2

Motion involves a change in position; it may be expressed as the distance (scalar) or displacement (vector).

A scalar has magnitude only; a vector has magnitude and direction.

Average speed (scalar) is distance traveled divided by elapsed time.

Average velocity (vector) is displacement divided by total time.

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Summary of Chapter 2

Instantaneous velocity is evaluated at a particular instant.

Acceleration (vector) is the time rate of change of velocity.

Kinematic equations for constant acceleration:

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Summary of Chapter 2

An object in free fall has a = –g.

Kinematic equations for an object in free fall:

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