p1370 10s unit 1 motion

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Math Methods in Phys, Unit 1 1

Lecture 1

Math Review


Linear Equation

Linear equation

Graph: a strait line

m: the slope

b: the y-intercept

Solving a linear equation

Example:

bmx +=

8453 =+ xx


Quadratic Equation

Quadratic equation

To solve the equation in the form

Find roots (quadratic formula)

Special case: b = 0

cbxaxy ++= 2

02 =++ cbxax

a

acbbx

2

42 =


Solving Simultaneous Equations If you have Nunknown quantities, you

need N independent equations to solve for

them

Example: Solve the following simultaneous

equations

24

072

=

=

yx

x


Class Example 1.1: Solving

Simultaneous Equations

Solve the following equations simultaneously

=+

=+=++

4

02

)(

zyx

zyxzyx

a

=++

=+

yxx

yxb

2

2)(

2


Exponents Definition

Calculation:

Example: Find the value of the following expression

...

1

2

1

0

aaa

aa

a

=

=

=

xyyx

yxyx

yxyx

aa

aaa

aaa

=

=

=

+

)(

/

...

1

1

2

2

1

aa

aa

=

=

...

33/1

2/1

aa

aa

=

=

42/1

12

27

394

2

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Geometric Relationship

Rectangle

Area

Triangle

Area

Circle

Circumference

Area

ab=

abA2

1=

rS 2=2

rA =


Class Example 1.2: Area of Trapezoid

Find the area of trapezoid ifa = 3, b = 4, and h =

2.


Trigonometric Relationship

Pythagorean theorem

Trigonometric function

Sine

Cosine

Tangent

Inverse trigonometric function

For example:

222 hba =+

h

b=cos

h

a=sin

b

a=tan

)(tan 1

b

a=


Class Example 1.3: Application of

trigonometry

(a) If the opposite a = 3 and the

hypotenuse h = 5, what is the angle ?

(b) The length of the sides of an

equilateral triangle is 2 m. Calculatethe altitude of the triangle.


Lecture 2

Measurement

Problem Solving


Physical Quantity

We use physical quantities to characterize the systemin study

A complete description of a physical quantity iscomposed of Symbol, number, unit

Example: the height of a person h = 6 feet

Comment about symbol The same quantity can be represented by different symbols

The same symbol may represent different quantities

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Scientific Notation

A number are said in scientific notion if it is expressed

as some power of 10 multiplied by another numberbetween 1 and 10.

Example:

Advantages Easy to deal with very large or small number

Easy to carry out calculation

9

12

102000000002.0

1032.4000,000,000,320,4=

=


Class Example 2.1: Calculation

with Scientific Notation

Calculate

(a) a xb

(b) a + b

(c)

000,000,000,000,000,000,000,000,200000,000,000,000,000,000,000,000,000,4

==

ba

22

ba +


SI Unit

SI base units in mechanics Length: meter (m)

Mass: kilogram (kg)

Time: second (s)

Examples of SI Derived units Area: square meter (m2)

Volume: cubic meter (m3)

Speed: meter per second (m/s)

Advantage to use SI units In a calculation, if all the input quantities are in SI units, the output

quantity will automatically in the corresponding SI unit.


Prefixes

Frequently used prefixes 109: giga (G)

106: mega (M)

103: kilo (k)

10-2: centi (c)

10-3: milli (m)

10-6: micro ()

10-9: nano (n)

10-12: pico (p)

Examples centimeter (cm): 1 cm = 10-2 m

kilogram (kg): 1 kg = 103 g

Where g is Abbreviation ofgram


Unit Conversion

You want to memorize:

Conversion involving

kilo, centi, milli, micro, nano, pico

Conversion between

year, month, day, hour, minute, second


Class Example 2.2: Unit Conversion

(a) How many seconds are in a day?

(b) The standard SI unit of force is calledNewton, which is kgm/s2. Some engineer,however, use the unit of gcm/s2. Express aforce of 2.0 gcm/s2 in Newtons.

(c) An area is 5.0 m2. What is this area in squarecentimeters (cm2)?

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Problem Solving Steps

Understand the problem and draw the appropriate

diagram

Write down all the know quantities and quantities to befound

Determine the principle to be used to solve the problem

Perform the calculation

Consider whether the results are reasonable


Class Example 2.3: Finding Distance

You walk 0.01 kilometer north, then 500centimeter east, and finally 15 meter south.

(a) What is the straight-line distance from youroriginal position to your final position?

(b) What is the direction of the of your finalposition relative to your original position? (givethe value of the angle)


Lecture 3

Kinematic Quantities for 1-D

Motion


Scalar and Vector

Scalar: Has only magnitude, but not direction

Can be specified by one positive number

Example:

Vector: Has both magnitude and direction

How many numbers need to specify a vector?

Two-dimension: two numbers

One-dimension: one number (can be positive or negative)

Example:

tm ,

vFv

v

,


Distance and Displacement Distance: length between the initial and final position, along the actual path of

motion

Distance is a scalar

Displacement: length of the straight-

line directly connecting the initial andfinal position

Displacement is a vector

SI unit: m

Example: You drive from Beaumont to Houston along I-10 and back toBeaumont. Suppose the distance from Beaumont to Houston is 80 m iles What is the distance for the round trip?

what is the displacementfor the round trip?

12xxx =


Speed and Velocity

Speed/Velocity

Measure how fast an object moves

Speed is a scalar and Velocity is a vector

Average speed/velocity

Measure how fast an object moves during an interval of time

Instantaneous speed/velocity

Measure how fast an object moves at a parti cular instant of time

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Average Speed/Velocity

Average speed:

Speed is a scalar

Average velocity:

Velocity is a vector

SI unit: m/s

timetravelingdistancespeedaverage =

12

12

tt

xx

t

xv

=

=

timetraveling

ntdisplacemevelocityaverage =

12tt

dt

dv

=

=


Instantaneous Speed/Velocity

Instantaneous speed (or simply speed) Its the limit of the average speed as the time interval goes to

infinitesimally small

Instantaneous velocity (or simply velocity) Its the limit of the average velocity as the time interval goes to

infinitesimally small

t

dv

t =

0lim

t

xv

t

=

0lim


Speed vs. Velocity

A car travels 10 meters to the east in 1 second and another cartravels 10 meters to the west in 1 second Do they have the same speed?

Do they have the same velocity?

You drive from Beaumont to Houston along I-10 and back toBeaumont. Suppose the distance from Beaumont to Houston is80 miles, and it takes you 1 hour each way.

What is the average speed for the round trip?

what is the average velocity for the round trip?


Class Example 3.1: Average Speed

A man moves along a straight-line:

he walks 60 meters for the first minute

For the next two minutes, he walks with a

constant speed of 0.5 m/s

Calculate the average speed for the whole

trip.


Acceleration

MeaningMeasures how fast the velocity changes

Average acceleration

Instantaneous acceleration

SI unit: m/s2

12

12

ttvv

tva

=

=

t

va

t

=

0lim


Direction of Acceleration

Direction of acceleration is in the direction of which thevelocity increases Determined by the direction of velocity change

Not by the direction of velocity itself

Example: determine the direction of the acceleration forthe following initial velocity v1 and final velocity v2 (thetime interval is 1 s): V1 = 4 m/s toward east, V2 = 5 m/s toward east

V1 = 5 m/s toward west, V2 = 4 m/s toward west

V1 toward east, V2 toward west

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Direction and Sign

Choose a positive direction

Any vector (x, v, a) along this direction is

represented by a positive number. Any vector

along the opposite direction is represented by a

negative number

Redo the example on the previous slide


Accelerated and Decelerated

Motion

Accelerated motion The direction ofaand vare the same

aand vhave the same sign

Decelerated motion The direction ofaand vare opposite

aand vhave the opposite sign

Note that its not directly determined by the direction ofa

Consider the previous example again


Graphical Analysis

x-tplot:Horizontal distance: time interval

Vertical distance: displacement

Slope: velocity

v-tplot :

Horizontal distance: time interval

Vertical distance: velocity change

Slope: acceleration


Class Example 3.2: v- tplot

The plot is velocity versustime for an object in a linearmotion

(a) Describe how the objectmoves during each phase

(b) Compute theacceleration for each phaseof motion

(c) Find the totaldisplacement


Lecture 4

Motion with Constant Acceleration:

Kinematic Equations


Kinematic Equations

There are five quantitiesx = x - x0, v, v0, a, and tfor amotion with constant acceleration:

Relate v, v0, a, and t:

Relate x - x0, v0, a, and t:

Relate x - x0, v, v0, and a:

Relate x - x0, v, v0, and t:

atvv += 0 atvv += 0 atvv += 0

tvvx )(2

10

+=

xavv += 220

2

2

0

2

1attvx +=

atvv += 0

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Class Example 4.1: Bullet Through a

Board

A bullet traveling horizontally at a speed of 300m/s hits and passes through a 0.1 m thickboard. The bullet emerges from the other sidetraveling at 100 m/s.

(a) How long did the bullet take to pass throughthe board?

(b) What was the acceleration of the bullet


Class Example 4.2: Stopping a Car

A car traveling at 30 m/s stops on a 50-meter-long section the roadway with aconstant deceleration.

(a) What was the required time to stop?

(b) What was the acceleration of the car?


Free-Fall

Objects in motion solely under the influence ofgravity are said to be in free fall

The downward Acceleration due to gravity nearthe Earths surface a= -g = -9.80 m/s2

So all the kinematic equations apply if youreplace x x0 byy y0, and aby -g


Class Example 4.3: Free Fall from Rest

Shanghai World Financial Center, oneof the tallest building in the world, has aheight of 490 m. A ball is dropped fromrest from the top of it.

(a) How long did it take to hit theground?

(b) What was the balls speed justbefore hitting the ground?

www.getwonder.com


Class Example 4.4: Free Fall Up and Down

A worker on a scaffold in front of a billboard throws aball straight up. The ball has an initial speed 11.2 m/swhen it leaves the workers hand at the top of thebillboard.

(a) What is the maximum height the ball reachesrelative to the top of the billboard?

(b) How long does it take ball to reach this height?

(c) What is the position of the ball after 2.0 seconds?


Class Example 4.5: Two Runners

Two Runners approach each other on a straight

track with different constant speeds, when they

are 100 m apart. If the speed of the runner

towards right is 4.50 m/s and it takes 12.5 s for

them to meet, what is the speed of the runner

towards left?

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Lecture 5

Vectors


Graph of Vectors

Two ways to represent a vector

GeometricallyBy components

Geometrically, a vector is represented by an

arrow

Length of the arrow Magnitude of the vector

Direction of the arrow Direction of the vector

Location of the arrow: does NOT matter


Adding Vectors Geometrically

Question: If you have two vector

What is the magnitude of the sum

Adding vectors geometrically:

1. Sketch the first vector to some convenient scale and at the properangle

2. Sketch the second vector to the same scale, with its tail at the headof the first vector, again at the proper angle

3. The vector sum is the vector that extends from the tail of the firstvector to the head of the second vector

4magnitudewithand3,magnitudewith BAvv

BACvvv

+=


More About Adding Vectors

Properties of vector addition:

Commutative law

Associate law

Vector Subtraction

Where is a vector with the same magnitude asbut the opposite direction

ABBAvvvv

+=+

)()( CBACBAvvvvrv

++=++

)( BABAvvvv

+=

Bv

Bv


Components of Vectors

A Component of a vector is the projection of the vector on an axis

The process of finding components of a vector is called resolving(decomposing) the vector

For An vector with magnitude A and orientation angle :

X-component:Ax= A cos

Y-component:Ay= A sin

Inverse transformation

Magnitude:

Angle:

22

yxAAA +=

)(tan1

x

y

A

A=


Adding Vectors by Components

Rule: add the x and y components separately

Subtracting vectors by components

),(),,(yxyx

BBBAAA ==vv

),( yyxx BABABA ++=+v

),(yyxx

BABABA =

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Class Example 5.1: Adding and SubtractingVectors

For three vectors F1 , F2 and F3, find F1+F2+F3 and F1-F2-F3 :

F1 has magnitude of 50.0 m and makes an angle of 20.0o with

respect to the positive x axis.

F2 has magnitude of 30 .0 m and

makes an angle of 60.0o with respect

to the positive x axis.

F3 has magnitude of 20 .0 m and is

directed along the negative y axis.


Class Example 5.2: Finding Distance

Revisited

You walk 10 meter north, then 5 meter east, and

finally 15 meter south.

(a) What is the straight-line distance from youroriginal position to your final position?

(b) What is the direction of the of your final positionrelative to your original position? (give the value of theangle)


Lecture 6

2-D Motion


Strategy to Solve 2-D Problem

Choose appropriate Coordinate System Usually, we choose the x and y direction so that one of them is

along the direction ofa

Decompose the motion into x and y components That is, decompose all the vectors

Write the equation for x and y components separately Motion along x and y direction are independent to each other

Establish the relationship between components ofmotion For example: they have the same time t

Solve the equations of both components together


Class Example 6.1: 2-D Motion

A particle has an initial velocity of 20 m/swith an angle of 60o north of east. It moveswith a constant acceleration of 5 m/s2 with

an angle of 30o north of east.

What is the particles velocity after 5 seconds?

How far did the particle travel?


Projectile Motion

The horizontal motion and the

vertical motion are independent

of each other

Horizontally, it is a motion with constant velocity ax = 0

Vertically, it is motion with constant acceleration ay = -g = -9.8 m/s

2

Gravitational constant g = 9.8 m/s2

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Class Example 6.2: Projection from

Ground to Ground

A ball is projected from the ground and falls back to the

ground, with the initial velocity of 10 m/s and an angle of36.9o above the ground.

(a) How long does it take for the ball to reach themaximum height?

(b) What is the maximum height of the ball?

(c) How long does it to take for the ball to falls back tothe ground?

(d) What is the horizontal range?


Lecture 7

2-D Motion: Applications


Class Example 7.1: Horizontal

Projection I

A ball is thrown horizontally with the speed of

15 m/s from the top of a 6.0-m-tall hill.

How far from the point on the ground directly

below the launch point does the ball strike the

ground?


Class Example 7.2: Horizontal

Projection II

A ball rolls horizontally with a speed of 5.0m/s off the edge of a tall platform. If the

ball lands 8.0 m from the point on the

ground directly below the edge of theplatform, what is the height of the platform.


Class Example 7.3: Hitting the Roof

A ball is thrown up onto a roof, landing atthe roof 1.0 s after leaving the hand. The

initial velocity is 10 m/s with an angle 60oabove the horizontal.

(a) What is the horizontal distance it traveled?

(b) What is the height of landing point

(c) What is the velocity when it hits the roof?


Class Example 7.4: Stone Thrown

from a Bridge I

A stone thrown off a bridge 20 m above a river

has an initial velocity of 10 m/s at an angle of

36.9o above the horizontal

(a) What is the horizontal range of the stone?

(b) At what velocity does the stone strike the

water?

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Class Example 7.5: Stone Thrown

from a Bridge II

A stone thrown off a bridge above a river has aninitial velocity of 10 m/s at an angle of 36.9obelow the horizontal. It travels 15 m horizontallywhen striking the water.

(a) What is the height of bridge from the water?

(b) At what velocity does the stone strike thewater?

p1370 10s unit 1 motion

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