Chapter 2 - Motion

Part I - KINEMATICSphysics

how objects movewhy objects movehow objects interact (environment)

KINEMATICS - how objects move (not why)

Description of motion - change in positioncarssports: baseball, football, soccer, etc.world: rotates and revolves

How to measure position?

PHYSICAL QUANTITIESdescribe the physical universetwo types of physical quantities:

SCALARS - described by a magnitude or quantity

how much, how far just describe amountmass, time, volume, length, temperature, density, speed

Vectors: magnitude and direction!

quantities for which direction is important i.e., where? (displacement - distance and direction)

velocity, acceleration, force, momentum, magnetic &electric field

displacement - like directions on map

Difference between scalars and vectors

20 feet

1 step = 2 ft

Distance= displacement =

How to describe vectors - distance and direction

Vector variables - A A A A A~~

Geometric description - represent with a directed line

scale factor - magnitude length - rulerdirection - points along arrow - protractor

A

C?BA

Scale 1 inch = 1mile

tail

head

magnitude =

direction =

VECTOR ALGEBRA - adding vectors

+ =

head-to-tail method: head of first to tail of second

result: from beginning to endC

magnitude :direction :

displacement from adding displacements

Resolution of Vectors - component description

Break vectors into componentsup/down - left/rightscale built incomponents give direction

x

y

A Ax Ay= +

A

Ax

Ay

x-component and y-component specifies vectoreasy component directions (perpendicular)like treasure map

minmiles

Looks like two perpendicular rulers

Vector - TWO SCALAR COMPONENTS

Can treat each direction separately as a vector

Rectilinear Kinematics

MOTION - changes in positionhow objects move without regard to whyone-dimensional motion

Kinematic physical quantities-how you moveposition, how fast, speed up

SCIENTIFIC MODELstraight line - start and endpoint particle- all mass and volume a point

start end

to=0

do=0

t

d

time (t) : use to keep track of the object at a particular instant - synchronize stop watches

time interval or an instant in time

start at t=0 toPosition (d) : rectilinear - displacement, distance

where the object is

start-position at t=0 do

stop watch at end time t, position d

Speed and Velocity

SPEED (scalar) – rate of change in positionhow far in a given time – how fast

AVERAGE SPEED – OVER A TIME INTERVAL

vAVG = = d / t

can speed up or slow down during tripfaster – cover more distance in a given time

what constant speed to cover a distance in a given time

distancetime

INSTANTANEOUS SPEED – AT A PARTICULAR TIMElike looking at speedometer at an instant v measure: average speed in a short interval

Speed of sound – uniform motion (constant speed)v = vAVG = 1100 ft/s

d

Hear thunder t=5s

See lightning

to=0s

Light faster than sound

How far away is the lightning strike?

Velocity – how fast and what direction

VECTOR!Magnitude – speedDirection - which way it’s going

Rectilinear – speed is magnitude of velocitydirection – left/right or up/down

x

y

+

+-

-

Direction using normal coordinate directions

AVERAGE VELOCITY – time interval again

vavg = = d / t

INSTANTANEOUS VELOCITY

displacement

time

20 feet

at each instant during the interval -- short piece of time interval

Speed and velocityt = 10 min.

Not same for 2D

CHANGES IN VELOCITY

speeding upslowing downchanging direction }

acceleration – rate of change in velocitycan feel – force (cars, elevators,..)

VECTOR

rectilinear – change in speed

Average and instantaneous – UNIFORM (CONSTANT) ACCELERATION

a= = (v-vo)

to=0

do=0

t

d

vo =v =

Final instantaneous velocity

initial instantaneous velocity

change in speed

timet

a

Two types of problems: Uniform motionUniform acceleration

Tools for 1D Uniform Acceleration - formulae

DEFINITIONS : vavg = d / t

a = (v-vo)/ t

DERIVATIONS : not magic – use math to rearrange definitions

UNIFORM ACCELERATION FORMULA

d=vot + ½ at2

v=vo + at

vavg= (v+vo)/2 vavg=d / tcan find out how objects move

Only works for constant acceleration.

to=0

do=0

t

d

vo =v =

a

Fill out line diagram and apply formula

How to pick right formula-pick two with variable of interest (v, t, a, d)-find variable you don’t care about

Identifies when acceleration begins

1D uniform acceleration - examples

1. An airplane starts from rest and reaches the final take-off velocity in 50 s. What is its acceleration?

2. Space shuttle rocket accelerates 2 m/s2. a) What is the velocity 90 s after lift-off?

b) How far did it go? Finish description

3. A car travels 75 ft/s east and slams on brakes. a) If it stops in a distance of 200 ft, then how long did it take to stop?

b) What is the acceleration?

4. Car accelerates on I-10 uniformly along ramp. The speed is 30 ft/s at one instant and 5 s later it is going 110 ft/s. What is the acceleration of the car?

2D Motion (planar) - projectiles

Unnatural motion to Aristotlefour elementsnatural state – rest (he said so)medium adds resistance to speed

slow / fast – depends on object

unnatural – medium pushes object

DEMO – weight vs. crumpled paperARISTOTLE RIGHT?

Galileo – scientific methodmeasure the motion -

kinematicscompare quantitatively

inclined planeall objects fall the same

rate: ay = - g = -10 m/s2

(ay = - g g = 10 m/s2)natural state – constant velocity

everything we need to treat complex 2D motion projectile motion :objects projected

supported by Church 2000 yrs

Planar Motion

two dimensional motionin the plane of paper x and y graph

describe vector position - vector velocityEXAMPLES

Uniform Circular Motion: ball on a string

Moves at a constant speed in circle

At time t : position given by x and y

, xy

Projectile Motion: cannonball, arrow

voProjectile: - projected (thrown) with initial velocity - falls in earth’s gravity

VERTICAL DIRECTION

Galileo: ALL objects fall the same

ay= -g g =10 m/s2 approx.

HORIZONTAL DIRECTION

No acceleration if no force (friction)ax=0 (ignore air resistance)

Projectile motion thrown in the earth’s gravity

separate into component directions (x,y)work with each component separately!

to=0yo=0voy =

t =

vy =y =

ay= - g =-10 m/s2

to=0xo=0vox =

t =

vx =x =

ax=0

Two Cases We Will Deal With:

Vertical Projectile - Jump ball FREE FALL - freely falling in gravity

thrown straight up-fall straight down

rectilinear - y direction

to=0yo=0voy =

t =

vy =y =

ay

a) time to highest point?b) how high?

EXAMPLES A baseball is thrown straight up with a speed of 35 m/s. a)How long does it take to reach the highest point? b) How high does the ball go?

An egg is thrown straight downward with aspeed of 12 m/s. What is its speed 3 seconds later?

Horizontal projectile - cannonball, soccer, gun

Projected with a horizontal velocity -no initial velocity in vertical (y) direction -must work in two directions - separate ! -time connects the directions

Separate componentsvertical free fallhorizontal uniform motiontwo rectilinear problems

12 m

vo=40 m/s

rangea) How much time to hit ground? what direction?

to=0yo=0

voy =

t =

vy =y =

B) What is the range of the projectile? direction?

time connects position for fall in both directions: how far does it go in x-direction in the time it takes to fall?

to=0xo=0vox =

t =

vx =x =

NOTE: time of fall has nothing to do with x-direction! falls same independent of horizontal speed!!!!

Motion - Part 2: Dynamics (MECHANICS)-Why objects move!

not unnatural motion* force not required to keep object going* well-defined laws of motion

Sir Isaac Newton - 1st theoretical physicist

Great mathematician

bubonic plague sent him home to orchard

developed theories in 18 months-

algebra, calculus, motion, gravitation

fluid motion, optics (Principia, 1687)

looked at results of others---Galileo, Keppler

“on the shoulders of great men”

Newton’s Laws explain events in everyday life!

Newton’s First Law of MotionA body in uniform motion will remain in uniform motion unless acted on by an external force

new natural state - uniform motionchange motion with force - acceleration

inertia - tendency of an object to remain in uniform motion LAW OF INERTIA

Decelleration – turning a curve PROJECTILE MOTION

so objects travel in a straight line at a constant speed unless force (push or pull) acts

– natural state

Galileo knew – but Newton published

Newton’s Second Law of Motion (Force Law)

The acceleration of a body is proportional to theforce and inversely proportional to the mass

a=F/m m proportionality constant

(inertial) mass (kg) – resistance to a change

in uniform motion-or force

-ability to remain in uniform motion

big mass – small acceleration

small mass – accelerates easily

train vs. bicycle

Example:How much force is required to accelerate

a 1 kg block at 1 m/s2?

F=? a=1 m/s2

m=1 kg

a=F/m F=ma

F= ma = (1 kg)(1 m/s2) = 1 kg m/s2

SI force: kg m/s2 = 1 Newton = 1 N

MODEL

F=ma not whole truth

F=ma vector equation –

direction

2N left gives 2 m/s2 left

Also talk about

net force – add all forces on object

Fnet=Fpush + (-Ffriction)friction opposes motion

Second Law Examples:

m=1000 kg1.F=? a=1 m/s2

m=1000 kg2. two people

F1=500 N a=?

F2=800 N

Fnet =

m=1000 kg3.

a=?F1=500 N

F2= 800 N

Direction – be careful!!!

Weight and mass

Mass – intrinsic property of matter

- doesn’t change

-always resistance to force

Weight – force of gravity on an object

- depends on location

-difficulty in lifting an object

-weightless in space, but F=ma still

no effort to lift!

weight – force of gravity

W= F = ma = mg on Earth surface

g=10 m/s2 acc. due to gravity

Surface gravity : depends on gsurfaceMoon gmoon=1/6 gearth

=1.66 m/s2

M=100 kg Wearth= 1000 N

Wmoon= 166 N

Wspace= 0

g=0 weightlessF=ma still

Newton’s laws good for more than 200 yearsNO DISCREPANCIES

Tribute to Newton 20th century-

new observationsmeasurement techniques improvedfailures in F=ma

Jet planes, rockets – very very fast scaleEinstein’s Theory of Special Relativity

E-Microscope, scattering – very very small scaleQuantum Mechanical Theory (Bohr, Scrodinger)

Telescopy (BH, Neutron stars) – very very massive scaleEinstein’s Theory of General Relativity

ALL theories reduce to F=ma in the scale

of everyday experience:

- use Newton’s 2nd law for our purposes

- F=ma valid for cars, buildings, etc.

less complicated math!!!!

- use other theories when needed….

Forces of Nature: accelerate objects

Gravitational – force between masses

suns, planets, people

Electromagnetic – force between charges

opposites attract-likes repel

“contact force”

Strong Nuclear – nucleus of an atom

keeps atom together

“ likes repel”

Weak Nuclear – nuclear decay

gamma rays, beta-decay, nuclear reactions

UNIFYING THEORY –

binds all forces together at times beginning

-all forces have same form HOLY GRAIL

Newton’s Third Law of Motion (Action-Reaction)

If one body exerts a force on a second, then the second exerts a force back on the first which is equal in magnitude and opposite in direction

Easy statement:

How objects push on one another

- Forces exerted in pairs

-Always exerts force back

-Large mass, small acceleration

pendulum toy

leaning on wall

walking

FhandFwall

Application of Newton’s Laws: Circular Motion

Uniform Circular motion object traveling in a circle of constant radius at uniform speed

R

Like planet motionOr a ball on a string

1st law – inertial movementconstant speed in straight line - velocity

2nd law – forced falls in toward center – direction change - acceleration

Centripetal (center-seeking) acceleration - FORCE

acceleration required to keep object on circletoo fast, spirals in – too slow, spirals out

ac=v2 / R

depends on particular circle and speed

Velocity – tangentAcceleration - radial

Another Application: MOMENTUM

momentum – difficulty in stopping an object

p = mv linear momentum

mass and velocityvector – direction!

BASEBALLm=0.2 kg v=40 m/sp=mv=(0.2 kg)(40 m/s) = 8 kg m/s SI units

kg m/s is almost N

TRAINm=100,000 kg v=1 m/sp=mv=(100,000 kg)(1 m/s) = 100,000 kg m/s

Heavy or moving fast harder to stop!no motion—no momentum

You can change the motion by changing momentum

accelerate-fell the force from momentum changes fastball hurts more than slider

DERIVATION: Impulse

Second law definition:acceleration

Fext=ma a= (v-vo)/ t

F=ma=m{(v-vo)/ t}

I = Ftc = mv-mvo Impulse-

change in momentum

tc contact time -during which force applied

external force – accelerates as long as force applied

IMPULSE-MOMENTUM THEOREM

Consequences: sports – tennis golf

baseball

Hit ball as hard as possibleand

Follow-through (increase tc)}

safety (cars) -- metal dash padded dash airbags

SAME IMPULSE-

increase

tc}Io=Ftc

IMPULSE – force applied for a time

external force produces acceleration

accelerates for time tc

to=0xo=0vox =

t =tc

vx =x =

a

Impulse momentum theorem includes acceleration

I= F tc = mv-mvo

EXAMPLE:

A baseball is initially pitched toward the batter

at 40 m/s, and the batter hits it straight back to

the pitcher at 30 m/s.

a) What impulse is imparted to the ball?

b) What is the force on the bat?

The bat applies the external force which

changes the motion of the ball

external – connected to body

– connected to ground - etc

Conservation of Momentum - COLLISIONS

Momentum important in collisions

COLLISION MODEL

m1 m2

v1v2

m1 m2

m1 m2

v1v2

IsolatedSystemNo external forces

F12 F21

internal forces3rd lawaction-reaction pair

individual impulses cancel –equal & opposite

DURING

BEFORE

AFTER

Momentum exchanged : I is change in momentum

I12=-I21 one gains, other loses momentum

accelerated

Conservation laws conserved – same before as after

constant if assumptions true

Conservation of momentum

ptot=m1v1+m2v2

if no external forces

INTERACTING OBJECTSFor collisions, conserved before and after collision

ptot= (m1v1+m2v2)before =(m1v1+m2v2)after

ISOLATED FROM OUTSIDE FORCESNo momentum lost transferred

EXAMPLE:

Internal forces transfer momentum

perfectly inelastic – stick together after colliding

Newton’s Law of Universal Gravitation

Newton described how a gravitational force would act

MOTIVATION:ASTRONOMY – circular motion

R

Earth

inertial- - linear

MOON

centripetalFalls toward center of earth

What caused moon to fall?

APPLES and the MOON fall due to same force -- gravity

LAW OF UNIVERSAL GRAVITATIONAll objects with mass attract all other objects with mass

-attractive force-smallest force in nature-universal (all objects the same) falling apples-orbiting planets-satellites

Moon falls like apple

EXPLAINED Heliocentric Model!

Gravitation Model – picture to understand

m1 m2

dpoint mass-centers

F=G m1m2 / d2

m1, m2masses (kg)

d separation (m)G universal constant

same for everything

G must be measuredNewton couldn’t do that, but he could:

1. explain motions of planets around Sun(satellites, comets)

2. explain the tides from moon 3. explain why g changes w/ altitude

(distance from center of earth) 4. orbital perturbations – deviations

from predicted path

Cavendish experiment

Established the universal gravitation constant GG = 6.67 x 10-11 N m2/kg2

Can do things like:

- calculate forces between ordinary objects

- ’weigh’ the earth

- predict new planets (perturbations)

- put man-made satellites into orbit

centripetal force equals gravity force

TV, mapping, weather, spy

geosynchronous – same period as earth