mechanics lecture 1, slide 1 1d -kinematics-summary main points of 1d-kinematics again hints on...

Mechanics Lecture 1, Slide 1

1d -Kinematics-summary

Main points of 1d-kinematics againHints on doing homeworkHomework Examples

Homework Deadline extended (Sunday 1/25 11:30PM)

Main Points of Unit 1


1d-Kinematic Equations for constant acceleration


))((2)))((

2

1)(

)(

)(

020

2

002

0

00

0

xtxavtv

xtvtatx

vtatv

ata

Basic Equations to be used for 1d – kinematic problems.

Need to apply to each object separately sometimes with time offset

When acceleration changes from one constant value to another say a=0 The problem needs to be broken down into segments

Homework Hints

Start Early and allow yourself enough time Write out solutions on paper BEFORE entering answers online Ensure that stated problem is understood

Identify the useful equations Identify the information to be used

Sometimes the required information is not stated explicitly Make diagrams and rough graphs of quantities Break the problem up into parts Solve algebraically first Check that equations are reasonable Carefully perform calculations Stay calm…


Interactive Example V vs T


20

0

0

/66667.65.1//10)05.1/())0()5.1(()0(

0)0(

0)0(

smssmssstvstvtaaa

tvvv

txxx

i

i

i

?

Interactive Example V vs T


Region 1: 0<t<1.5 s0)0( stxxi

Region 2: 1.5s<t<3.0s2

2 /05.1/)/0(/ smssmtvaa

Region 3: 3.0s<t<5.0ssssttt if 0.20.30.5

Region 1 Region 2 Region 3

What happened in region 3?Smartphysics Conclusion

ssttt if 05.1 2

1 /667.65.1/)/10(/ smssmtvaa smssmtatavtastv /0.10)5.1)(/667.6(00)0()5.1( 2

1101

mssmtatvxstx 5.7)5.1)(/667.6(2

1)(

2

1)5.1( 222

100

smsmtstvtatv /0.10/100)5.1()( 2

mmmssmmssstvstxstx 5.220.155.7)5.1)(/0.10(5.7)5.10.3)(5.1()5.1()0.3( smstvtstv /0.10)5.1(0)0.3(

23 /100.2/)/0.10/10(0.2/)(/ smssmsmsvvtvaa if

smsmssmstvtastv /0.10/10)2)(/10()0.3()5( 22

mssmsmtstvtastxstx 0)0.2)(/10()0.2)(/10(2

1)0.3()(

2

1)0.3()0.5( 222

3

mstxmmstxstx 5.22)0.3()20()20()0.3()0.5(

smsmsmstvtastv /0.10/10)0.2)(/10()0.3()0.5( 22

)5( stxx final

https://www.smartphysics.com/Course/PlaySlideshow?unitItemID=192805

https://www.smartphysics.com/Course/PlaySlideshow?unitItemID=192805

I-74 problem


I-74 problem


mihrhrmitvx

hrhrmihrmi

mimi

vv

xxt

meetameet

ca

acmeet

66.73)9821.0)(/75(

9821.0)/65(/75(

05.137

)(00

dcdcc

ccc

aaa

tvttxx

xtvtx

xtvtx

)(

)(

)(

0

0

0

00

)()(;@

cmeetcameeta

meetcmeetameet

xtvxtv

ttxttxtt

)(

)(

00

00

ca

acmeet

accameet

vv

xxt

xxvvt

delaycdcc tvttxx )(0

hrmivhrmiv

hrtmittxx

ca

delaydca

/65;/75

;5.0;105)(;00

Facts Equations

Where are the cars when they meet? The same place

Solve for meeting time Where was Chuck when Anna passed prospect avenue?

)(00

ca

acmeet vv

xxt

Express meeting time in terms of separation of Chuck and

Anna at t=0:

mihrhrmimitvttxx dcdcc 5.137)5.0()/65(105)(0

Solve for meeting time and location

Be careful with your facts!

I-74 Problem


x

t

Anna

Chuck

tmeet

xa0

td

xc0

xc (t=td)

Car-ride problem


3.5 9.8

Yellow carBlue car

Car-ride problem


smsmssmvtavstv bb /4.4/0)1(/4.4)1( 20

ystopy

b

bstopstopb

b

b

atta

mx

smatta

smta

smta

f

)0(

27.159

;/)8.9(

;/0)8.95.3(

;/4.4)5.30(

2

2

2

)5.3()1.6( tvtv bb

22

/359.3))8.9((2

)/4.15(sm

txx

sma

bbstopbstop

mtvssssmtx bb 97.123)5.3(*)5.38.9()5.3)(/4.4(2

1)8.9( 22

220

2 )/4.15(0))8.9((2 smvvtxxa fbbbstop f

smssmtvb /4.15)5.3)(/4.4()5.3( 2

sm /4.15

Gather factsAnd equations

))((2)))((

2

1)(

)(

)(

020

2

002

0

00

0

xtxavtv

xtvtatx

vtatv

ata

Car-ride problem


stst

ssm

smsmsmt

sm

msmsmsmt

a

xtxaatv

t

xtxttvta

tt

tattvtxttx

stop

bstop

bbbstopbstopb

bbbbstop

stop

bstopbbstopb

f

f

f

38.148.9

58.4/359.3

/2.237/2.237/4.15

/359.3

3.35/68.14/4.15/4.15

)8.9(21

421

)8.9(

0)8.9())(8.9()(2

1

8.9

)(2

1))(8.9()8.9()(

2

2222

2

22

2

2

2

2

22

2

/539.1)385.14(

27.1592

)(

27.1592

)(2

127.159

sms

m

t

ma

tamx

stopy

stopyy f

Quadratic Equation-which solution


2

2

2222

2/

2

2

hck

abh

khc

ahb

aa

cbxaxkhahxaxkhxa

22

2

/)(

///2

1

2

1'

0)(2

1

00

000

00

gvxxhck

tgvgvavh

gaa

xxtvat

bbb

fbbb

bbfbf

f

f

t

a

acbbx

cbxax

2

4

0

2

2

Two thrown balls problem




mmsssmsssmstx

mxstvstatx

ttt

r

rrr

delayr

04.192.27)7.251.3)(/1.6()7.251.3)(/81.9(2

1)51.3(

04.19)7.2()7.2(2

1)(

22

2

00

2/81.9;2.27;/1.6;7.2;6.0;/4.220000

smamxsmvstmxsmv rrdelaybb Facts

smxxvv bbbb f/0)(

max

ssm

sm

a

vvt

bb

ff 283.2

/81.9

/4.2202

0

mx

mssmssmxtvatx

f

f

b

bfbfb

174.26

6.0)283.2)(/4.22()283.2)(/81.9(2

1

2

1 222

00



ssm

m

sm

mmmt

smsms

mmssmssm

vvsa

xxsvsat

meet

br

rbr

meet

638.3/01.2

3126.7

/01.2

6.2647.1675.35

/1.6/4.22)7.2)(81.9(

2.276.0)7.2)(/1.6()29.7)(/81.9(21

)7.2(

)7.2()29.7(21 222

00

000

)()( meetbmeetr ttxttx SAME HEIGHT MEET

Use relevant kinematic equations

0000)()(

2

1)7.2()7.2(

2

1 22bmeetbmeetrmeetrmeet xtvtaxstvsta

Simplify equation

0)7.2()29.7(2

1))7.2(

2

1(

0)()(2

1)7.2()29.7(

2

1)7.2)(2(

2

1

2

1

00000

00000

2

222

brrmeetbr

bmeetbmeetrrmeetrmeetmeet

xxsvsatvvsa

xtvtaxsvtvsatsaat

Solve for tmeet

Two thrown balls problem-Alternative Solution


ssm

sm

a

vt

sm

smsm

a

xxavvt

xxtvat

tvatxx

bf

bbbb

f

bbfbf

fbfbb

f

f

f

283.2/81.9

/4.220

/81.9

)574.25)(81.9(2)/4.22(/4.22)(2

0)(2

12

1

2

2

22

2

2

0

000

00

00

mmmsm

ma

vvxx

vvxxa

bb

bb

bbbb

f

f

ff

174.26574.256.0)81.9(2

)/4.22(6.0

2

)(2

222

22

0

0

00

Use equation relating velocity and position to solve for height

Then solve for time at maximum height

Continue solving the rest of the problem as before…

The meeting Problem in General


0000 222

112

21

1

)()(2

1)()(

2

1

)()(

0

_

xtvtaxttvtta

ttxttx

t

launchedfirstobject

meetmeetdmeetdmeet

meetmeet

d

Whenever possible solve symbolically.Then plug in the numbers!!!! You can understand what is going on better if you keep the equations organized.

02

1

2

1)2(

2

1

2

100000 22

2111

22 xtvatxtvtvatttaat meetmeetdmeetdmeetdmeet

0)2

1())((

00000 2112

21 xxtvattvvat ddmeetd

Define your problem; Identify useful equations

Start solving…

)(

)(

))((

)21(

00

000

21

2112

d

d

d

dd

meet ttv

ttx

vvat

xxtvatt

Gather terms…Does the equation look reasonable?

Solve for desired quantity…

)2

1(

00 112 xtvat dd

Position of object 1 when object 2 is launched

)(01vatd

Speed of object 1 when object 2 is launched

This time is w.r.t object 2. To obtain time w.r.t object 1 need to add delay time

dmeetmeet ttt 1)(

Two thrown balls problem #2




mxmxsmvsmvstsma

ttv

ttx

vvat

xxtvatt

d

d

d

d

dd

meet

1;26;/8.23;/2.1;4.0;/81.9

)(

)(

))((

)21(

0000

00

000

2121

21

2112

Position of object 1 at moment object 2 is launched

Velocity of object 1 at moment object 2 is launched

Use your equation in symbolic form and gather the values to be used….

mmssmssmxtvat dd 735.2426)4.0)(/2.1()4.0)(/81.9(2

1()

2

1( 22

112

00

Solve for individual elements of equation…can check if things make sense

smsmssmvatd /124.5)/2.1()4.0)(/81.9()( 210

smsmsmttv

mmmttx

d

d

/94.28/8.23/124.5)(

735.231735.24)(

Solve for displacement and relative velocity at time when object 2 launches

Divide to obtain time when objects meet

sttt

sttv

ttxt

dmeetred

d

dmeet

22.1

82.0)(

)(

Note that acceleration term after 2nd object is released drops out!!!



mstx

msssmsssmstx

stt

msm

smm

a

vvxx

vvxxa

smssmsmatvv

ssm

smsmt

sm

mmsmsmsmt

a

xxavvt

xxtvatxtvatx

mtx

smamxsmvstmxsmv

rb

rb

rb

bb

bb

bbbb

frr

f

f

fr

rrdelaybb

f

f

ff

f

70.24)8.1(

0.1)4.08.1)(/8.23()4.08.1)(/81.9(2

1)8.1(

4.0

87.29)/81.9(2

)/8.23(00.1

2

)(2

/586.22)18.2)(/81.9(/2.1

18.2/81.9

/617.22/2.1/81.9

)00.26)(/81.9(2)/2.1()/2.1(

)(2

02

1

2

1

0.0)(

/81.9;0.26;/2.1;4.0;0.1;/8.23

22

2

222

22

2

2

2

22

0200

002

002

2

0

0

00

0

0000

Schaum’s Outline


Displacement,Velocity & Acceleration


Need to become comfortable with displacement, velocity and acceleration and how they are related!!!

Constant Acceleration


Hyperphysics-Motion


http://hyperphysics.phy-astr.gsu.edu/hbase/mot.html


Hyperphysics-Motion




HyperphysicsMotion


Displacement vs timet

Velocity vs timet

Acceleration vs timet

HyperphysicsMotion


Enter Question Text

A. TrueB. False


6%

94%

I experienced some degree of “frustration” with the Homework Problems

Enter Question Text

A. TrueB. False


21%

79%

I experienced a sense of “accomplishment” solving the Homework Problems

mechanics lecture 1, slide 1 1d -kinematics-summary main points of 1d-kinematics again hints on...

Documents

pm slide

calm mechanics lecture

vs t mechanics lecture

dkinematic equations

d kinematics

homework hints

main points of unit

useful equations