How to Plot Graphs

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MODULE

GRAPHS

AND

EQUATIONS OF MOTION

NAME: CLASS:

Hillcrest High School, Private Bag 1012, Hillcrest, 3650All rights reserved. This module or ortio!s thereo" ma# !ot $e reroduced

$# a!# mea!s %ithout the school&s ermissio!.


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GRAPHS AND EQUATIONS OF MOTION Page 2

Objectives

'! comletio! o" this module #ou should(

1. u!dersta!d a!d $e a$le to de"i!e the "ollo%i!g terms( rectili!ear motio!) u!i"orml#

accelerated motio!) direct roortio!) sloe o" a grah) area u!der a grah)2. $e a$le to ma!iulate a!d al# the e*uatio!s o" motio!)3. $e a$le to dra% grahs rerese!ti!g various t#es o" motio!)+. $e a$le to a!al#se grahs rerese!ti!g various t#es o" motio!.

Tass

1. Stud# the theor# sectio! o" this module beforeattemti!g the %orsheet.2. A!s%er the re- a!d ost-la$. *uestio!s "ou!d i! the Practical sectio! o" this module.

These %ill $e mared i! class.3. arr# out the e/erime!ts i! the Practical sectio! %ori!g i! groups no larger than four

pupils. The ticer-tae e/erime!t is e/ami!a$le a!d it is esse!tial that ever# uil !o%sho% to a!al#se a ticer tae. ach grou mem$er must there"ore dro a! o$ect o"di""ere!t mass a!d a!al#se his her o%! tae. ach mem$er must the! lot his her dataa!d co# the grahs o" the other mem$ers o!to the same iece o" grah aer "orcomariso! uroses.

+. /erime!t 13.1 must $e %ritte! u "ull# accordi!g to the guideli!es give! i! M!"#$eC%&'2.

5. omlete the orsheet sectio! i! #our lass%or a!d Home%or Boo.6. The Tutorial must $e comleted a!d ha!ded i! searatel#.

Res!#(ces a)" S#**!(tive Mate(ia$

1. omuter( The 1445 rolier ultimedia !c#cloedia 789:!"oedia 789S !carta 45 46 789!c#cloedia o" Scie!ce 789The a# Thi!gs or 789

2. ;ideos ( Seed, ;elocit# a!d Acceleratio!Simle grahs o" s, a a!d tAt The dge 7Part 2( g-


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GRAPHS AND EQUATIONS OF MOTION Page ,

',-' INTRODUCTION

:! this module #ou %ill stud# the grahs o" $odies movi!g i! straight li!es. he! motio! islimited to a straight li!e, $oth seed a!d velocit# have the same mag!itude, as do dista!ce a!ddislaceme!t. otio! i! a straight li!e is re"erred to as (ecti$i)ea( .!ti!). '!ce the grahs

a!d e*uatio!s relati!g to this t#e o" motio! have $ee! u!derstood, more comle/ ro$lemsca! $e solved $# resolvi!g the motio! i!to como!e!ts a!d treati!g each como!e!t as asimle rectili!ear ro$lem.

',-2 GENERAL CONCEPTS

The grah i! Fig#(e ',-' sho%s the relatio!shi$et%ee! a h#sical *ua!tit# / a!d a!other relatedh#sical *ua!tit# 0. The grah is a straight li!eassi!g through the origi! o" the a/es.

The t%o *ua!tities / a!d # are said to $e "i(ect$0 *(!*!(ti!)a$to o!e a!other, $ecause i"

the o!e is dou$led, the other %ill also dou$le, etc. This relatio!shi is statedmathematicall# as(

/ # 7or / #9.

The sloe 7or gradie!t9 o" the grah is give! $# the e/ressio!(

Sloe cha!ge i! # # m,

cha!ge i! / /

%here m is the c!)sta)t !1 *(!*!(ti!)a$it0. =e-arra!gi!g the a$ove e/ressio!(

# m/, or more siml#, # m/.

omare this e/ressio! %ith the sta!dard e*uatio! "or a straight li!e( # m/ c. Thevalue o" the #-i!tercet 7i.e. c9 is o$viousl# Cero i! the grah a$ove. e ca! thus co!cludethat t%o *ua!tities / a!d # %ill $e directl# roortio!al i" their grah has the e*uatio! #

m/ c %here c is Cero.

Graph of X vs Y

0

1

2

3

4

5

6

7

8

0 1 2 3 4

X

Y

Fig#(e ',-'


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GRAPHS AND EQUATIONS OF MOTION Page

:! ge!eral, theslopeo" the grah o" t%o related h#sical *ua!tities alwaysrerese!ts a

third h#sical *ua!tit#. This third h#sical *ua!tit# ca! $e most easil# ide!ti"ied $#carr#i!g out a "i.e)si!)a$ a)a$0siso" the sloe - i.e. $# determi!i!g the u!its o" thesloe.

The t%o *ua!tities / a!d # are i! ge!eral nevere*ual, eve! i" the# have the same !umericalvalue, si!ce their u!its %ill almost al%a#s di""er.


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The u!it o" the sloe is that o" velocity. :t should $e o$vious that the sloe rerese!ts

velocit#, si!ce to calculate the sloe %e have divided dislaceme!t $# time.

The e*uatio! o" the grah is there"ore s v.t, %here v is the co!sta!t o" roortio!alit#.

The area u!der the grah is e*ual to D E + s E 20 m +0 m.s. The metre seco!d is notthe

u!it o" a!# h#sical *ua!tit#, so the area u!der this grah does !ot rerese!t a "ourthh#sical *ua!tit#.

:t ca! there"ore $e co!cluded that(

The sloe o" a"is*$ace.e)t3ti.e grah rerese!ts velocit#.The area u!der a"is*$ace.e)t3ti.e grah does !ot rerese!t a!# h#sical *ua!tit#.

',- 5ELOCIT63TIME GRAPHS

o!sider a car %hich starts "rom rest a!d gai!s seed u!i"orml# 7i.e. it u!dergoes a co!sta!tacceleratio!9 alo!g a straight road. :" the car&s i!sta!ta!eous velocit# %as measured ever#seco!d, a set o" results similar to those ta$ulated $elo% might $e o$tai!ed.

ts vm.s-1

0 0

1 2

2 +

3 6

+ F

A! a!al#sis o" the grah reveals that(

m v F m.s-1 2 m.s-2

t + s

0

1

2

3

45

6

7

8

0 1 2 3 4

t/s

v/m/s!

Figure 13.3: Graph of v vs t


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The u!it o" the sloe is that o" acceleration. :t should $e o$vious that the sloe rerese!ts

acceleratio!, si!ce to calculate the sloe %e have divided velocit# $# time.

The e*uatio! o" the grah is there"ore v a.t, %here a is the co!sta!t o" roortio!alit#.

The area u!der the grah is e*ual to D E + s E F m.s -1 16 m. The area u!der the grah

there"ore rerese!ts dislaceme!t.

:t ca! there"ore $e co!cluded that(

The sloe o" ave$!cit03ti.e grah rerese!ts acceleratio!.The area u!der ave$!cit03ti.e grah rerese!ts dislaceme!t.

Go% co!sider a car %hich has a! initial velocity#a!d %hich accelerates u!i"orml# to afinalvelocityvi! a time o" tseco!ds. Assumi!g that the o$server o!l# starts his sto-%atch at the

mome!t the car starts to accelerate, a set o" results similar to those ta$ulated $elo% ma# $eo$tai!ed.

ts vm.s-1

0 2

1 +

2 6

3 F+ 10

A! a!al#sis o" this grahs reveals that(

m v v - u 710 - 29 m.s-1 2 m.s-2 7as $e"ore9.

t t + s

The e*uatio! o" the grah is v a.t u, %here a is the co!sta!t o" roortio!alit# a!d u

7the i!itial velocit#9 is the #-i!tercet.

Figure 13.4: Graph of v vs t

0

2

4

6

8

10

0 1 2 3 4

t/s

v/m/s!


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The area u!der the grah must agai! rerese!t dislaceme!t. Ho%ever, the area !o%

co!sists o" twodisti!ct shaes, each rerese!ti!g a dislaceme!t. The rectangulararea isthe dislaceme!t the car u!dergoes as a result o" its i!itial velocit# u. ve! i" the car did!ot accelerate, it %ould still $e dislaced $# this amou!t. The triangular area is thedislaceme!t %hich the car u!dergoes as a result o" its acceleratio!, as i! the revious

e/amle 7%here the car did !ot have a! i!itial velocit#9.

The total dislaceme!t 7sT9 is there"ore rerese!ted $# the sum o" the recta!gular area 7s19a!d the tria!gular area 7s29) i.e.(

sT s1 s2 72 m.s-197+ s9 D710 m.s-1- 2 m.s-197+ s9 F m 16 m 2+ m.

',-4 ACCELERATION3TIME GRAPHS

Si!ce %e are limited to the stud# o" u!i"orm acceleratio! o!l#, there are o!l# three t#es o"

acceleratio!-time grahs that %e !eed to co!sider. Because the sloes o" these grahs are allCero 7i.e. the rate o" cha!ge o" acceleratio! is Cero9, %e !eed o!l# evaluate the area u!der thegrahs. This area is "ou!d $# multil#i!g time $# acceleratio! a!d is thus e*ual to velocit#.

5 u!i"orm acceleratio! i! the ositive directio!

Cero acceleratio!

a7ms29 0 ts

u!i"orm acceleratio! i! the !egative directio! -5

Fig#(e ',-4: rah o" a vs t

The area u!der a!acce$e(ati!)3ti.e grah rerese!ts velocit#.

',-7 THE EQUATIONS OF MOTION

The "our e*uatio!s o" motio! are derived "rom the velocit#-time grah "or a $od# u!dergoi!gu!i"orm acceleratio! "rom a! i!itial velocit# 7#9 to a "i!al velocit# 7v9 i! a time o" tseco!ds7see Fig#(e ',-7$elo%9.

The sloe o" the velocit#-time grah is o$tai!ed as "ollo%s(

Sloe cha!ge i! velocit# cha!ge i! time


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i.e. a v - u t

=e-arra!gi!g(

v u at --------------------------- *uatio! 1I

v

v7ms9 s2

u

s1

0 ts 0 t

Fig#(e ,-7: rah o" v vs t

*uatio! 1 is our "irst e#ati!) !1 .!ti!). The remai!i!g three e*uatio!s are derived "romco!sideratio! o" the dislaceme!t o" the $od# duri!g this same time eriod. This dislaceme!tis give! $# the area u!der the grah. The di""ere!t methods o" calculati!g this area result i!

the remai!i!g e*uatio!s.

'!e method o" determi!i!g the area u!der the grah 7s9 is to co!sider it to $e the sum o" arecta!gular area 7s';a!d a tria!gular area 7s2;.

s s1 s2 ut D7v - u9t

Ho%ever, "rom *uatio! 1 %e have( v - u at. Su$stituti!g "or v - u i! the a$ove e/ressio!7"or reaso!s that %ill $ecome clear laterJ9 %e get(

s ut Dat2 --------------------------- *uatio! 2I

The area u!der the grah ma# also $e co!sidered that o" a traeCium. The area o" a traeCiumis give! $# the e/ressio!(

Area o" traeCium 7hal" the sum o" the arallel sides9 E 7dista!ce $et%ee! the arallel sides9.

:! this case,

s 7u v9.t --------------------------- *uatio! 3I 2

Go% u v is also the average velocit# o" the o$ect. He!ce, *uatio! 3 simli"ies to(2


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GRAPHS AND EQUATIONS OF MOTION Page &

s vav.t %hich is a! e*uatio! eve! a rimar# school uil usesJ

A "i!al e*uatio! o" motio! ca! $e derived $# com$i!i!g *uatio!s 1 a!d 3 as "ollo%s.


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GRAPHS AND EQUATIONS OF MOTION Page '%

3. :! all other su$stitutio!s "or u a!d v, ositive values are used %he! these velocities are i!thesamedirectio! as #our ositive re"ere!ce directio!, a!d !egative values are used %he!the# are i! the oppositedirectio!.

+. :" the $od# is u!dergoi!g a !o%! acceleratio! i! the same directio! as #our ositivere"ere!ce directio!, su$stitute this value "or a as a ositive value.

5. :" the $od# is u!dergoi!g a !o%! acceleratio! i! the oosite directio! to that o" #ourositive re"ere!ce directio!, su$stitute this value "or a as a !egative value. A !egativeacceleratio! %hich results i! a $od# slo%i!g do%! is also re"erred to as a "ece$e(ati!)ora (eta("ati!). Ho%ever, !ot all !egative acceleratio!s are !ecessaril# deceleratio!s. A!o$ect %hich is goi!g "aster a!d "aster i! the !egative re"ere!ce directio! is alsou!dergoi!g a !egative acceleratio!, $ut this is !ot a deceleratio! or a retardatio!.

6. :" a $od# is dislaced i! the ositive directio! the! the dislaceme!t itsel" is also ositive.:" the dislaceme!t is i! the oosite directio! to the o!e #ou have chose! as ositive, the!the dislaceme!t has a !egative value.

L. :" the value #ou o$tai! as a solutio! "or a! u!!o%! vector *ua!tit# is ositive, the! thedirectio! o" the vector is the same as #our ositive re"ere!ce directio!. :" the value is

!egative, the! the vector has a directio! oosite to that o" #our ositive re"ere!cedirectio!.

F.


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',-& =OR?ED E@AMPLES

E/a.*$e ': A car travelli!g alo!g a straight road %ith a! i!itial velocit# o" 2 m.s-1

accelerates "or 5 seco!ds a!d reaches a "i!al velocit# o" L m.s-1. alculate(

7a9 the acceleratio! o" the car duri!g the 5 seco!d i!terval)7$9 the average velocit# o" the car duri!g this eriod) a!d7c9 the dislaceme!t o" the car duri!g the same eriod.

A)s+e(s:

7a9 a v - u L m.s-1 - 2 m.s-1 1 m.s-2in the direction of motion. t 5 s

7$9 vav u v 2 m.s-1 L m.s-1 +,5 m.s-1in the direction of motion. 2 2

7c9 ethod 1( s vav.t 7+,5 m.s-1975 s9 22,5 m.s-1in the direction of motion.

ethod 2( s ut Dat2

72 m.s-1975 s9 D71 m.s-2975 s92

10 m 12,5 m 22,5 m in the direction of motion.

ethod 3( s v2 - u2 7L m.s-1 92 - 72 m.s-1 92 7+4 - +9m2.s-2

2a 271 m.s-19 2 m.s-2 22,5 m in the direction of motion.

E/a.*$e 2: A car accelerates u!i"orml# "rom rest a!d reaches a velocit# o" 30 m.s -1 i! atime o" 10 seco!ds, a"ter %hich it travels at this velocit# "or 150 m. The driverthe! alies the $raes a!d the car decelerates u!i"orml# at 5 m.s -2 u!til it

comes to rest. alculate(

7a9 the car&s i!itial acceleratio!)7$9 the time tae! "or the car to come to rest a"ter the $raes are alied)7c9 the total time duri!g %hich the car is i! motio!) a!d7d9 the total dislaceme!t o" the car.

A)s+e(s:

7a9 a1 v1- u1 730 - 09m.s-1 3 m.s-2in the direction of motion. t1 10 s

7$9 t3 v3- u3 70 - 309m.s-1 6 s a3 -5 m.s-27c9 Time to cover 150 m at 30 m.s-1(

t2 s2 150 m 5 s v2 30 m.s-1tTotal t1 t2 t3 10 s 5 s 6 s 21 s

7d9 s1 Da1t12 D73 m.s-29710 s92 150 ms3 u3t3 Da3t32 730 m.s-1976 s9 D7-5 m.s-2976 s92 40 msTotal s1 s2 s3 150 m 150 m 40 m 340 m in the direction of motion.

',-'% ACCELERATION DUE TO GRA5IT6


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GRAPHS AND EQUATIONS OF MOTION Page '2

'!e o" the most imorta!t e/amles o" u!i"orml# accelerated motio! is that o" a "ree-"alli!g$od#. A "alli!g $od# is attracted to%ards the ce!tre o" the earth $ecause $oth the earth a!d the$od# have mass.

The .asso" a $od# is the !ame give! to that roert# o" the $od# %hich e!a$les it to attract

a!d $e attracted $# other $odies ossessi!g the same roert#.

:! the a$se!ce o" air, all o$ects "all %ith the same acceleratio! at the same locatio! !ear theearth&s sur"ace. The acceleratio! is due to the gravitatio!al "orce o" attractio! $et%ee! theearth a!d the o$ect. This "orce is o"te! re"erred to as the 1!(ce !1 g(avit0, or more siml#,g(avit0. The acce$e(ati!) "#e t! g(avit07s#m$ol g9 varies slightl# "rom lace to lace, $ut ithas a! average value at the earth&s sur"ace o" a$out 4,F m.s -2 i! a vacuum. :! calculatio!s #ou%ill use the eve! more aro/imate value o" 10 m.s -2. Acceleratio! due to gravit# is a vector*ua!tit# - its directio! is alwaysverticall# do%!%ards to%ards the ce!tre o" the earth.

A! o$ect "alli!g i! air e/erie!ces a! i!creasi!gl# greater %i!d resista!ce as its velocit#

i!creases. This resista!ce or drag "orce causes a gradual decrease i! the acceleratio! o" theo$ect u!til a velocit# is reached %here the o$ect&s %eight a!d the oosi!g drag "orce acti!go! it $ala!ce o!e a!other 7see Ne+t!)s La+s !1 M!ti!)9. he! this hae!s, the o$ect&sacceleratio! is Cero a!d it "alls at a co!sta!t ma/imum velocit# called its te(.i)a$ ve$!cit0.

Provided the o$ect u!der stud# has a relativel# large mass a!d its velocit# does !ot $ecometoo high, it ca! $e reaso!a$l# assumed that the o$ect %ill "all %ith a co!sta!t acceleratio! o"a$out 10 m.s-2to%ards the ce!tre o" the earth. A! o$ect "alli!g %ith a co!sta!t acceleratio! issaid to $e i! 1(ee 1a$$. Gote that a s#diver does !ot "ree "all accordi!g to the scie!ti"icde"i!itio! o" the term.

',-'' =OR?ED E@AMPLES

E/a.*$e ': A $all is roected verticall# u%ards %ith a! i!itial velocit# o" 3 m.s-1.:g!ori!g air "rictio!, calculate(

7a9 the ma/imum height reached $# the $all) a!d7$9 the time tae! "or the $all to retur! to the grou!d.

A)s+e(s: Taking up as positive.

7a9 M s M M v2 - u2M 70 m.s-1 92 - 73 m.s-1 92 -4 m 0,+5 m 2g 27-10 m.s-29 -20

7$9 tTotal 2 E 7Time tae! to reach ma/imum height9 2.7v - u9 2.70 m.s-1 - 3 m.s-1 9 0,6 s

g 7-10 m.s-29


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GRAPHS AND EQUATIONS OF MOTION Page ',

E/a.*$e 2: A $all is roected horiCo!tall# "rom a to%er %ith a velocit# o" 20 m.s -1. :ttaes 5 seco!ds "or the $all to reach the grou!d. :g!ori!g air "rictio!,calculate(

7a9 the height o" the to%er) a!d

7$9 the horiCo!tal dislaceme!t o" the $all.

A)s+e(s: Taking down as positive.

7a9 o!sider the vertical como!e!t o" the $all&s velocit#(

M svM M uv.t Dgt2M D710 m.s-2975 s92 125 m

7$9 Go% co!sider the horiCo!tal como!e!t o" the $all&s velocit#. The $all&s "light time isthe same "or $oth como!e!ts 7$ecause the $all ca!!ot reach the grou!d at t%odi""ere!t timesJ9. Si!ce %e are ig!ori!g air "rictio!, there are !o u!$ala!ced "orces to

cause the $all to seed u or slo% do%! i! the horiCo!tal. There"ore(

sh uh.t 720 m.s-1975 s9 100 m

E/a.*$e ,: A $all is thro%! verticall# u%ards at 5 m.s-1"rom a lat"orm 30 m a$ove thegrou!d. :g!ori!g air "rictio!, calculate(

7a9 the time tae "or the $all to reach the grou!d) a!d7$9 the seed %ith %hich the $all stries the grou!d.

A)s+e(s: Taking up as positive.

7a9 s ut Dgt2 Dgt2 ut - s 0

t -u 7u2 2gs9D -5 725 27-1097-3099Ds -5 25 s

g -10 -10

t 3 s or t -2 s

Si!ce time ca!!ot $e !egative, t 3 s.

7$9 v2 u2 2as 752 27-1097-3099 m2.s-2 625 m2.s-2

M v M 25 m.s-1


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GRAPHS AND EQUATIONS OF MOTION Page '

=OR?SHEET

1. Stud# the "ollo%i!g grahs a!d descri$e the t#e o" motio! %hich each rerese!ts(

F

sm 0 ts

-F

2. Stud# the "ollo%i!g grahs a!d descri$e the t#e o" motio! %hich each rerese!ts(

5

v7ms9 0 ts

-5

3. =e%rite each o" the "our e*uatio!s o" motio!, mai!g each o" the "our varia$les i!each e*uatio! the su$ect o" the "ormula. @ou should e!d u %ith si/tee! di""ere!te*uatio!sJ Gote that %he! a varia$le is the su$ect o" a "ormula, it o!l# aears o! thele"t-ha!d side o" the e*uatio! a!d it must !ot $e s*uared, etc.

+. =e%rite the "our e*uatio!s o" motio! "or the case %here u 0.

5. 8ra% s-t, v-t, a-t, s-t a!d v-t setch grahs "or a sto!e %hich is thro%! verticall#u%ards i!to the air. Tae t 0 to $e the i!sta!t that the sto!e leaves the thro%er&sha!d, a!d tae u as the ositive directio!. Geglect the e""ects o" air "rictio! a!dassume that the acceleratio! a!d deceleratio! o" the sto!e are u!i"orm.

77

22

''

,

4

88

4

'

2

7

89

,


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6. =eeat *uestio! 5 "or a $ou!ci!g $all. o!sider o!l# o!e $ou!ce, a!d assume that the$all retur!s to the height "rom %hich it %as droed. Tae do%! as the ositivedirectio!.

L. The "ollo%i!g acceleratio!-time ta$le %as o$tai!ed "rom a car road test, i! %hich the

car %as accelerated "rom rest.

t s 1 2 3 + 5 6 L

a m.s-2 + + + + + + +

7a9 Plot the acceleratio!-time grah.7$9 8ra% u a velocit#-time ta$le a!d lot the velocit#-time grah.7c9 Nse #our velocit#-time grah to "i!d the dista!ce covered $# the car i! the "irst 5

seco!ds.7d9 8ra% u a dislaceme!t-time ta$le a!d lot the dislaceme!t-time grah.

F. A car movi!g at 5 m.s-1is accelerated "or 10 s at 2 m.s-2. Ho% "ar does the car travel i!this time a!d %hat is its "i!al velocit#O

Answers: 1! m" # m$s-1in the direction of motion

4. A! o$ect moves "rom rest %ith a u!i"orm acceleratio! o" 10 m.s-2. alculate(

7a9 the seed a"ter 5 s)7$9 the seed a"ter it has covered F0 m) a!d7c9 the dista!ce travelled i! 20 s.

Answers: %a& ! m$s-1 %b& '! m$s-1 %c& # !!! m

10. :! order to tae o"", a! aircra"t re*uires a ru! o" 600 m. :" a"ter starti!g "rom rest ittaes the aircra"t 30 s to tae o"", %hat is the average acceleratio! a!d tae-o"" seedO

Answers: 1,(( m$s-#in the direction of motion" '! m$s-1

11. A $ullet leaves the $arrel o" a ri"le %ith a velocit# o" 1 000 m.s-1. :" the $arrel is 1 mlo!g, calculate the acceleratio! o" the $ullet $e"ore it leaves the ri"le.

Answer: ) 1!m$s-#down the barrel

12. A motor c#clist ridi!g at 20 m.s-1sudde!l# sees a large tree i! the road 120 m ahead.He alies the $raes a!d the motor c#cle decelerates at 2 m.s -2. Ho% "ar a%a# "romthe tree does he come to restO Answer: #! m

13. A car, travelli!g at +0 m.s-1, sudde!l# has its $raes alied a!d is decelerated at10 m.s-2. Ho% "ar %ill it travel $e"ore it comes to restO Answer: *! m

1+. A motor c#clist o! a horiCo!tal road travels %ith a co!sta!t velocit# o" 5 m.s-1a!d the!accelerates u!i"orml# at 2 m.s-2. alculate(

7a9 his seed a"ter 5 s)7$9 the dista!ce travelled duri!g the 5 s) a!d7c9 his seed a"ter travelli!g 150 m "rom the oi!t %here he started to accelerate.

Answers: %a& 1 m$s-1 %b& ! m %c& # m$s-1

15. A car accelerates "rom F m.s-1

to 20 m.s-1

over a dista!ce o" 336 m. alculate(


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7a9 the acceleratio! o" the car) a!d7$9 the time tae! "or the car to cover this dista!ce.

Answers: %a& !, m$s-#in the direction of motion %b& #' s

16. A! o$ect starti!g "rom rest accelerates "or + s a!d reaches a velocit# o" 20 m.s-1. A"ter

travelli!g at a co!sta!t velocit# "or a "urther 3 s it is $rought to rest i! 2 s. 8ra% avelocit#-time grah o" the o$ect&s motio!. alculate its acceleratio!, deceleratio! a!dthe dista!ce covered duri!g the 4 s i!terval.

Answers: m$s-#in the direction of motion" 1! m$s-#in the opposite direction" 1#! m

1L. A! o$ect accelerates u!i"orml# "rom rest at 3 m.s-2"or 5 s, a"ter %hich the acceleratio!ceases. alculate the o$ect&s dislaceme!t a"ter L seco!ds.

Answer: +, m in the direction of motion

1F. A car accelerates a%a# "rom oi!t K at 10 m.s-2over a dista!ce o" +5 m a!d reaches avelocit# o" 50 m.s-1. alculate the seed o" the car as it assed the oi!t K.

Answer: '! m$s-1

14. A uil rolls a $all u a lo!g, i!cli!ed ram. The $all leaves the uil&s ha!d %ith avelocit# o" 16 m.s-1 u the ram, a!d therea"ter its acceleratio! is -2 m.s-2u the ram.

7a9 hat, aart "rom air "rictio! a!d sur"ace "rictio!, causes the $all to decelerate as itrolls u the ramO

7$9 Ho% lo!g %ill it tae "or the $all to reach a oi!t 55 m "rom the uil&s ha!dO7c9 /lai! the h#sical sig!i"ica!ce o" the t%o solutio!s o$tai!ed i! 7$9 a$ove.7d9 alculate the ossi$le velocities o" the $all %he! it is 55 m "rom the $ottom o" the

ram.

7e9 alculate the time tae! "or the $all to come to rest $e"ore rolli!g $ac, a!d he!cecalculate ho% "ar it rolls u the ram.7"9 State, %ithout a!# "urther calculatio!, the time tae! "or the $all to retur! to its

starti!g oi!t, a!d its velocit# at this mome!t.7g9 8ra% a "ull#-la$elled velocit#-time grah "or the $all "rom the time that it leaves

the uil&s ha!d to the time that it retur!s.7h9 Nsi!g the aroriate "eature o" the a$ove grah, deduce - %ith reaso!s - the

acceleratio! o" the $all %he! it is "urthest "rom its starti!g oi!t.Answers: %b& s and 11 s %d& + m$s-1and -+ m$s-1 %e& * s" +' m %f& 1+ s" -1+ m$s-1

%h& -# m$s-#

20. A car, travelli!g i!itiall# at a co!sta!t seed o" 10 m.s-1

, u!dergoes a u!i"ormacceleratio! o" + m.s-2a!d travels L2 m %hile doi!g so.

7a9 alculate the time tae! to travel the L2 m.7$9 '!e o" the t%o solutio!s o$tai!ed i! 7a9 is h#sicall# i!admissi$le. /lai! the

sig!i"ica!ce o" the !egative sig! a!d suggest %h# this other solutio! - %hile $ei!gmathematicall# correct - is imossi$le.

7c9 alculate the seed o" the car a"ter it has travelled the L2 m.Answers: %a& 's and -. s %c& #+ m$s-1

21. A sto!e, "alli!g "rom rest "rom the to o" a $ridge, stries the %ater $elo% 5 s later.Ho% high is the $ridge a$ove the %ater, a!d %ith %hat seed does it e!ter the %aterO

Answers: 1# m" ! m$s-1


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22. Assumi!g a streamli!ed $om$ "alls %ithout air resista!ce, ho% "ar %ill it !eed to "all,a!d "or ho% lo!g, i! order to $rea the sou!d $arrier o" 330 m.s-1O

Answer: '' m" (( s

23. he! a $o# thro%s a $all straight u i!to the air he "i!ds that it taes 2,5 s "or the $all

to reach its ma/imum height.

7a9 At %hat seed did the $all leave his ha!dO7$9 hat height did the $all reachO7c9 ith %hat seed does the $all retur! to his ha!dO7d9 Ho% lo!g does the do%!%ard tri taeO7e9


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PRACTICALS

E@PERIMENT ',-'

PRE3LA


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GRAPHS AND EQUATIONS OF MOTION Page '&

APPARATUS:

Ticer timer a!d mou!ti!g $racet) car$o!ised ticer tae) la$orator# o%er sul#)2 co!!ecti!g leads) masi!g tae) so!ge ru$$er mat) metal $locs o" various masses $ut !otless tha! 100 g.

PROCEDURE:

1. Puils are to %or i! grous o" three or "our, %ith each grou mem$er reso!si$le "oro$tai!i!g a!d a!al#si!g a ticer tae "or a! o$ect o" di""ere!t mass to those droed $#the other mem$ers o" the grou.

2. Place a ta$le o! to o" the cu$oards do%! the side o" the la$orator# a!d mou!t a ticertimer to it usi!g the $racet sulied. Place the mat o! the "loor $elo% the timer.

3. Nsi!g the co!!ecti!g leads, co!!ect the ticer timer to the o%er sul#&s a.c. termi!alsa!d select the correct voltage 7see voltage rati!g o! ticer timer9.

+. Nsi!g the mi!imum amou!t o" masi!g tae, attach a stri o" car$o!ised ticer tae o"

a$out 1,5 metres i! le!gth to the o$ect that is to $e droed.5.


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GRAPHS AND EQUATIONS OF MOTION Page 2%

+. alculate the average velocit# 7i! ce)ti.et(es *e( sec!)"9 "or each i!terval $# dividi!gthe dislaceme!t "or that i!terval $# 0,1 s.

5. Nse the li!ear regressio! rogramme 7availa$le at the omuter e!tre9 to a!al#se #ouro%! data set a!d "it the $est straight li!e to it. =emem$er to record the values o" aa!d bi! the e*uatio! 0 B a b/determi!ed $# the regressio! rogramme.

6. 8ra% the a/es "or a grah o" average velocit# 7i! cms9 o! the #-a/is versus totalelasedtime 7i! seco!ds9 o! the /-a/is, usi!g the grah aer sulied. ver# mem$er o" #ourgrou must use the same scale a!d dra% the a/es i! e/actl# the same %a# i! order toreduce #our %or load 7see $elo%9.

L. ach mem$er must the! lot the data oi!ts "or his her o$ect a!d the t%o oi!ts at thee!ds o" the $est straight li!e "itti!g these data oi!ts. B# laci!g the searate ieces o"grah aer over each other, the data oi!ts "or the other mem$ers& o$ects ca! $etra!s"erred $# rici!g a hole through the data oi!ts o! the uermost sheet o" grah

aer usi!g a comass or divider oi!t. To avoid co!"usio!, each set o" oi!ts should $eclearl# mared %ith crosses, circles, tria!gles or s*uares $e"ore the !e/t set is tra!s"erred.

F. The value o" bis the sloe o" the grah a!d rerese!ts the acceleratio! due to gravit# o"

the o$ect co!cer!ed. o!vert the values "or bto metres er seco!d s*uared 7ms2

9 a!dta$ulate them i! #our reort. @ou should "i!d that these values are all ver# similar 7asi!dicated $# the "act that #our grahs should all $e ver# !earl# arallel9. 8etermi!e a!average value "or the acceleratio! due to gravit# 7g9.

I)te(va$N#.be(

Dis*$ace.e)ts; c.

Ti.e *e(I)te(va$t; s

T!ta$ E$a*se"Ti.et; s

Ave(age5e$!cit0

st; c.s;

1 0,1 0,1

2 0,1 0,2

3 0,1 0,3+ 0,1 0,+

5 0,1 0,5

6 0,1 0,6

L 0,1 0,L

Plot o! /-a/is

Plot o! #-a/is

CONCLUSION:

@ou must state the value o" ga!d comme!t o! the "act that the grahs are arallel 7see AIMa$ove9.


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GRAPHS AND EQUATIONS OF MOTION Page 2'

POST3LA


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ALTERNATI5E METHODS FOR DETERMINING g

Several di""ere!t methods ca! $e used to determi!e a value "or the acceleratio! due to gravit#.Si!ce g varies "rom o!e locatio! to a!other, it is imorta!t to $e a$le to determi!e its value*uicl# a!d accuratel# so that it ca! $e used i! other e/erime!ts %hich re*uire the value o"

g i! calculatio!s. Provided #ou have su""icie!t time, #our teacher ma# sul# #ou %ith thee*uime!t #ou !eed to er"orm o!e or more o" the "ollo%i!g e/erime!ts.

E@PERIMENT ',-2: THE GALILEO METHOD

This is the least accurate method $ecause it relies o! huma! re"le/es to start a!d sto asto%atch. :t is there"ore most accuratel# er"ormed over a ver# great height. The methodi!volves droi!g a massive o$ect through a !o%! height a!d measuri!g the time tae! "orit to strie the grou!d.


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GRAPHS AND EQUATIONS OF MOTION Page 2,

The time tae! to comlete several comlete oscillatio!s ca! $e measured %ith a! ordi!ar#sto%atch.

E@PERIMENT ',-4: FREE3FALL APPARATUS

This method emlo#s a digital timer %hich starts cou!ti!g %he! a $all $eari!g is released $#a! electromag!et at the to o" a ga!tr#. he! the $all $eari!g stries a!d oe!s a gate at the

$ottom o" the ga!tr#, the timer stos cou!ti!g. The dista!ce "alle! ca! $e accuratel#measured, a!d the timer ca! record thousa!dths o" a seco!d. Ho%ever, this method iscomlicated $# the "act that $all $eari!gs are !ot made o" so"t iro! a!d there"ore do !ot de-mag!etise the mome!t the electromag!et is s%itched o"". The rete!tivit# o" the metal causesthe $all $eari!g to ha!g u "or a "e% thousa!dths o" a seco!d. This ro$lem ca! $e overcome

$# droi!g the $all $eari!g over several di""ere!t dista!ces a!d "i!di!g the sloe o" a grah o"average velocit# versus time 7as i! the Ticer Tae e/erime!t9. This method is the simlesta!d most accurate o" all the methods outli!ed here. As #our teacher "or the ri!ted

i!structio!s.

E@PERIMENT ',-7: STRO


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NAME:

TUTORIAL

omlete the "ollo%i!g ta$le $# recordi!g #our a!s%ers i! the $la! saces. :t isrecomme!ded that #ou esta$lish a! algorithm "or each t#e o" ro$lem. 8etach a!d su$mitthis tutorial "or mari!g.

Go.

:!itial velocit#

7#9i! m.s-1


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SUMMAR6

EQUATIONS OF MOTION

v u at

v2 u2 2as

s ut Dat2

s 7u v9.t vav.t 2

GRAPHS

GRAPH SLOPE AREA GI5ES DOES NOTGI5E

s vs t v Go mea!i!g.Positio! relativeto re"ere!ce

oi!t) directio!o" travel)directio! o"acceleratio!.

v vs t a s8irectio! o"travel) directio!o" acceleratio!.

Positio!.

a vs tGot alica$leto u!i"orml#acceleratedmotio!

v8irectio! o"acceleratio!o!l#.

Positio!)directio! o"travel.

NOTE:

6!# +i$$ !)$0 c!.e ac(!ssparabolic curves!) "is*$ace.e)t3ti.e g(a*s- These are o"

the "orm # a/2 $/ c 7i.e. s Dat2 ut9. These curves are *uadratic i! to!l# i" theacceleratio! is !o!-Cero. :" the acceleratio! isCero the! the e*uatio! simli"ies to s ut.Thus, a ara$olic curve o! a dislaceme!t-time grah is a result o" u!i"orml# acceleratedmotio!. 0isplacement-time graphs will therefore be parabolic if acceleration is taking

place, or straight lines if the is no acceleration.

6!# +i$$ !)$0 c!.e ac(!ss ve$!cit03ti.e g(a*s +ic a(e st(aigt $i)es-

6!# +i$$ !)$0 e)c!#)te( acce$e(ati!)3ti.e g(a*s +ic a(e *a(a$$e$ t! te ti.e a/is-

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