11 mechanics 1
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Chapter 11:
Mechanics 1: Linear Kinematics &
Calculus
Ian ParberryUniversity of North Texas
Fletcher DunnValve Software
3D Math Primer for Graphics & Game Development
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What Youll ee in !his Chapter
!his chapter "i#es a taste o$ linear %inematics an calculus' It is
i#ie into ei"ht sections'
ection 11'1 "i#es an o#er#ie( o$ (hat (e hope to achie#e'
ection 11') tal%s about basic *uantities an units' ection 11'+ introuces a#era"e #elocity'
ection 11', loo%s at instantaneous #elocity an the eri#ati#e'
ection 11'- is about acceleration'
ection 11'. iscusses motion uner constant acceleration'
ection 11'/ loo%s at acceleration an the inte"ral'
ection 11'0 eamines uni$orm circular motion'
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Wor Clou
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ection 11'1:
4#er#ie(
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5 Moest Proposal
5$ter reain" this chapter6 you shoul %no(:
!he basic iea o$ (hat a eri#ati#e
measures an (hat it is use $or' !he basic iea o$ (hat an inte"ral
measures an (hat it is use $or'
Deri#ati#es an inte"rals o$ tri#ialepressions containin" polynomials an
tri" $unctions'
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7o( Much Calculus is 2eee8
1' I %no( absolutely nothin" about eri#ati#es or inte"rals'
)' I %no( the basic iea o$ eri#ati#es or inte"rals6 but probably couln9t
sol#e any $reshman calculus problems (ith a pencil an paper'
+' I ha#e stuie some calculus'
Le#el ) %no(le"e o$ calculus is su$$icient $or this boo%6 an our
"oal is to mo#e e#eryboy (ho is currently in cate"ory 1 into
cate"ory )' I$ you9re in cate"ory +6 our calculus iscussions (ill
be a hope$ully entertainin"; re#ie(' We ha#e no elusions that
(e can mo#e anyone into cate"ory + (ho is not alreay there'
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Doublethin% 5bout Discreteness &
Continuity
!here is stron" e#ience that the
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Classical Mechanics
We are "oin" to stuy classical mechanics6 also %no(n as
Newtonian mechanics6 (hich has se#eral simpli$yin"
assumptions that are incorrect in "eneral but true in
e#eryay li$e in most (ays that really matter to us:
!ime is absolute
pace is =ucliian
Precise measurements are possible
!he uni#erse ehibits causality an complete preictability!he $irst t(o are shattere by relati#ity6 the secon t(o by
*uantum mechanics' !han%$ully6 these t(o sub>ects are not
necessary $or #ieo "ames'
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Particles an Dimensions
We aim6 in this chapter6 to o the math to "et e*uations
that preict the position6 #elocity6 an acceleration o$ a
particle at any "i#en time t'
?ecause (e are treatin" our ob>ects as particles6 (e (illnot consier their orientation or rotational e$$ects until
Chapter 1)'
When rotation is i"nore6 all o$ the ieas o$ linear
%inematics eten into +D in a strai"ht$or(ar (ay6 anso $or no( (e (ill be limitin" oursel#es to )D an 1D;'
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ection 11'):
?asic Auantities an
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Len"th6 !ime6 an Mass
Mechanics is concerne (ith the relationship amon" three $unamental
*uantities in nature: length6 time6 an mass'
engthis a *uantity you are no oubt $amiliar (ith' We measure len"th
usin" units li%e centimeters6 inches6 meters6 an $eet'
Timeis another *uantity (e are #ery com$ortable (ith measurin"' Wemeasure time usin" units li%e secon6 minute6 an hour'
!he *uantity massis not *uite as intuiti#e as len"th an time' !he
measurement o$ an ob>ect9s mass is o$ten thou"ht o$ as measurin" the
amount o$ stu$$ in the ob>ect'
!his is not a ba e$inition6 but its not *uite ri"ht'
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Mass an Wei"ht
Mass is o$ten con$use (ith weight6 especially since the units use to
measure mass are also use to measure (ei"ht: the "ram6 poun6
%ilo"ram6 ton6 etc'
!he mass o$ an ob>ect is an intrinsic property6 (hile its (ei"ht is a local
phenomenon that epens on the stren"th o$ the "ra#itational pull eerte
by a nearby massi#e ob>ect'
Your mass (ill be the same (hether you are in Chica"o6 or on the moon6 or
near Eupiter6 or li"ht years a(ay $rom the nearest hea#enly boy6 but in
each case your (ei"ht (ill be #ery i$$erent'
In this boo% an in most #ieo "ames our concerns are con$ine to a
relati#ely small patch on a $lat =arth6 an (e (ill approimate "ra#ity by a
constant o(n(ar pull'
It (on9t be too harm$ul to con$use mass an (ei"ht because "ra#ity $or us
(ill be a constant'
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ection 11'+:
5#era"e elocity
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tory o$ the !ortoise & the 7are Mathersion;
4nce upon a time there (as a tortoise an a hare'
!he a#era"e #elocity o$ the tortoise is "reater than thea#era"e #elocity o$ the hare'
!he =n'
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!he !ortoise an the 7are
!he "un "oes o$$ at time tB'
!he hare sprints ahea to time t16 then slo(s'
5t time t) a istraction passes by in the opposite irection' !he hare
turns aroun an (al%s (ith her'
5t time t+ he "i#es up on her an be"ins to pace bac% an $orth
alon" the trac% e>ectely until time t,6 (hen he ta%es a nap'
Mean(hile the tortoise has been ma%in" slo( an steay pro"ress6
an at time t- he catches up (ith the sleepin" hare'
!he tortoise plos alon" an crosses the tape at t.'
!he hare (a%es up at time t/ an hurries in a $renGy to the $inish'
5t time t0 the hare crosses the $inish line'
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5#era"e elocity
(here is the position o$ the hare at time '
I$ (e ra( a strai"ht line throu"h any t(opoints on the "raph o$ the hare9s position6
then the slope o$ that line measures the
hares a#era"e #elocity o#er the timeinter#al bet(een the t(o points'
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=ample
Consier the a#era"e
#elocity o$ the hare as he
ecelerates $rom time t1
to t)6 as sho(n here'
!he slope o$ the line is
the ratio '
!his slope is also e*ualto the tan"ent o$ an"le
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i"n o$ 5#era"e elocity
5#era"e #elocity can e#en be ne"ati#e or
B'
It is Gero (hen '
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ection 11',:
Instantaneous elocity & the
Deri#ati#e
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What is Instantaneous elocity8
Instantaneous #elocity is #elocity at a sin"le
point in time'
o $ar (e#e only e$ine a#era"e #elocity
o#er a time perio' Hecall:
?ut this $ails (hen i#ie by Gero error;' oho( are (e "oin" to e$ine instantaneous
#elocity8
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5n =asy Case
Instantaneous #elocity is easy(hen #elocity is a constant $ora nonGero perio o$ time'
!he #elocity "raph (ill be astrai"ht line'
!he har part is(hen #elocityis chan"in"'
!he #elocity "raph (ill notbea strai"ht line'
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ir Isaac 2e(ton to the Hescue
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Ima"e: Wi%imeia
Commons
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7ere We 3o
Put some concrete units o$ time an
space on it minutes an $urlon"s;'
What (as the hares instantaneous
#elocity at min8
For a small enou"h inter#al the "raph isnearly a strai"ht line se"ment an the
#elocity is nearly constant'
o the instantaneous #elocity at any "i#en
instant (ithin the inter#al (ill be near the
a#era"e #elocity o#er the (hole inter#al'
Lets try #aryin" '
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Calculus
7ope$ully you#e ta%en Freshman Calculus'
I$ not6 theres a summary in the boo% ections
11',') to 11','/;'
11','): =amples o$ Deri#ati#es
11','+: Calculatin" Deri#ati#es $rom the De$inition
11',',: 2otation
11','-: 5 Fe( Hules an hortcuts
11','.: Deri#ati#es (ith !aylor eries
11','/: !he Chain Hule
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ection 11'-:
5cceleration
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What is 5cceleration8
5cceleration is rate o$ chan"e o$ #elocity'
5cceleration is a #ector'
For eample6 the acceleration ue to "ra#ity is about
+) $ts)6 e*ui#alently @'0 ms) o(n(ars'
!he #elocity at an arbitrary time to$ an ob>ect uner
constant acceleration a is "i#en by the simple linear
$ormula vt; J vB at6 (here vB is the initial #elocity
at time tJ B'
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4bser#ations
Where the acceleration is Gero6 the #elocity is constant an the
position is a strai"ht but possibly slope; line'
Where the acceleration is positi#e6 the position "raph is cur#e
li%e 6 an (here it is ne"ati#e6 the position "raph is cur#e li%e '
!he most interestin" eample occurs on the ri"ht sie o$ the"raphs' 2otice that at the time (hen the acceleration "raph
crosses aJ B6 the #elocity cur#e reaches its ape6 an the
position cur#e s(itches $rom to '
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More 4bser#ations
5 iscontinuity in the #elocity $unction causes a %in% in the position "raph' Furthermore6 it
causes the acceleration to become in$inite actually6 une$ine;' uch iscontinuities on9t
happen in the real (orl'
!his is (hy the lines in the #elocity "raph are connecte at those iscontinuities6 because
the "raph is o$ a physical situation bein" approimate by a mathematical moel'
5 iscontinuity in the acceleration "raph causes a %in% in the #elocity "raph6 but notice that
the position "raph is still smooth' In $act6 acceleration can chan"e instantaneously6 an $or
this reason (e ha#e chosen not to bri"e the iscontinuities in the acceleration "raph'
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ection 11'.:
Motion
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Motion
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Pro>ectile Motion
Pro>ectile motion is acceleration uner
"ra#ity'
For simplicity6 (e i"nore (in resistance' 4ut "oal is a $unctionxt; $or the position
o$ a pro>ectile at time t'
Its con$usin"6 but (ere "oin" to usex$or#ertical istance here'
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2umerical 5pproimation
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2umerical 5pproimation )
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Con#er"ence
!he approimations "et better as the
number o$ time slices increase'
We say that it convergesto the correct
#alue'
5cceleration is the areauner the #elocity
"raph' We "et a better approimation as the
number o$ slices increases'
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5rea
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=ample
Auestion: 7o( $ar (ill
an ob>ect thro(n
o(n(ars $rom the top
o$ a tall builin" at -
$tsec tra#el in )',
secons8
5ns(er: !he area uner
vt; $rom tJB to tJ)','
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Hemember !his Formula
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o the 5ns(er Is
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ection 11'/:
!he Inte"ral
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ection 11'0:
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!hat conclues Chapter 11' 2et6 Chapter 1):
Mechanics ): Linear & Hotational
Dynamics