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'

Chapter 11 2otes +D Math Primer $or3raphics & 3ame De#

)

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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'


1)

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1+

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ection 11'+:

5#era"e elocity


1,

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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'


1-

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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'


1.

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1/

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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


1@

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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 '


)B

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ection 11',:

Instantaneous elocity & the

Deri#ati#e


)1

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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


),

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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).

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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'


+B

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+1

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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 )


+0

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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)','


,1

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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

11 Mechanics 1

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