CHAPTER
11 Kinematics of Particles
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icle
Motion
Deendent Motion
Pro!lems
Acceleration
Rectanular *omonents of Velocity and
Acceleration
anential and .ormal *omonents
Radial and rans%erse *omonents
Samle Pro!lem 11"1/
Samle Pro!lem 11"12
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0 Dynamics includes:
-  Kinematics: study of the eometry of motion" inematics is used to
relate dislacement, %elocity, acceleration, and time ithout reference to
the cause of motion"
-  Kinetics: study of the relations e3istin !eteen the forces actin on a
 !ody, the mass of the !ody, and the motion of the !ody" inetics is used
to redict the motion caused !y i%en forces or to determine the forces
re4uired to roduce a i%en motion"
0  Rectilinear  motion: osition, %elocity, and acceleration of a article as it
mo%es alon a straiht line"
0  Curvilinear  motion: osition, %elocity, and acceleration of a article as it
 
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Rectilinear Motion: Position, Velocity & Acceleration
0 Particle mo%in alon a straiht line is said
to !e in rectilinear motion"
0  Position coordinate of a article is defined
 !y ositi%e or neati%e distance of article
from a fi3ed oriin on the line"
0 he motion of a article is 5non if the
 osition coordinate for article is 5non for
e%ery %alue of time t " Motion of the article
may !e e3ressed in the form of a function,
e"", #26   t t  x   −=
 
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0 Instantaneous %elocity may !e ositi%e or
neati%e" Manitude of %elocity is referred
to as particle speed "
 P  at time t  and P’  at t 7t ,
t 
dt 
dx
t 
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v’  at t 7t ,
 Instantaneous acceleration t 
0 Instantaneous acceleration may !e:
-  ositi%e: increasin ositi%e %elocity
or decreasin neati%e %elocity
- neati%e: decreasin ositi%e %elocity
or increasin neati%e %elocity"
 
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0 *onsider article ith motion i%en !y
#26   t t  x   −=
dx v   −==
t  dt 
−===
0 at t  9 /,  x 9 /, v 9 /, a 9 12 ms2
0 at t  9 2 s,  x 9 16 m, v 9 vmax 9 12 ms, a 9 /
 
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Determination of te Motion of a Particle
0 Recall, motion of a article is 5non if osition is 5non for all time t "
0 yically, conditions of motion are secified !y the tye of acceleration
e3erienced !y the article" Determination of %elocity and osition re4uires
to successi%e interations"
0 hree classes of motion may !e defined for:
 
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Determination of te Motion of a Particle
( ) ( ) ( )
( ) ( ) ( )
( ) ( ) ( )
( ) ( ) ( )∫ ∫ ∫ 
∫ ∫ ∫ 
=−===
=−====
v
dt t v xt  xdt t vdxdt t vdxt v dt 
dx
dt t  f vt vdt t  f dvdt t  f dvt  f a dt 
dv
/ /
/
/ /
/
/
/
( )
( ) ( )
( ) ( ) ( )∫ ∫ ∫    =−==
=====
 x
 x
 x
 x
 xv
v
dx x f v xvdx x f dvvdx x f dvv
 x f  dx
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Determination of te Motion of a Particle
( ) ( ) ( )
( )
( )
( )
( ) ( )
( )
( )
( )
( ) ( )
( )
∫ 
∫ ∫ 
∫ 
∫ ∫ 
=−
====
=
====
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time t ,
corresondin time, and
corresondin %elocity"
from indo 2/ m a!o%e round"
S)@$I).:
0 Interate tice to find v<t = and y<t ="
0 Sol%e for t  at hich %elocity e4uals
ero <time for ma3imum ele%ation=
and e%aluate corresondin altitude"
0 Sol%e for t  at hich altitude e4uals
ero <time for round imact= and
e%aluate corresondin %elocity"
 
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t v dt 
   
   − 
   
   +=
S)@$I).:
 
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!am"le Pro#lem $$%
0 Sol%e for t  at hich %elocity e4uals ero and e%aluate
corresondin altitude"
   −=   t t v
s/1>"1=t 
0 Sol%e for t  at hich altitude e4uals ero and e%aluate
corresondin %elocity"
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!am"le Pro#lem $$%
0 Sol%e for t  at hich altitude e4uals ero and
e%aluate corresondin %elocity"
( ) / s
m >/'"
s
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recoil consists of iston attached to !arrel
mo%in in fi3ed cylinder filled ith oil"
As !arrel recoils ith initial %elocity v0,
 iston mo%es and oil is forced throuh
orifices in iston, causin iston and
cylinder to decelerate at rate roortional
to their %elocity"
kva   −=
S)@$I).:
0 Interate a 9 dv/dt  9 -kv to find v<t ="
0 Interate v<t = 9 dx/dt   to find x<t ="
0 Interate a 9 v dv/dx 9 -kv to find
v< x="
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S)@$I).:
0 Interate a 9 dv/dt  9 -kv to find v<t ="
( ) ( ) kt 
v
( )   kt evt v   −= /
( )
( )
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!am"le Pro#lem $$%'
0 Interate a 9 v dv/dx 9 -kv to find v< x="
kxvv
( )   ( )kt e k 
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(niform Rectilinear Motion
+or article in uniform rectilinear motion, the acceleration is ero and
the %elocity is constant"
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+or article in uniformly accelerated rectilinear motion, the acceleration of
the article is constant"
at t v x xdt at vdxat v dt 
dx   t  x
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0 +or articles mo%in alon the same line, time
should !e recorded from the same startin
instant and dislacements should !e measured
from the same oriin in the same direction"
=−=   A B A B   x x x relati%e osition of B 
ith resect to A  A B A B   x x x   +=
 
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in ele%ator shaft ith initial %elocity of
1; ms" At same instant, oen-latform
ele%ator asses ' m le%el mo%in
uard at 2 ms"
Determine a! hen and here !all hits
ele%ator and "! relati%e %elocity of !all
and ele%ator at contact"
S)@$I).:
eneral e4uations for uniformly
%elocity of ele%ator into e4uation for
uniform rectilinear motion"
 !all ith resect to ele%ator and sol%e
for ero relati%e osition, i"e", imact"
0 Su!stitute imact time into e4uation
for osition of ele%ator and relati%e
%elocity of !all ith resect to
ele%ator"
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0 Su!stitute initial osition and %elocity and constant
acceleration of !all into eneral e4uations for
uniformly accelerated rectilinear motion"
t at vv
ele%ator into e4uation for uniform rectilinear
motion"
v
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!am"le Pro#lem $$%*
0 Brite e4uation for relati%e osition of !all ith resect to
ele%ator and sol%e for ero relati%e osition, i"e", imact"
( )   ( ) /2'>/'"1;12 2 =+−−+=   t t t  y  #  B
( )
0 Su!stitute imact time into e4uations for osition of
ele%ator and relati%e %elocity of !all ith resect to ele%ator"
( )6'"#2'+= #  y
m#"12= #  y
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Motion of !e)eral Particles: De"endent Motion
0 Position of a article may depend  on osition of one
or more other articles"
0 Position of !loc5 B deends on osition of !loc5 A$
Since roe is of constant lenth, it follos that sum of
lenths of sements must !e constant"
=+   B A   x x 2 constant <one deree of freedom=
0 Positions of three !loc5s are deendent"
=++   C  B A   x x x 22 constant <to derees of freedom=
0 +or linearly related ositions, similar relations hold
 !eteen %elocities and accelerations"
/22or /22
/22or /22
aaa dt 
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is ulled don at # in"s" At t  9 /,
collar A starts mo%in don from K  
ith constant acceleration and ero
initial %elocity" noin that
%elocity of collar A is 12 in"s as it
 asses &, determine the chane in
ele%ation, %elocity, and acceleration
of !loc5 B hen !loc5 A is at &"
S)@$I).:
0 Define oriin at uer horiontal surface
ith ositi%e dislacement donard"
0 *ollar A has uniformly accelerated
rectilinear motion" Sol%e for acceleration
and time t  to reach &"
0 Pulley % has uniform rectilinear motion"
*alculate chane of osition at time t "
0 ?loc5 B motion is deendent on motions
of collar A and ulley %" Brite motion
relationshi and sol%e for chane of !loc5
 B osition at time t "
0 Differentiate motion relation tice to
de%elo e4uations for %elocity and
acceleration of !loc5 B"
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!am"le Pro#lem $$%+ S)@$I).:
0 Define oriin at uer horiontal surface ith
 ositi%e dislacement donard"
0 *ollar A has uniformly accelerated rectilinear
( ) ( )[ ]
aa
 
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!am"le Pro#lem $$%+ 0 Pulley % has uniform rectilinear motion" *alculate
chane of osition at time t "
( )
t v x x
0 ?loc5 B motion is deendent on motions of collar
 A and ulley %" Brite motion relationshi and
sol%e for chane of !loc5 B osition at time t "
otal lenth of ca!le remains constant,
( ) ( ) ( )
( )[ ]   ( )[ ]   ( )[ ]
( ) in"16/   −=−   B B   x x
 
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e4uations for %elocity and acceleration of !loc5 B"
/ s
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ra"ical !olution of Rectilinear-Motion Pro#lems
0 (i%en the x't  cur%e, the v't  cur%e is e4ual to
the x't  cur%e sloe"
0 (i%en the v't  cur%e, the a't  cur%e is e4ual to
the v't  cur%e sloe"
 
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ra"ical !olution of Rectilinear-Motion Pro#lems
0 (i%en the a't  cur%e, the chane in %elocity !eteen t ( and t ) is
e4ual to the area under the a't  cur%e !eteen t ( and t )"
0 (i%en the v't  cur%e, the chane in osition !eteen t ( and t ) is
 
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0  *oment'area met+od  to determine article osition
at time t  directly from the a't  cur%e:
( )∫    −+=
−=−
( )∫    −+=− 1
( )   =−∫  1
1
v
v
dt at t  first moment of area under a't  cur%e
ith resect to t  9 t ( line"
( ) ( )
 centroidof a!scissa
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from v'x cur%e:
 
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Cur)ilinear Motion: Position, Velocity & Acceleration
0 Particle mo%in alon a cur%e other than a straiht line
is in curvilinear motion"
0  Position vector  of a article at time t  is defined !y a
%ector !eteen oriin - of a fi3ed reference frame and
the osition occuied !y article$
0 *onsider article hich occuies osition P
defined !y at time t  and P’  defined !y at t .
t ,
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=
=
0 *onsider %elocity of article at time t  and %elocity
at t . t ,
0 In eneral, acceleration %ector is not tanent to
 article ath and %elocity %ector"
 
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( ) ( )
du
d 
du
 P  f d    +=
0 Deri%ati%e of roduct of scalar and %ector functions,
( )
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Rectan0ular Com"onents of Velocity & Acceleration
0 Bhen osition %ector of article P  is i%en !y its
rectanular comonents,
0 Velocity %ector,
k v 1viv
d   1
  d   1
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0 Rectanular comonents articularly effecti%e
hen comonent accelerations can !e interated
indeendently, e"", motion of a roCectile,
//   ==−====    a g  ya xa  0  y x  
ith initial conditions,
( )   ( )
( )   ( ) /
/
v gt vvvv
0 Motion in horiontal direction is uniform"
0 Motion in %ertical direction is uniformly accelerated"
0 Motion of roCectile could !e relaced !y to
indeendent rectilinear motions"
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Motion Relati)e to a /rame in Translation 0 Desinate one frame as the fixed frame of reference"
All other frames not riidly attached to the fi3ed
reference frame are moving frames of reference"
0 Position %ectors for articles A and B ith resect to
the fi3ed frame of reference -xy  are "and  B A   r r 
0 Vector Coinin A and B defines the osition of
 B ith resect to the mo%in frame Ax’y’’  and  A Br 
 A B A B   r r r    +=
0 Differentiatin tice,
= A Bv %elocity of B relati%e to A" A B A B   vvv   +=
= A Ba acceleration of B relati%e
to A"  A B A B   aaa   +=
0 A!solute motion of B can !e o!tained !y com!inin
motion of A ith relati%e motion of B ith resect to
mo%in reference frame attached to A"
 
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Tan0ential and 1ormal Com"onents
0 Velocity %ector of article is tanent to ath of
 article" In eneral, acceleration %ector is not"
Bish to e3ress acceleration %ector in terms of
tanential and normal comonents"
0   are tanential unit %ectors for the
 article ath at P  and P’ " Bhen dran ith
resect to the same oriin, and
is the anle !eteen them"
t t    ee   ′and
( )
( )
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t evv =0 Bith the %elocity %ector e3ressed as
the article acceleration may !e ritten as
dt 
ds
ds
d 
d 
chane of seed and normal comonent reflects
chane of direction"
0 anential comonent may !e ositi%e or
neati%e" .ormal comonent alays oints
toard center of ath cur%ature"
 
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0 Relations for tanential and normal acceleration
also aly for article mo%in alon sace cur%e"
0 Plane containin tanential and normal unit
%ectors is called the osculating plane"
nt "   eee   ×=
"inormal e
normal  principale
 
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Radial and Trans)erse Com"onents 0 Bhen article osition is i%en in olar coordinates,
it is con%enient to e3ress %elocity and acceleration
ith comonents arallel and erendicular to -P "
r  r  e
θ    θ ==
0 Similarly, the article acceleration %ector is
( )   ( )   θ 
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Radial and Trans)erse Com"onents 0 Bhen article osition is i%en in cylindrical
coordinates, it is con%enient to e3ress the
%elocity and acceleration %ectors usin the unit
%ectors "and,,   k ee R   θ 
0 Position %ector,
r d  v  R   ++==   θ θ 
0 Acceleration %ector,
vd  a  R   +++−==   θ θ θ θ  22
 
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section of hihay at 6/ mh" he
motorist alies !ra5es causin a
constant deceleration rate"
 !een reduced to ' mh, determine
the acceleration of the automo!ile
immediately after the !ra5es are
alied"
S)@$I).:
comonents of acceleration"
direction ith resect to tanent to
cur%e"
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acceleration"
ith resect to tanent to cur%e"
( ) 2222 1/"#8'"2   +−=+=   nt    aaa 2s
ft 1"=a
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Rotation of the arm a!out ) is defined
 !y θ  2 /"1't 2here θ  is in radians and t  
in seconds" *ollar ? slides alon the
arm such that r  9 /"> - /"12t 2 here r  is
in meters"
determine a! the total %elocity of the
collar, "! the total acceleration of the
collar, and c! the relati%e acceleration
of the collar ith resect to the arm"
S)@$I).:
0 %aluate radial and anular ositions,
and first and second deri%ati%es at
time t "
cylindrical coordinates"
arm"
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0 %aluate time t  for θ  9 #/o"
s;6>"1rad'2"/#/
/"1' 2
and second deri%ati%es at time t "
2
2
sm2"/
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( )( )
 
© 2007 The McGraw-Hill Companies, Inc. All rights reserve.
 
Motion of collar ith resect to arm is rectilinear
and defined !y coordinate r "
2sm2/"/−== r a -A B