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Fundamental of Physics
PETROVIETNAM UNIVERSITY
FUNDAMENTAL SCIENCES DEPARTMENT
Hanoi, August 2012
Pham Hong QuangE-mail: [email protected]
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References
Pham Hong Quang Fundamental Science Department 2
[1] Cơ sở Vật lý, Tập I, II, III: Cơ học, nhiệt học, David
Halliday, Robert Resnik, Jearl Walker, bản dịch tiếng
Vit, !"# $i%o d&c '1((()*
[+] Fundamentals o ph!sics, "th ed#, $%tended,
David Halliday, Robert Resnick and Jearl Walker, John
Wiley -ons '+../)*
[0] Vt l2 34i c56ng, t7 1, 869!hit, :56ng D;y<n
#=nh ch> bi<n
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Chapter 1
Pham Hong Quang Fundamental Science Department 3
1.1 Measurement, Standards of
Length, Mass, and Time
1.2 Dimensional Analysis
1.3 on!ersion of "nits
1.# oordinate Systems
1.$ %e&tors and s&alars
1.' The Dis(la&ement, %elo&ity,
and A&&eleration %e&tors
1.) "niform ir&ular Motion
1.* Angular %elo&ity and
A&&eleration %e&tor
1.* Pro+e&tile Motion
inemati&s of a (oint-lie
o/+e&t
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1.1 Standards of Length, Mass, and Time
Pham Hong Quang Fundamental Science Department 5
Length@nits
-A B Ceter, C
Dened in terCs oE a CeterB the distance traveled bylight in a vac;;C d;ring agiven tiCe
-ee Fable 1*1 Eor soCeeGaC7les oE lengths
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Length, Cont.
Pham Hong Quang Fundamental Science Department 6
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1.1 Standards of Length, Mass, and Time
Pham Hong Quang Fundamental Science Department 7
Mass@nits
-A B kilograC, kgDened in terCs oE akilograC, based on as7ecic cylinder ke7tat the Anternational
#;rea; oE -tandards-ee Fable 1*+ EorCasses oE vario;sobects
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Mass,Cont.
Pham Hong Quang Fundamental Science Department 8
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1.1 Standards of Length, Mass, and Time
Pham Hong Quang Fundamental Science Department 9
Time@nits
seconds, sDened in terCs oE theoscillation oE radiation EroCa cesi;C atoC
-ee Fable 1*0 Eor soCea77roGiCate tiCe intervals
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Time,Cont.
Pham Hong Quang Fundamental Science Department 10
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1.1 Standards of Length, Mass, and Time
Pham Hong Quang Fundamental Science Department 11
System ofMeasurements
@- 8;stoCaryeveryday ;nits:ength is Ceas;red inEeet
FiCe is Ceas;red in
secondsIass is Ceas;red insl;gs
oEten ;ses eight, in
7o;nds, instead oECass as a
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1.1 Standards of Length, Mass, and Time
Pham Hong Quang Fundamental Science Department 12
Pre0esLreGes corres7ondto 7oers oE 1.Mach 7reG has as7ecic naCe
Mach 7reG has as7ecic abbreviation
Fhe 7reGes can be;sed ith any base;nits
Fhey are C;lti7liersoE the base ;nitMGaC7lesN
1 CC O 1.90 C1 Cg O 1.90 g
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1.2 Dimensional Analysis
Pham Hong Quang Fundamental Science Department 13
•DiCension has as7ecic Ceaning B it
denotes the 7hysicalnat;re oE a K;antity•DiCensions aredenoted ith sK;arebrackets
:ength [:]Iass [I]
FiCe [F]
12D lA l C t
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1.2 Dmensonal Analyss, Cont.
Pham Hong Quang Fundamental Science Department 14
• FechniK;e to check the correctness oE an eK;ation orto assist in deriving an eK;ation
•DiCensions 'length, Cass, tiCe, coCbinations) can betreated as algebraic K;antities
• add, s;btract, C;lti7ly, divide•#oth sides oE eK;ation C;st have the saCediCensions•DiCensions oE soCe coCCon K;antities are givenbelo
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14C d t S t
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1.4 Coordnate Systems
Pham Hong Quang Fundamental Science Department 16
•@sed to describe the 7osition oE a 7oint in
s7ace
•
8oordinate systeC consists oE P Ged reEerence 7oint called the origin
-7ecic aGes ith scales and labels
Anstr;ctions on ho to label a 7ointrelative to the origin and the aGes
e
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1.4 Coordinate Systems
Pham Hong Quang Fundamental Science Departmentdinate 17
ye
x y zr OM xe ye ze.= = + +
;;;;rr r r r
Des&artesoordinate System
Loints are labeled
' % , !,& )
θ
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1.4 Coordinate Systems
Pham Hong Quang Fundamental Science Department 18
θ
S(heri&al oordinateSystem
G O r sinTcosUyO rsinTsinU
O rcosTPndr O
cosT O
U O arctg
222 z y x ++
222 z y x
z
++
x
y
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1.5 Vectors and Scalars
Pham Hong Quang Fundamental Science Department 19
•P scalar quantity is coC7letely
s7ecied by a single val;e ith an
a77ro7riate ;nit and has no direction*
•P vector quantity is coC7letely
described by a n;Cber and
a77ro7riate ;nits 7l;s a direction*
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1.5 Vectors and Scalars
Pham Hong Quang Fundamental Science Department 20
%e&tor am(le•
P 7article travels EroC Pto # along the 7athshon by the dotted redline Fhis is the distance
traveled and is ascalar
• Fhe displacement isthe solid line EroC P to #
Fhe dis7laceCent isinde7endent oE the7ath taken beteenthe to 7ointsDis7laceCent is a
vector
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1.5 Vectors and Scalars
Pham Hong Quang Fundamental Science Department 21
%e&tor otation• FeGt ;ses bold ith arro to denote a vectorN
•Plso ;sed Eor 7rinting is siC7le bold 7rintN A
•When dealing ith ;st the Cagnit;de oE a vector,
an italic letter ill be ;sedN ' or
Fhe Cagnit;de oE the vector has 7hysical ;nits
Fhe Cagnit;de oE a vector is alays a 7ositive
n;Cber
•When handritten, ;se an arroN
A
A
A
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1.5 Vectors and Scalars
Pham Hong Quang Fundamental Science Department 22
Adding %e&tors 4ra(hi&ally
•8hoose a scale
•Dra the rst vector, , ith the a77ro7riate length
and in the direction s7ecied, ith res7ect to a
coordinate systeC
•Dra the neGt vector ith the a77ro7riate length
and in the direction s7ecied, ith res7ect to a
coordinate systeC hose origin is the end oE vector
and 7arallel to the coordinate systeC ;sed Eor
A
AA
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1.5 Vectors and Scalars
Pham Hong Quang Fundamental Science Department 23
Adding %e&tors
4ra(hi&ally, &ont.•8ontin;e draing the
vectors Xti79to9tailY
• Fhe res;ltant is dran
EroC the origin oE to
the end oE the last
vector
•Ieas;re the length oEand its angle
@se the scale Eactor
to convert length to
act;al Cagnit;de
A
R
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1.5 Vectors and Scalars
Pham Hong Quang Fundamental Science Department 24
om(onent Method of Adding%e&tors
•$ra7hical addition is not recoCCended hen
High acc;racy is reK;ired
AE yo; have a three9diCensional 7robleC
•8oC7onent Cethod is an alternative Cethod
At ;ses 7roections oE vectors along
coordinate aGes
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1.5 Vectors and Scalars
Pham Hong Quang Fundamental Science Department 25
om(onents of a
%e&tor, 5ntrodu&tion•P &om(onent is a 7roection
oE a vector along an aGis
Pny vector can be
coC7letely described by its
coC7onents
•At is ;seE;l to ;se
re&tangular &om(onents
Fhese are the 7roections oE
the vector along the G9 and
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1.5 Vectors and Scalars
Pham Hong Quang Fundamental Science Department 26
om(onents of a%e&tor
•Pss;Ce yo; are given a
vector•At can be eG7ressed in
terCs oE to other
vectors, and
• Fhese three vectors
EorC a right triangle
•
A
x A y A
x y = +A A A
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1.5 Vectors and Scalars
Pham Hong Quang Fundamental Science Department 27
om(onents of a
%e&tor, 0nal
• Fhe coC7onents can be
7ositive or negative and
ill have the saCe ;nits
as the original vector
• Fhe signs oE the
coC7onents ill de7end
on the angle
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1.5 Vectors and Scalars
Pham Hong Quang Fundamental Science Department 28
"nit %e&tors
•P unit vector is a vector ith a
Cagnit;de oE eGactly 1*•@nit vectors are ;sed to s7eciEy a
direction and have no other 7hysical
signicance
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1.5 Vectors and Scalars
Pham Hong Quang Fundamental Science Department 29
"nit %e&tors, &ont.• Fhe syCbols
re7resent ;nit vectors• Fhey EorC a set oE
C;t;ally 7er7endic;lar
vectors in a right9handed
coordinate systeC
•ReCeCber,
kZand, HZ,iZ
= = =ˆˆˆ 1i j k
5 d l
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1.5 Vectors and Scalars
Pham Hong Quang Fundamental Science Department 30
Adding %e&tors "sing "nit
%e&tors
@sing
Fhen
and so ( % O ' % ) % and ( ! O ' ! ) !
= +R A B
( ) ( )
( ) ( )
= + + +
= + + += +
r
r
r
ˆˆˆˆ
ˆˆ
ˆˆ
x y x y
x x y y
x y
A A B B
A B A BR R
R i j i j
R i j
R i j
2 2 1tan
y
x y
x
R R R R
R θ
−= + =
5 d l
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1.5 Vectors and Scalars
Pham Hong Quang Fundamental Science Department 31
S&alar (rodu&t of t6o !e&tors
Fhe dot product oE to vectors is the s;C oE the7rod;cts oE their corres7onding coC7onents* AE
aO\a1, a+ and /O\b1, b+, then a^/O a1b1a+b+ *
MG* AE aO\1, and /O\0,/, then
a./O00+O0Q
AE T is the angle beteen vectors a and /, then
!oteN these are ;st to ays oE eG7ressing the dot
7rod;ct
!ote that the dot 7rod;ct oE to vectors 7rod;ces a
θ cosbaba =⋅
15V dS l
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1.5 Vectors and Scalars
Pham Hong Quang Fundamental Science Department 32
%e&tor (rodu&t of t6o !e&tors
Fhe cross 7rod;ct a G / oE to
vectors a and /, ;nlike the dot 7rod;ct,
is a vector*
_or this reason, it is also called the vector
7rod;ct*
!ote that a G / is dened only hen a and /
are three9diCensional '09D) vectors*AE a O ‹a1, a+, a0› and / O ‹*1, *+, *0›, then
the cross 7rod;ct oE a and / is the vector
a G / O ‹a+*0 9 a0*+, a0*1 9 a1*0, a1*+ 9 a+*1›
15V t dS l
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1.5 Vectors and Scalars
Pham Hong Quang Fundamental Science Department 33
%e&tor (rodu&t of t6o !e&tors, &ont.
An order to Cake Denition 1 easier
to reCeCber, e ;se the notation oE deterCinants*
P deterCinant oE order 0 can be dened in terCs oE
second9order deterCinants as EollosN
1 2 3
2 3 1 3 1 2
1 2 3 1 2 32 3 1 3 1 2
1 2 3
a a ab b b b b b
b b b a a ac c c c c cc c c
= − +
15V t dS l
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1.5 Vectors and Scalars
Pham Hong Quang Fundamental Science Department 34
%e&tor (rodu&t of t6o !e&tors, &ont.
We see that the cross 7rod;ct oE the vectorsa O a1i a+ + a0 and / O *1i *+ + *0
isN
2 3 1 3 1 2
2 3 1 3 1 2
a a a a a ab b b b b b
× = − +a b i !
1 2 3
1 2 3
a a a
b b b
× =
i ! a b
15V t dS l
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1.5 Vectors and Scalars
Pham Hong Quang Fundamental Science Department 35
%e&tor (rodu&t of
t6o !e&tors, &ont.Dire&tion7AE the ngers oE yo;r righthand c;rl in the directionoE a rotation 'thro;gh anangle less than 1/.`)EroC a to /, then yo;rth;Cb 7oints in thedirection oE a G /*
Length7AE + is the angle beteena and / 'so . + ), then
a G / O a/ sin +
. , ,A l ti V t
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. , , Acceleration Vectors
Pham Hong Quang Fundamental Science Department 36
Position and
Dis(la&ement
• Fhe 7osition oE an obect
is described by its
7osition vector,
• Fhe dis(la&ement oE
the obect is dened as
the change in its
position
∆ ≡ −f i r r r
1.6TheDsplacement,Velocty,and
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1.6 The Dsplacement, Velocty, and Acceleration Vectors
Pham Hong Quang Fundamental Science Department 37
A!erage !elo&ity
• Fhe average velocity is the ratio oE the dis7laceCent to
the tiCe interval Eor the dis7laceCent
• Fhe direction oE the average velocity is the direction oE
the dis7laceCent vector
∆≡ ∆
ravg
t r v
1.6TheDsplacement,Velocty,and
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1.6 The Dsplacement, Velocty, and Acceleration Vectors
Pham Hong Quang Fundamental Science Department 38
5nstantaneous
!elo&ity• Fhe instantaneo;s velocityis the liCit oE the averagevelocity as Δt a77roaches
ero
Ps the tiCe interval
becoCes sCaller, thedirection oE thedis7laceCenta77roaches that oE the
line tangent to the
∆ →
∆≡ =
∆
r
0
limt
d
t dt
r r v
1.6TheDsplacement,Velocty,and
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1.6 The Dsplacement, Velocty, and Acceleration Vectors
Pham Hong Quang Fundamental Science Department 39
5nstantaneous %elo&ity, &ont.• Fhe direction oE the instantaneo;s
velocity vector at any 7oint in a
7articles 7ath is along a line tangentto the 7ath at that 7oint and in the
direction oE Cotion
• Fhe Cagnit;de oE the instantaneo;svelocity vector is the s7eed
Fhe s7eed is a scalar K;antity
1.6TheDsplacement,Velocty,and
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1.6 The Dsplacement, Velocty, and Acceleration Vectors
Pham Hong Quang Fundamental Science Department 40
− ∆≡ =− ∆
r f i avg
f i t t t v v va
A!erage a&&eleration
• Fhe average acceleration oE a
7article as it Coves is dened as
the change in the instantaneo;s
velocity vector divided by the
tiCe interval d;ring hich that
change occ;rs*
1.6TheDsplacement,Velocty,and
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1.6 The Dsplacement, Velocty, and Acceleration Vectors
Pham Hong Quang Fundamental Science Department 41
A!erage
A&&eleration, &ont•Ps a 7article Coves, the
direction oE the change in
velocity is Eo;nd by
vector s;btraction
• Fhe average acceleration
is a vector K;antity
directed along
∆ = −f i v v v
∆v
1.6TheDsplacement,Velocty,and
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1.6 The Dsplacement, Velocty, and Acceleration Vectors
Pham Hong Quang Fundamental Science Department 42
5nstantaneous a&&eleration
• Fhe instantaneo;s acceleration is the liCiting
val;e oE the ratio as Δt a77roaches ero
Fhe instantaneo;s eK;als the derivative oE
the velocity vector ith res7ect to tiCe
∆ →
∆≡ =
∆r
0
limt
d t dt v va
∆ ∆t v
1.6TheDsplacement,Velocty,and
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1.6 The Dsplacement, Velocty, and Acceleration Vectors
Pham Hong Quang Fundamental Science Department 43
Produ&ing An A&&elerationVario;s changes in a 7articles Cotion Cay
7rod;ce an acceleration
•
Fhe Cagnit;de oE the velocity vector Caychange
• Fhe direction oE the velocity vector Cay
changeMven iE the Cagnit;de reCains constant
• #oth Cay change siC;ltaneo;sly
1.7Unform CrcularMoton
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1.7 Unform Crcular Moton
Pham Hong Quang Fundamental Science Department 44
•Uniform circular motion occ;rs hen an
obect Coves in a circ;lar 7ath ith a
constant s7eed
• Fhe associated analysis Cotion is a particle
in uniorm circular motion
•Pn acceleration eGists since the direction oE
the Cotion is changing
• Fhis change in velocity is related to an
acceleration
• Fhe velocity vector is alays tangent to the
7ath oE the obect
V
1.7 Unform Crcular Moton
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. U o C cua oto
Pham Hong Quang Fundamental Science Department 45
hanging %elo&ity in
"niform ir&ular
Motion
• Fhe change in the velocityvector is d;e to the
change in direction
• Fhe vector diagraC shosf i
= + ∆v v v
1.7 Unform Crcular Moton
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Pham Hong Quang Fundamental Science Department 46
entri(etal A&&eleration
• Fhe acceleration is alays
7er7endic;lar to the 7ath oE the
Cotion
• Fhe acceleration alays 7oints
toard the center oE the circle oE
Cotion• Fhis acceleration is called the
centripetal acceleration
1.7 Unform Crcular Moton
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Pham Hong Quang Fundamental Science Department 47
entri(etal A&&eleration, &ont.• Fhe Cagnit;de oE the centri7etal acceleration
vector is given by
• Fhe direction oE the centri7etal acceleration
vector is alays changing, to stay directedtoard the center oE the circle oE Cotion
2
C var =
1.7 Unform Crcular Moton
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Pham Hong Quang Fundamental Science Department 48
Period
• Fhe period , T , is the tiCe
reK;ired Eor one coC7lete
revol;tion
• Fhe s7eed oE the 7article o;ld
be the circ;CEerence oE the
circle oE Cotion divided by the7eriod
• FhereEore, the 7eriod is dened
as
2 r T
v
π ≡
1.7 Tangental and Radal Acceleraton
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g
Pham Hong Quang Fundamental Science Department 49
• Fhe Cagnit;de oE the velocity co;ld also be
changing•An this case, there o;ld be a tangential
acceleration
• Fhe Cotion o;ld be ;nder the in;ence oE
both tangential and centri7etal accelerations
• !ote the changing acceleration vectors
17TangentialandRadialAcceleration
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1.7 Tangential and Radial Acceleration
Pham Hong Quang Fundamental Science Department 50
Total A&&eleration
• Fhe tangential acceleration ca;ses the
change in the s7eed oE the 7article
• Fhe radial acceleration coCes EroC a
change in the direction oE the velocity
vector
17TangentialandRadialAcceleration
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1.7 Tangential and Radial Acceleration
Pham Hong Quang Fundamental Science Department 51
Total A&&eleration, e8uations
• Fhe tangential accelerationN
• Fhe radial accelerationN
• Fhe total accelerationN
Iagnit;de
Direction-aCe as velocity vector iE - is increasing,o77osite iE - is decreasing
t
dv a
dt =
2
r C
v a a
r = − = −
2 2
r t a a a= +
18AngularVelocityandAccelerationVector
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1.8 Angular Velocity and Acceleration Vector
Pham Hong Quang Fundamental Science Department 52
z ze eω = θ = ω
r r rγ = θ = γ z ze e
r r r
v R= ω ∧;rrr
1.9ProjectileMotion
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1.9 Projectile Motion
Pham Hong Quang Fundamental Science Department 53
•Pn obect Cay Cove in
both the % and ! directions
siC;ltaneo;sly
• Fhe EorC oE to9
diCensional Cotion e ill
deal ith is called
(ro+e&tile motion
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1.9ProjectileMotion
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1.9 Projectile Motion
Pham Hong Quang Fundamental Science Department 55
Pro+e&tile Motion Diagram
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1.9ProjectileMotion
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1.9 Projectile Motion
Pham Hong Quang Fundamental Science Department 57
;ange and Maimum
Height of a Pro+e&tile
•When analying 7roectile
Cotion, to characteristics
are oE s7ecial interest
• Fhe range, (, is the
horiontal distance oE the
7roectile
• Fhe CaGiC;C height the
7roectile reaches is h
1.9ProjectileMotion
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1.9 Projectile Motion
Pham Hong Quang Fundamental Science Department 58
Height of a Pro+e&tile,
e8uation
• Fhe CaGiC;C height oE the 7roectile
can be Eo;nd in terCs oE the initial
velocity vectorN
• Fhis eK;ation is valid only Eor
syCCetric Cotion
2 2sin
2
i i v h
g
θ =
1.9ProjectileMotion
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1.9 Projectile Motion
Pham Hong Quang Fundamental Science Department 59
More A/out the ;ange of a
Pro+e&tile
1.9 Projectile Motion
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.9 j
Pham Hong Quang Fundamental Science Department 60
;ange of a Pro+e&tile, 0nal
• Fhe CaGiC;C range occ;rs at θ i O
Qo
•
8oC7leCentary angles ill 7rod;cethe saCe range
Fhe CaGiC;C height ill be
dierent Eor the to angles Fhe tiCes oE the ight ill be
dierent Eor the to angles
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