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KINEMATICS
University of Southeastern Philippines
General Physics Lec
Ernel D !a"#a"
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Kine$atics % &uantitative 'escription of $o
reference to its physical causes
Scalar )no 'irection* Vector )(+ 'irection*
Distance )'* Displacement ) d*
Speed )s* Velocity )v)
Acceleration )a*
,o( far you travelChan"e in position
),o( far you travel in a "iven
'irection*
,o( fast you travel ,o( fast you travel )in a "iven
'irection*
-ate of chan"e of velocity
)'escri#in" how thin"s $ove*
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Reference frame and Position
. -eference fra$e is the physical entity to$otion or position of an o#/ect is #ein" refe
. Position refers to the location of an o#/erespect to so$e reference fra$e
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Describing Motion
There are lots of 'ifferent (ays to '
$otion0
1 2or's
3 S4etches
5 Ti$e elapse' photo"raphs6 Physical E7pressions )E8uations*
9 Graphical -epresentation
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Kinematics E!ations
Average speed: sav ; ' + chan"e in t
sav
" d # $t " d # tf
% ti
Average velocity& vav ; $d + $t S
vav " 'df % di) # t
Average acceleration& aav ; $v + $t SI unit: $
aav ; )vf % vi) + ∆t
df ; di < vav ∆t
vf " vi < aav ∆t
Note: if the ti$e intervals are very s$all (e call these 8uantities ins
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Sa$ple pro#le$ 1: Spee' an' =elo
. Every $ornin"> you /o" aroun' a 39? $ trac4 four ti$e$inutes 2hat is your )a* avera"e spee' an' )#* averavelocity@
a Total 'istance you /o""e' is : 6 39? $ ; 1>??? $> t
ave spee' ; '+ t > 1>??? + 1B?? s
; ? $+s
# ou have no resultant 'isplace$ent since you are #ac
(here you starte' Therefore> your avera"e velocity is ;
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Sa$ple pro#le$ 3
. A car $ovin" at constant spee' travels 5? $ in 9 s )a* (hat spee' of the car@ )#* ,o( far (ill the o#/ect $ove in 1? s@
Solution:
"iven: t ; 9 s> '; 5? $ at constant spee'
)a* s ; '+t > 5?+9 ; $+s
)#* s; '+t> $+s ; ' + 1? s
' ; ) $+s*)1?s*
; ? $
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Acceleration
. Chan"e in velocity over ti$e
. 2hen 'oes an o#/ect is acceleratin"@
% 2hen it is $ovin" (ith chan"in" spee'
% 2hen $ovin" (ith constant spee' #ut (ith ch
'irection% 2hen $ovin" (ith chan"in" spee' as (ell as c
'irection
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Sa$ple pro#le$ 1: Acceleration
. A Nissan Sentra is stoppe' at a traffic li"ht 2hen the "reen> the 'river accelerates so that the carFs spe
rea's 1? $+s after 9 s 2hat is the carFs acceleration
it is constant@
Solution:
"iven: =i ; ? > =f ; 1? $+s at t ; 9 s>
a ; =f =i + t > 1? $+s ? $+s + 9
; 3 $+s3
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(sing Split imes*
Position 'm) ?%9 9%1? 1?%19 19%3?
3?%3
Split ime 's) 1 36 5? 59 6?Av+ Velocity'm#s)
51 31 1H 16 13
Deter$ine the avera"e velocity for each 'istance inte
Deter$ine avera"e velocity of the o#/ect over the ti$e
Deter$ine the avera"e acceleration over the ti$e rec
vav ; df % di + ∆t ; 39$ % ?$ + 169 s ; ,+- m # s
a ; vf % vi + ∆t ; )139 $+s % 51$+s* + 169 s ;
Note: the a is ne"ative #ecause the change in v is ne"at
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More e7a$ples Practice $a4es perfect r
. An'y Green in the carThrustSSC set a (orl' recor'
of 5611 $+s in 1JJH To
esta#lish such a recor'> the
'river $a4es t(o runs throu"h
the course> one in each
'irection> to nullify (in'effects ro$ the 'ata>
'eter$ine the avera"e velocity
for each run
Ans
Ans(
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Deter$ine the avera"e acceleration of the plane
v ; 3? 4$+h t ; 3Js
aav ; )vf % vi* + )tf ti*
aav ; )3? 4$+h ? 4$+h* + )3Js ?s*
aav ; J? 4$+h +s
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1raphical Representation of Motion
Kinematics Relationships hro!gh 1raphing&
1 The slope of a '%t "raph at any ti$e tells you theavera"e velocity of the o#/ect
3 The slope of a v%t "raph at any ti$e tells you th
avera"e acceleration of the o#/ect
5 The area un'er a v%t "raph tells you the'isplace$ent of the o#/ect 'urin" that ti$e
6 The area un'er a a%t "raph tells you the
chan"e in velocity of the o#/ect 'urin"
that ti$e
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Constant Motion
n the '%t "raph at any point in
ti$e 0 vav ; $d + $t
vav ; )2. % .)m + '2 % .)s
vav ; ,. m#s
The slope is constant on this "raph
so the velocity is constant
n the v%t "raph at any point in
ti$e0 aav ; vf % vi + ∆t
aav ; ),. % ,.)m#s + '2 % .)s
aav ; . m#s0
Loo4in" at the area #et(een the
line an' the 7%a7is0
Area of rectan"le ; # 7 h
Area ; 9s 7 1? $+s ; 2. m
2hich is of course 'isplace$ent
n the a%t "raph the area #et(e
the line an' the 7%a7is is0
Area of rectan"le ; # 7 h
Area ; 9s 7 ? $+s3 ; . m#s
The area thus represents0
$v ; aav $t
Chan"e in velocity
0
10
20
30
40
50
60
0 1 2 3 4 5 6
time (s)
0
2
4
6
8
10
12
14
16
18
20
0 1 2 3 4 5 6
time (s)
0
2
4
6
8
10
0 1 2 3 4 5
time (s)
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Chan"in" Motion
n the '%t "raph at any point inti$e 0 vav ; $d + $t
The slope is constantly increasin"
on this "raph so the velocity is
increasin" at a constant rate
The slope of a tan"ent line 'ra(n
at a point on the curve (ill tell you
the instantaneous velocity at this
position
n the v%t "raph at any point inti$e0 aav ; vf % vi + ∆t
aav ; )0. % .)m#s + '2 % .)s
aav ; 3 m#s0
Loo4in" at the area #et(een the
line an' the 7%a7is0
Area of trian"le ; 1+3 )# 7 h*
Area ; 1+3 )9s 7 3? $+s* ; 2. m2hich is of course 'isplace$ent
n the a%t "raph the area #etthe line an' the 7%a7is is0
Area of rectan"le ; # 7 h
Area ; 9s 7 6 $+s3 ; 0. m#s
The area thus represents0
Chan"e in velocity
0
10
20
30
40
50
60
0 1 2 3 4 5 6
time (s)
-10
0
10
20
30
40
50
60
0 1 2 3 4 5 6
time (s)
0
5
10
15
20
25
0 1 2 3 4 5 6
time (s)
0
5
10
15
20
25
0 1 2 3 4 5 6
time (s)
0
1
2
3
4
5
6
7
8
0 1 2 3 4
time (s)
0
1
2
3
4
5
6
7
8
0 1 2 3 4
time (s)
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To 'eter$ine the velocity at any point in t
nee' to fin' the slope of the 'istance%ti$e
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Slope #et(een 1s an' 5s
sho(s the velocity at 3s
The velocity at 3s is ∆p+∆t ; )1B$ % 3$*+
The acceleration is "iven #y
velocity%ti$e "raph Therefo
a ; ∆v # ∆t ; 3?$+s + 9s ; 3m
-10
0
10
20
30
40
50
60
0 1 2 3 4 5 6
time (s)
0
5
10
15
20
25
0 1 2 3
time (s)
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E4ample Problem
A 0nd year 5S Economics st!dent is late for the P
r!ns east down the road at / m#s for /.s7 then thi
she has dropped her calc!lator so stops for ,.s toShe 9ogs bac8 west at 0 m#s for ,.s7 stops for 2
accelerates !niformly from rest to 3 m#s east ov
second period+
a* S4etch the velocity%ti$e "raph of the stu'entFs $otion
#* Deter$ine the total 'istance an' 'isplace$ent of the
'urin" this ti$e
c* Deter$ine the stu'entFs avera"e velocity 'urin" this ti
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Velocity%ime 1raph of the St!dent:s Mo
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otal distance traveled by the st!dent is;+
dtotal = d1 + d2 + d3 + d4 + d5
dtotal = s1∆t1 + s2∆ t2+ s3∆ t3 + s4∆ t4 + s5∆ t5
dtotal = (3m/s)(30s) + (0m/s)(10s) + (2m/s)(10s) + ………
(0m/s)(5s) + (1/2(4m/s)(10s)
dtotal = 130 m
otal displacement by the st!dent is;+
dtotal = d1 + d2 + d3 + d4 + d5
dtotal = v1∆t1 + v2 ∆ t2+ v3∆ t3 + v4∆ t4 + v5∆ t5
dtotal = (3m/s)(30s) + (0m/s)(10s) + (-2m/s)(10s) +
………(0m/s)(5s) + (1/2(4m/s)(10s)
dtotal = 90 m (East)
+ east -
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Average velocity of the st!dent is;++
vav = dtotal / ∆ttotal = + 90m East / 65s
vav = 1.4 m/s East
90 m
- 20 m
20 m
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'f ; 'i < vav ∆t
'f ; 'i < )vi < vf * +3 ∆t #ut vf ; vi < aav ∆t
'f ; 'i < )vi < )vi < aav∆t* +3 ∆t
df " di < vi t < ,#0 aav t0
'f ; 'i < )vi < vf * +3 ∆t #ut ∆t ; )vf % vi * + aav
'f ; 'i < )vi < vf * +3 )vf % vi* +aav
∆' ; )vi < vf * +3 )vf % vi* +aav So0 ∆' ; )vf 3 % vi
3 * +3aav
an'0 vf 0 " vi
0 < 0aav d
d " vi t < ,#0 aav t0 or0
#ut vav " 'vi < vf ) # 0
or0 0 aav d " vf 0 % vi
0
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A = HB
A=1/2HB
vt t vd d
tt at vd d
t at vd d
ii f
avii f
avii f
∆+∆+=
+∆+=
∆+∆+=
2
1
21
21 2
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Sa$ple pro#le$
. A racer accelerates from rest at a constant rate m/s/s. How fast will the racer be going at the ens! "b# How far has the racer tra$elle %ring this
&ol%tion'
gi$en' $i = 0 $f = !
t = 6.0 s f = !
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. "a# (sing e)%ation a = *f + *i / t, 2.0 m/s/s = *f +s
*f = 0 - "2.0 m/s/s#"6.0 s#
= 12 m/s
. "b# f = $i t - at2, f = 0"6.0# - "2.0m/s/s #
= 36 m
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. A car has %niforml accelerate from rest to a s25 m/s after tra$elling 5 m. hat is its accelera
&ol%tion'
gi$en' *i = 0, *f = 25 m/s, f = 5 m
2a = vf 3 % vi
3 > 3a)H9$* ; 39 )$+s*3 ?
a ; 63 $+s3