kinematics ernel

8/19/2019 KINEMATICS ernel

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KINEMATICS

University of Southeastern Philippines

General Physics Lec

Ernel D !a"#a"


2/26

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*


3/26

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


4/26

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


5/26

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


6/26

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 ;


7/26

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*

; ? $


8/26

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


9/26

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


10/26

(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


11/26

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(


12/26

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


13/26

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


14/26

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 ; 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)


15/26

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 ; 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)


16/26

To 'eter$ine the velocity at any point in t

nee' to fin' the slope of the 'istance%ti$e


17/26

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)


18/26

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


19/26

Velocity%ime 1raph of the St!dent:s Mo


20/26

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 -


21/26

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


22/26

'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


23/26

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


24/26

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 = !


25/26

. "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


26/26

. 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

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