A Geometrical Approach to Speed and Acceleration (complete)

8/11/2019 A Geometrical Approach to Speed and Acceleration (complete)

http://slidepdf.com/reader/full/a-geometrical-approach-to-speed-and-acceleration-complete 1/21

A geometrical

approach to the

concepts of speed

and acceleration



!"#$%&'($

!" $"%"&'( )*&+&"',- ."'/"01+2*& $"3+4+0+'4 '5 06"2'42"(0- '5 -(""$ *4$ *22"&"1*0+'4 061'7.6 $+-27--+'4

'5 6+- "8("1+/"40*& 3+4$+4.- *9'70 06" /'0+'4 '5

5*&&+4. '9:"20-; <*06"1 06*4 0*=+4. 06" 2'42"(0- '5

2*&27&7- *4$ 06"4 *((&>+4. 06"/ 0' $"3+4" -(""$ *4$

*22"&"1*0+'4 +4 ."4"1*&? @" $' 06" 1"%"1-"? 0*=+4.

)*&+&"',- *4*&>-+- '5 /'0+'4 *4$ *((&>+4. +0 0'

+401'$72" 06" 2'42"(0- '5 2*&27&7-; A6+- (1'%+$"- *

4'%"& ('+40 '5 %+"@ @+06+4 * 21+0+2*& *4$ +4B7+1>C9*-"$*((1'*26;



)*) ,-$ %./%$0&1 .2 10$$3 4/3 4%%$5$64&'./

D'@ .''$ +- >'71 74$"1-0*4$+4. '5 -(""$ *4$

*22"&"1*0+'4E !'7&$ +0 -0*4$ 7( 74$"1 21'--C"8*/+4*0+'4

+4 2'710E D"1" *1" -'/" B7"-0+'4- 06*0 >'7 /+.60 9"

*-="$ 9> * 2'710 (1'-"270'1 0' 0"-0 >'71 21"$+9+&+0> *- *4

"8("10 @+04"--F

G; H(""$ +- 74$"1-0''$ *- $+-0*42" $+%+$"$ 9> 0+/";

!6*0 $+-0*42" +- 9"+4. 1"5"11"$ 0' 6"1"E !6*0 0+/" +-

9"+4. 1"5"11"$ 0' 6"1"E

I; J- 06+- @6*0 >'71 2*1,- -(""$'/"0"1 +- /"*-71+4.

@6"4 +0 1".+-0"1-? 5'1 "8*/(&"? KL =/M61E A' @6*0

$+-0*42" *4$ 0+/" @'7&$ +0 9" 1"5"11+4.E

N; O40+& 06" +401'$720+'4 '5 $+.+0*& 0"264'&'.> +4 06"

GPQL-? 2*1- 7-"$ "$$> 2711"40 -(""$'/"0"1- R>'7 2*4

.''.&" 06+-S; T+$ 06"-" /"*-71" *4> *207*& $+-0*42"- *4$

0+/"-E !6*0 $+$ 06"> /"*-71"E

U; D'@ @'7&$ >'7 /"*-71" 06" 26*4." +4 -(""$ '5 *

%"6+2&"E

V; !6*0 $' >'7 06+4= *22"&"1*0+'4 /"*4-E RW)"00+4.

5*-0"1, +-4,0 *4 "8("10 *4-@"1 *4$ @'7&$ 1"3&"20 9*$&> '4>'71 21"$+9+&+0>; X'7 @'7&$ 9" "8("20"$ 0' .+%" *

47/"1+2*& $"3+4+0+'4;S

7$%&'./ )

8/&6.39%&'./

2



)* : ;45'5$. ;45'5$' 4/3 &-$ 1&93< .2 26$$ 2455 =.&'./

A6" 1"4*+--*42" J0*&+*4 -2+"40+-0 )*&+&"' )*&+&"+ RGVKU Y

GKUIS +401'$72"$ 06" /'$"14 4'0+'4- '5 -(""$ *4$

*22"&"1*0+'4 +4 06" 2'40"80 '5 01>+4. 0' 74$"1-0*4$ 06"

/'0+'4 '5 5*&&+4. '9:"20-; D" $"-21+9"$ 06" /'0+'4 '5 *

$1'(("$ '9:"20 9> -*>+4. 06*0 06" $+-0*42" '5 +0- 5*&& @*-

(1'('10+'4*& 0' 06" -B7*1" '5 06" $71*0+'4 '5 +0- 5*&&; J4 6+-

@'1&$F

Z 06"1" @"1" 4' 2&'2=- '1 -0'( @*026"- *- @" =4'@ 06"/?

Z 06"1" @"1" 4' .1*(6-? 5'1/7&*- '1 57420+'4-?

Z 06"1" @*- 4' *&."91* '1 2*&27&7-;

H' @6*0 2'7&$ 6" 6*%" ('--+9&> /"*40 9> 6+- $"-21+(0+'4E

J4 06" &+.60 '5 06" $'0 ('+40- *9'%"? >'7 -6'7&$ *+/ *0 *

(71"&> ."'/"01+2*& +40"1(1"0*0+'4; [*4 >'7 $"%+-" *4

*4+/*0+'4 06*0 1"*&+-"- +0E

\1" )*&+&"',- @'1$- 1"*&&> * $"-21+(0+'4 '5 06" /'0+'4 '5

06" $1'(("$ '9:"20 '1 :7-0 -'/" &+/+0"$ *-("20 '5 +0E J-

06"1" -'/" 9"00"1 @*> 0' $"-21+9" 06" /'0+'4 '5 '9:"20-E

A' *4-@"1 06" *9'%" B7"-0+'4- +0 /*> 9" 6"&(57& 0' 0*&=

*9'70 '4" '5 06" /'-0 5*/'7- "8("1+/"40- 2'4$720"$ 9>

)*&+&"'? =4'@4 *- 06" '/%5'/$3 054/$ $>0$6'=$/& ;

3

Portrait of Galileo Galilei by Giusto Sustermans (Source: Wikipedia)

Gallery 1.1



!" *&& "8("1+"42" /'0+'4 +4 '71 $*+&> &+%"-; J4 (*10+27&*1?

"%"1>9'$> 6*- 5"&0 '1 @+04"--"$ /'0+'4 $7" 0' .1*%+0>;

A6" 4"80 -+07*0+'4- *1" "8*/(&"- '5 @6*0 @" *&& =4'@ *-

?6$$ ?455 @.&'./;

GSA61'@+4. * 9*&& -01*+.60 7(?

IST1'((+4. * 2'+4 51'/ 06" 0'( '5 * 97+&$+4.

]1"" 5*&& /'0+'4 6*- 9""4 -07$+"$ +4 *&& =+4$- '5 @*>-

-+42" *42+"40 0+/"-; D'@"%"1? *0 06" $*@4 '5 06"

-"%"40""406 2"4071> )*&+&"' +401'$72"$ * $"2+-+%"&> 4"@

*((1'*26? @6+26 &"$ 6+/ 0' 6+- 1"/*1=*9&" $+-2'%"1>

*9'70 06" 1"&*0+'4-6+( 9"0@""4 06" $+-0*42" '5 5*&& *4$

$71*0+'4 '5 5*&& '5 * $1'(("$ '9:"20;

)*&+&"' /*$" 06+- $+-2'%"1> +4$+1"20&>? 9"2*7-" +0 @*-

+/('--+9&" 0' /"*-71" *2271*0"&> 6'@ &'4. *4 '9:"20

0''= 0' 5*&& 7-+4. 06" 0"264'&'.> '5 6+- $*>; D" '9-"1%"$?

6'@"%"1? 06*0 * 5*&&+4. '9:"20 @*- *4 "801"/" 2*-" '5 *

9*&& 1'&&+4. $'@4 *4 +42&+4"$ (&*4" Y +4 06" 2*-" '5 *

5*&&+4. '9:"20? 06" (&*4" +- "55"20+%"&> %"10+2*&;

7$%&'./ :

,-$ '/%5'/$3 054/$ $>0$6'=$/&

4

A replica of Galileo’s inclined plane (Source: Museo Galileo gallery)

Gallery 1.2



H' )*&+&"' +4%"-0+.*0"$ 06" 1"&*0+'4-6+( 9"0@""4 !"#

%&'!()*# +, !-(.#/ ()% %0-(!&+) +, !-(.#/ +, ( 1(// -+//&)2

%+3) ( 4/()# (! ( -()2# +, &)*/&)(!&+)' 06*0 @"1" 4'0 0''

-0""(? @6"1" 06" 4"2"--*1> 0+/" /"*-71"/"40 2'7&$ 9"

/*$" @+06 1"*-'4*9&" *2271*2>;

A' /"*-71" 06" 0+/" +0 0''= 5'1 * 9*&& 0' 1'&& $'@4 06"

+42&+4"$ (&*4"? )*&+&"' 7-"$ * @*0"1 2&'2=; A6+- $"%+2"

2'4-+-0"$ '5 * &*1." %"--"& '5 @*0"1

(&*2"$ +4 *4 "&"%*0"$ ('-+0+'4 @+06 *

-/*&& 6'&" $1+&&"$ +4 06" 9'00'/

2'44"20"$ 0' * (+(" '5 -/*&& $+*/"0"1; \

06+4 -01"*/ '5 @*0"1 3&'@"$ +40' *

2'40*+4"1 9"&'@ 06" %"--"&; A' -0*10 06"

2&'2=? 6" *&&'@"$ @*0"1 0' 3&'@ +40' 06"

2'40*+4"1; A' -0'( 06" 2&'2=? 6" -0'(("$

06" 3&'@ '5 @*0"1; A' 1"-"0 06" 2&'2= 6"

"/(0+"$ 06" 2'40*+4"1;

^> @"+.6+4. 06" */'740 '5 @*0"1 +4 06"

2'40*+4"1? 6" 2'7&$ 06"4 2'/(*1" 06"

0+/"- +0 0''= 06" 9*&& 0' 01*%"& "*26

$+-0*42"; ]'1 "8*/(&"? +5 0@+2" *- /726

@*0"1 R+4 @"+.60S 3+&&"$ 06" 2'40*+4"1? 6" =4"@ 06*0 06"

0+/" /"*-71"$ @*- 0@+2" *- &'4.;

\&06'7.6 4'0 7-"$ 9> )*&+&"'? * %*1+*0+'4 '4 06" @*0"1

2&'2= +- 0' /"*-71" 06" 6"+.60 '5 06" @*0"1 2'&&"20"$ +4 *

079" '1 (+("? +4-0"*$ '5 @"+.6+4. @*0"1 2'&&"20"$ +4 *

2'40*+4"1; A6+- /"*-71"- 0+/" *- * &"4.06? @6+26 +- /'1"

*0074"$ 0' 06" @*> @" 1"(1"-"40 +0 06"-" $*>-;

A67-? >'7 2*4 2'/(*1" 06"

$+-0*42" 2'%"1"$ 9> 06" 9*&& *4$

06" 6"+.60 '5 06" @*0"1 +4 06"2'40*+4"1; J4 06" 4"80 -"20+'4 >'7

@+&& 3+4$ * $>4*/+2 1"(1"-"40*0+'4

'5 )*&+&"',- "8("1+/"40;

5

A sketch of the water clock based on Galileo’sdescription (Source: Juan Carlos Ponce Campuzano).

Gallery 1.3



!" 2*4 "/7&*0" 06" +42&+4"$ (&*4" "8("1+/"40 R@+06'70 2'4-+$"1+4. 51+20+'4S 0' .*+4 $""("1 +4-+.60- +40' )*&+&"',-

06'7.60 (1'2"--"- *4$ 06" $"%"&'(/"40 '5 /*06"/*0+2*& +$"*-;

A*) ;45'5$.B1 $>0$6'=$/&45 1$&90

A6" *((&"0 '4 06" 4"80 (*." -6'@- * ."'/"01+2 1"(1"-"40*0+'4 '5 )*&+&"',- "8("1+/"40; A6" +42&+4"$ (&*4" +- 1"(1"-"40"$

@+06 * 1+.60C*4.&"$ 01+*4.&"; A6" 9*&& +- 1"(1"-"40"$ @+06 * 2+12&"; ]+4*&&>? +4 06+- 2*-"? 06" @*0"1 2&'2= +- 1"(1"-"40"$ @+06

* 1"20*4.&" $"(+20+4. 06" 079" '1 (+(" 06*0 +- 2'&&"20+4. 1744+4. @*0"1; !" *1" 0' 2'/(*1" 06" 6"+.60 '5 06" @*0"1 @+06 06"$+-0*42" 2'%"1"$ 9> 06" 9*&&;

A6" .'*& '5 06+- 3+1-0 *((&"0 +- 0' &"0 >'7 5*/+&+*1+_" >'71-"&5 @+06 )*&+&"',- "8("1+/"40*& $"-+.4; A6" 9*&& 1'&&- $'@4 06"

(&*4"? @6+26 2*4 9" -"0 *0 $+55"1"40 +42&+4*0+'4-;

;$/$645 '/1&69%&'./1C

G; A' -"0 06" 9*&& +4 /'0+'4? (7-6 06" 7&46& 9700'4;

I; `7-6 06" D$1$& 9700'4 0' 9".+4 06" -+/7&*0+'4 *.*+4;

N; A6" -&+$"1? @+06 06" 0*. E41$ R6'1+_'40*& -"./"40 *0 06" 9'00'/S? 26*4."- 06" *4.&" '5 +42&+4*0+'4 '5 06" (&*4";

7$%&'./ A

F 3</4='% 6$06$1$/&4&'./

6



8/1&69%&'./1 2.6 '/&$64%&'/G H'&- &-$ 4005$&C

A6" *9'%" -+/7&*0+'4 *&&'@- >'7 0' /*4+(7&*0" 06" *4.&"

'5 +42&+4*0+'4 '5 06" (&*4"; A6" ('+40 +- 0' ."0 5*/+&+*1

@+06 06" *((&"0 +4 (1"(*1*0+'4 5'1 06" 4"80 *20+%+0>;

!6*0 -6'@- >'7 06*0 +0 0*="- &"-- 0+/" 5'1 06" 9*&& 0' 1'&&

$'@4 (&*4"- 06*0 *1" /'1" -0""(&> +42&+4"$E

I>&$6/45 5'/J 2.6 KL? ($61'./F

600(FMM@@@;."'."91*079";'1.M-07$"40M/GLVNVP

7

Interactive 1.1

http://www.geogebratube.org/student/m105359




A*: ;45'5$.B1 3'1%.($6<

A6" 4"80 *((&"0 *$$- 5"*071"- 0' 06" (1"%+'7- '4" 0'

$"-21+9" )*&+&"',- "8("1+/"40-; A6"-" 5"*071"- *1"F

G; ,'=$ 8/&$6(451C X'7 2*4 26''-" 06" 47/9"1 '5

+40"1%*&- '5 06" -*/" &"4.06 +40' @6+26 06" 0+/"

R2'40*+4"1S +- $+%+$"$; a4 06" 1+.60 -+$" '5 06"

2'40*+4"1 +- $+-(&*>"$ 06" 47/9"1 '5 2'/(&"0" 0+/"

+40"1%*&-? @6"4 06" 9*&& 6*- 2'%"1"$ * (*10+27&*1

$+-0*42";

I; ,'=$C b'%" 06+- '(0+'4 0' -"" @6"1" '4 06" (&*4" 06"

9*&& @'7&$ 9" *0 06*0 0+/";

N; A6" $+-0*42" 2'%"1"$ 9> 06" 9*&& +- $+%+$"$ +40' *

47/9"1 '5 +40"1%*&- '5 06" -*/" &"4.06? @6+26 $"("4$-

'4 06" 47/9"1 '5 0+/" +40"1%*&-; O4$"1 06" 9*&& +-$+-(&*>"$ 06" 47/9"1 '5 2'/(&"0" $+-0*42" +40"1%*&-

2'%"1"$ 9> 06" 9*&&;

8

Galileo's inclined plane experiment. Fresco by Giuseppe Bezzuoli, 1841. Tribuna di Galileo, Florence. (Source: Wikipedia)

Gallery 1.4



8/1&69%&'./1 2.6 '/&$64%&'/G H'&- &-$ 4005$&C

G; A6" *((&"0 -0*10- @+06 '4" ,'=$ 8/&$6(45? 06+- /"*4-

06*0 06" 0+/" +- 4'0 $+%+$"$; T+%+$" 06" 0+/" +40' 0@'

+40"1%*&- *4$ '9-"1%" 06" /'0+'4 '5 06" 9*&& 1'&&+4. $'@4

9> (7-6+4. 06" 7&46& 9700'4 '1 7-+4. 06" -&+$"1 ,'=$;

D'@ /*4> +40"1%*&- $'"- 06" 9*&& 2'%"1 +4 06" 3+1-0 0+/"

+40"1%*&E \4$ +4 06" -"2'4$E

I; [6''-" *4'06"1 47/9"1 '5 ,'=$ 8/&$6(451 *4$ '9-"1%"

@6*0 6*(("4- +4 "*26 2*-"; ]'1 "8*/(&"? +5 06" 0+/" +-

$+%+$"$ +40' 061"" +40"1%*&- 06"4 6'@ /*4> $+-0*42"

+40"1%*&- $'"- 06" 9*&& 2'%"1 +4 06" 3+1-0 0+/" +40"1%*&? +4 06"

-"2'4$ '4"? *4$ +4 06" 06+1$ '4"E

N; ]+4$ * 1"&*0+'4-6+( 9"0@""4 06" 47/9"1 '5 0+/"

+40"1%*&- *4$ 06" 47/9"1 '5 $+-0*42" +40"1%*&- 2'%"1"$ 9>

06" 9*&& *4$ '9-"1%" @6"06"1 06+- 1"&*0+'4-6+( +- -0+&&

/*+40*+4"$ @6"4 >'7 26*4." 06" *4.&" '5 +42&+4*0+'4 '5 06"

(&*4"; !6*0 +- 06" 1"&*0+'4-6+( 9"0@""4 06" $+-0*42"

2'%"1"$ 9> 06" 9*&& *4$ 06" "&*(-"$ 0+/"E

9

Interactive 1.2

External link for PDF version:






M*) ,-$ "'6&- .2 10$$3

O-+4. 06" +42&+4"$ (&*4" )*&+&"' 5'74$ 06*0F

,-$ 3'1&4/%$ %.($6$3 "< &-$ "455 H41 06.0.6&'./45 &.

&-$ 46$4 .2 4 1N946$ H-.1$ 1'3$ 5$/G&- H41 G'($/ "<

&-$ 3964&'./ O&'=$PQ 4/3 H'&- 4 3'22$6$/& %./1&4/& .2

06.0.6&'./45'&< 2.6 $4%- 3'22$6$/& '/%5'/4&'./*

H' 6" +45"11"$ 06*0 06+- @'7&$ 9" 06" -*/" 5'1 * 51""&>

5*&&+4. 9*&&; J5 @" /"*-71" 0+/" +4 -"2'4$- *4$ &"4.06 +4

/"01"- R@6+26 '5 2'71-" )*&+&"' $+$ 4'0S 06"4 06"

2'4-0*40 +- 1'7.6&> U;P? @6+&" 5'1 +42&+4*0+'4- &"-- 06*4

%"10+2*& 06" 2'4-0*40 +- -/*&&"1;

c*071*&&> )*&+&"' @'4$"1"$ @6> -B7*1"- -6'7&$ 6*%"

*4>06+4. 0' $' @+06 $"-21+9+4. 06" /'0+'4 '5 5*&&+4.

'9:"20-? *4$ @6*0 06" />-0"1+'7- 2'4-0*40 '5

(1'('10+'4*&+0> 2'7&$ /"*4; )*&+&"' W"8(&*+4"$, 06"

2'4-0*40 9> 06+4=+4. *9'70 1"20*4.&"- '5 * 3+8"$ -6*("?/"*4+4. @+06 * 3+8"$ 1*0+' '5 6"+.60 0' @+$06;

J5 @" 2'4-+$"1 1"20*4.&"- 06*0 *1" 0@+2" *- 6+.6 *- 06">

*1" @+$" 06"4 06"+1 *1"* @+&& 9" 0@+2" 06" *1"* '5 06"

-B7*1" 5'1/"$ '4 06"+1 9*-"; b'1" ."4"1*&&>? 1"20*4.&"-

'5 * 3+8"$ -6*(" @+&& "86+9+0 06" -*/" 1*0+' 9"0@""4 06"+1

7$%&'./ M

;45'5$.B1 G$.=$&6'%45 4/45<1'1

10

Figure 1.1



*1"* *4$ 06" *1"* '5 06" -B7*1" 5'1/"$ '4 06"+1 9*-" *-

06"> $' 9"0@""4 06"+1 6"+.60 *4$ @+$06;

)*&+&"' 06"1"5'1" 2'42"+%"$ 6+- ."4"1*& $+-2'%"1> *9'70

9*&&- 1'&&+4. $'@4 (&*4"- 0' -*> 06*0 0' * (&*4" @+06 *

.+%"4 +42&+4*0+'4 06"1" 2'11"-('4$"$ * .+%"4 -6*(" '5

1"20*4.&"; A6" $+-0*42" 06*0 * 9*&& 6*- 1'&&"$ $'@4 06"

(&*4" +- .+%"4 9> 06" *1"* '5 06" 1"20*4.&" '5 06+-

(*10+27&*1 -6*(" @6'-" @+$06 +- "B7*& 0' 06" $71*0+'4 '5

01*%"&;

A6"-" $*>- @" @'7&$ $"/'4-01*0" 06+- 7-+4. *4

*4+/*0+'4 '5 * 1"20*4.&" '5 06+- (*10+27&*1 -6*(" @6'-"

@+$06 +- "8(*4$+4. *0 06" 1*0" '5 '4" /"01" ("1 -"2'4$; \0

*4> +4-0*40 +0- @+$06 @'7&$ -*> 6'@ &'4. +0 @*- -+42" 06"

9*&& -0*10"$ 1'&&+4. *4$ +0- *1"* @'7&$ -*> 6'@ 5*1 +0 6*$

1'&&"$;

b'1" ."4"1*&&>? *4> +40"1%*& '5 0+/" 2'11"-('4$- 0' *

-"./"40 '4 06" 9*-" '5 06" "8(*4$+4. 1"20*4.&"; J4 06"2'71-" '5 -726 * 0+/" +40"1%*& 06" 1"20*4.&" "8(*4$- 51'/

* -/*&&"1 0' * &*1."1 '4" 061'7.6 *4 dC-6*(" *- -6'@4 +4

]+.71" G;I;

A6" -"./"40 '5 06" 9*-" 2'11"-('4$+4. 0' 06" 0+/"

+40"1%*& +- 06" W5''0, '5 06+- dC-6*("; A6" *1"* '5 06" dC

-6*("? 9"+4. 06" $+55"1"42" +4 06" *1"* '5 06" 1"20*4.&"- *0

06" -0*10 *4$ "4$ '5 06" 0+/" +40"1%*&? .+%"- 06" $+-0*42"

01*%"&&"$ 9> 06" '9:"20 $71+4. 06+- 0+/" +40"1%*&; A6"

$71*0+'4 '5 06" 0+/" +40"1%*& +- .+%"4 9> 06" &"4.06 '5 +0-

11

Figure 1.2





A6" *1"* '5 06+- -01+( -*>- 6'@ 5*1 06" '9:"20 6*- 5*&&"4

$71+4. 06+- 0+/"? *4$ +0- @+$06 .+%"- 06" $71*0+'4 '5 06"

0+/" +40"1%*&;

\ -01+( +- /726 -+/(&"1 06*4 *4 dC-6*("; ^70 +0 *&-' .+%"-

* $+1"20 ."'/"01+2*& +40"1(1"0*0+'4 '5 -(""$; A6" *1"* '5

06" -01+( +- "B7*& 0' 06" *1"* '5 * 1"20*4.&" @6'-" 6"+.60

+- 06" *%"1*." '5 06" 6"+.60- '5 +0- "$."-; X'7 2*4 -"" 06+-

+4 ]+.71" G;V? @6+26 -6'@- 06*0 06" -01+( +- '90*+4"$ 51'/

06+- 1"20*4.&" 9> 2700+4. '70 * 01+*4.&" '4 06" &"50? *4$

*$$+4. +4 * 2'4.17"40 01+*4.&" '4 06" 1+.60; c'@ 06"

$+-0*42" 01*%"&&"$ +- "B7*& 0' 06" *1"* '5 06+- 1"20*4.&"?

@6+26 +- +0- 6"+.60 0+/"- 06" @+$06 '5 +0- 9*-"; ^70 06+- +-

06" 6"+.60 0+/"- 06" $71*0+'4 '5 01*%"&? -' 06" 6"+.60 /7-0

.+%" 06" -(""$ '5 06" 5*&&+4. '9:"20;

A67- )*&+&"' 2'42&7$"$ 06*0 &-$ 10$$3 .2 &-$ 2455'/G .6

6.55'/G ."#$%& .($6 4/< &'=$ '/&$6(45 =91& "$ $N945 &.

&-$ 4($64G$ .2 &-$ -$'G-& ? *0 06" 9".+44+4. *4$ "4$ '5

06*0 0+/" +40"1%*&? '5 06" "8(*4$+4. 01+*4.&" 06*0

$"-21+9"- +0- /'0+'4;

J4 '1$"1 0' 6*%" * 9"00"1 +4-+.60 '5 @6*0 +- /"40+'4"$ +4

06" (1"%+'7- (*1*.1*(6-? (&*> 5'1 * @6+&" @+06 06"

8/&$64%&'($ )*A @6+26 +- 9*-"$ '4 )*&+&"',- @'1=; J4 06+-

*((&"0 >'7 @+&& 3+4$ 06" 1+.60 *4.&"$ 01+*4.&" /'$"& '5

)*&+&"' 5'1 1"(1"-"40+4. 06" -(""$ '5 06" 9*&&; A6" 9*-" '5

06" 01+*4.&" 1"(1"-"40- 0+/" *4$ 06" 6"+.60 1"(1"-"40-

-(""$;

H' >'7 @+&& 9" *9&" 0' -"" -+/7&0*4"'7-&> 06" /'0+'4 '5

06" 9*&& 1'&&+4. $'@4 06" +42&+4"$ (&*4" *4$ 06"

."'/"01+2*& 1"(1"-"40*0+'4 '5 06" -(""$;

13

Figure 1.5



I>&$6/45 5'/J 2.6 KL? ($61'./F

600(FMM@@@;."'."91*079";'1.M-07$"40M/GLKGLV

Interactive 1.3

14





J0 '22711"$ 0' )*&+&"' 06*0 06" 6"+.60 '5 *4> 01+*4.&" 2'7&$

*&-' 9" 1".*1$"$ *- -726 *4 *%"1*." @6"4 06" "*1&+"1 *4$

&*0"1 +4-0*40- @"1" 06" -*/"; A6+- &"$ 6+/ 0'

$"2+$" 06*0 06" 6"+.60 2'7&$ 9" +40"1(1"0"$?

&+=" 06" '06"1 *%"1*."-? *- * /"*-71" '5-(""$? 970 *0 * -+4.&" +4-0*40;

)*&+&"' @*- 4'0 -*>+4. 06*0 06" 6"+.60 @*-

06" -(""$ '5 06" 1'&&+4. 9*&& *0 *4 +4-0*40? +4

06" 7-7*& -"4-" '5 06*0 0"1/; D" 74$"1-0''$

06*0 -(""$ *0 *4 +4-0*40 /*$" 4' -"4-" +4

06" 7-7*& 0"1/- '5 $+-0*42" 01*%"&&"$

$+%+$"$ 9> $71*0+'4 '5 01*%"&? 5'1 +4 *4

+4-0*40 '4" 6*- 01*%"&&"$ 4' $+-0*42" *4$

0*="4 4' 0+/" 0' $' +0; D" @*- 4'0 -*>+4.

06*0 6" 6*$ -'/"6'@ 3+4"--"$ 06+- *4$

$+%+$"$ _"1' 9> _"1'? '1 -'/"06+4. +43+4+0"&>

-/*&& 9> -'/"06+4. +43+4+0"&> -/*&&; D" @*-

-+/(&> 4'0+2+4. 06*0 06" 6"+.60 2'7&$ 9"

1".*1$"$ *- *4 +4-0*42" '5 06" *%"1*."

6"+.60? *4$ -+42" *%"1*." 6"+.60- 2'11"-('4$"$ 0' -(""$-

06" 6"+.60 /+.60 06"1"5'1" 9" 2'4-+$"1"$ *&-' *- *

/"*-71" '5 -(""$e 970 2&"*1&> *4 "40+1"&> 4"@ =+4$ '5

/"*-71";

)*&+&"' -"+_"$ '4 06+- 4"@ 4'0+'4 '5 -(""$

9"2*7-"? 74&+=" 06" 7-7*& 4'0+'4? +0 @*- *

0+/" %*1>+4. B7*40+0> :7-0 &+=" $+-0*42"? *4$

6*$ * 1*0" '5 26*4." '@4; A6" 7-7*& 4'0+'4

'5 -(""$ +- * B7*40+0> $"0"1/+4"$ '%"1 *4

+40"1%*& '5 0+/" 1*06"1 06*4 *4 +4-0*40 *4$ +0

/*="- 4' -"4-" *0 *&& 0' 0*&= *9'70 06" 1*0"

'5 26*4." '5 -726 * B7*40+0>;

M*: ,-$ "'6&- .2 4%%$5$64&'./

A6" 1*0" '5 26*4." '5 06" +4-0*40*4"'7-

-(""$ +4 06" 4'4C+4-0*40*4"'7- -"4-" +-? +4

-'/" @*>? 06" -6*(" '5 06" "8(*4$+4.

01+*4.&"e +0 +- 06" 1*0+' '5 +0- 6"+.60 0' +0-

@+$06; A6+- +- 9"2*7-" +4 *4> 0+/" +40"1%*&?

@6+26 2'11"-('4$- 0' * -01+(? 06" 1*0" '5

26*4." '5 06" 6"+.60 '5 06" 01+*4.&" +- .+%"4

9> 06" 26*4." +4 6"+.60 $+%+$"$ 9> 06" @+$06 '5 06" -01+(;

^70 06+- +- :7-0 06" 1*0+' '5 6"+.60 0' @+$06 '5 06" 01+*4.&"

15

Galileo Galilei sculpted by Aristodemo

Costoli in 1851. Firenze, Italy. (Source:

Wikipedia)



*0 06" 0'( '5 06" -01+(? @6+26 6*- 06" -*/" -6*(" *- R+-

-+/+&*1 0'S 06" "8(*4$+4. 01+*4.&" +0-"&5;

)*&+&"' *1.7"- 06*0 06" +4-0*40*4"'7- -(""$ -6'7&$ 9"

1".*1$"$ *- 6*%+4. *4 +4-0*40*4"'7- 1*0" '5 26*4." 06*0

+- *&-' 2'4-0*40? 06" -*/" 2'4-0*40 *- 06" 4'4C

+4-0*40*4"'7- 1*0" '5 26*4."; D" -*@ 06+- *- ="> 0'

74$"1-0*4$+4. @6> 01+*4.&"-? *4$ 067- 1"20*4.&"- *4$

-B7*1"-? @'7&$ 6*%" *4>06+4. 0' $' 06" /'0+'4 '5 5*&&+4.

'9:"20-; \4$ 6" -*@ +4 +0 * 4"@ 2'42"(0+'4 '5 * 4*071*&

&*@; A6" &*@ '5 /'0+'4 5'1 * 5*&&+4. '9:"20 +- 06*0 06" 1*0"

'5 26*4." '5 +0- -(""$ +- 2'4-0*40;

A6" 4'%"&0> +4 06+- 4*071*& &*@ +- 06*0 +0 +- 51*/"$ +4

0"1/- '5 1*0"- '5 26*4."; J0 2*11+"- 06" +/(&+2*0+'4 06*0

=4'@+4. 06" 1*0" '5 26*4." '5 * B7*40+0> *&&'@-

$"0"1/+4*0+'4 '5 06" B7*40+0> +0-"&5; J0 +- 06" 9".+44+4. '5

* $>4*/+2*& 74$"1-0*4$+4. '5 06" 4*071*& @'1&$; J4

*$%*42"$ /*06"/*0+2*& 0"1/- @" @'7&$ -*> 06*0 +0

-+.4*&- 06" -'&70+'4 '5 $+55"1"40+*& "B7*0+'4-; A6" /"1+0 '5

)*&+&"'f- /'$"&? 6'@"%"1? +- 06*0 +0 2*4 9" 74$"1-0''$ +4

0"1/- '5 :74+'1 6+.6 -26''& ."'/"01>? *4$ >"0 2'4%">- *

('@"157& 4"@ @*> '5 74$"1-0*4$+4. 4*071*& (6"4'/"4*

+4 0"1/- '5 * 4'%"& 2'42"(0 '5 1*0" '5 26*4.";

A6" 1*0" *0 @6+26 06" +4-0*40*4"'7- -(""$ +421"*-"$ @*-

.+%"4 * 4"@ 4*/" 9> )*&+&"' g4%%$5$64&'./; A6" 2'42"(0

'5 /"*-71*9&" *22"&"1*0+'4 $+$ 4'0 "8+-0 9"5'1" 6+/?

*&06'7.6 74$'790"$&> ("'(&" 74$"1-0''$ 06" 5""&+4. '5

."00+4. 5*-0"1 '1 -&'@"1; A6" *22"&"1*0+'4 '5 * 5*&&+4.

'9:"20 @*- $"0"1/+4"$ 9> )*&+&"' +4$+1"20&> 51'/ 6+-

"8("1+/"40-; J4 '71 74+0- +0 +- P;Q5M'#*M'#*; J0 +- 06" 3+1-0

(6>-+2*& 2'4-0*40 '5 4*071" $+-2'%"1"$? 9> 2'401*-0 @+06

."'/"01+2*& 2'4-0*40- &+=" π ;

M*A ;45'5$.B1 =.3$5 G$/$645'R$3

d"0 7- 4'@ ."4"1*&+_" )*&+&"'f- /'$"& -' 06*0 @" *11+%" *0

06" +4$"3+4+0" +40".1*&; A6" "8(*4$+4. 01+*4.&" 2*4 9"

2'4-+$"1"$ *- $"0"1/+4"$ 9> 06" &+4" y = mx? 06" xC*8+-

*4$ * %"10+2*& &+4" /'%+4. *0 74+0 -(""$ 0' 06" 1+.60

R]+.71" G;KF (S; ^> 26*4.+4. 06" &+4" y = mx 0' *4'06"1

271%" @" '90*+4 * 4"@ /'$"& 5'1 * 0+/" %*1>+4. B7*40+0>

*4$ +0- 1*0" '5 26*4."; J4 06+- /'1" ."4"1*& /'$"& @" *&-'

3+8 * %"10+2*& &+4"? -' 06*0 /'$"& +- 06" "8(*4$+4. 1".+'4

16



74$"1 06" 271%"? '%"1 06" xC*8+- *4$ 9"0@""4 06" 3+8"$

*4$ /'%+4. %"10+2*& &+4"- R-"" ]+.71" G;KF 1S;

A6" *1"* '5 06+- "8(*4$+4. 1".+'4 +- 06" 0+/" %*1>+4.

B7*40+0> '5 +40"1"-0; J0- 1*0" '5 26*4." *0 *4> +4-0*40 +- 06"

6"+.60 '5 06" 271%" '%"1 06" xC*8+- *&'4. 06" /'%+4.

%"10+2*& &+4" *0 06*0 +4-0*40;

A6+- +- :7-0 *4 *4+/*0"$ %"1-+'4 '5 06" +4$"3+4+0" +40".1*&

'5 06" 57420+'4 @6'-" .1*(6 +- .+%"4 9> 06" 271%"; J0 @+&&

9" $+-0719+4. +4 06*0 +0 $"3+4"- *@*> 06" 574$*/"40*&

06"'1"/ '5 2*&27&7-; J0 +- -0+&& -'/"@6*0 2'/(70*0+'4*&&>

+401*20*9&"; D'@"%"1? +0 6*- 06" "4'1/'7- /"1+0 '5

-6'@+4. %"1> 2&"*1&> 06*0 =4'@+4. 06" 1*0" '5 26*4." '5 *

B7*40+0> *4$ 06" +4-0*40 @6"1" +0 6*- 06" %*&7" _"1'

$"0"1/+4"- +0 2'/(&"0"&>; ]'1 06" 1*0" '5 26*4."$"0"1/+4"- 06" 6"+.60 '5 06" 271%" *9'%" 06" xC*8+-? 06"

271%" +- -+/(&> 06" .1*(6 '5 06" 1*0" '5 26*4."? *4$ 06"

+4-0*40 @6"1" 06" B7*40+0> +- _"1' $"0"1/+4"- 06" 3+8"$

%"10+2*& R&"50 6*4$S &+4";

Figure 1.6

17



[The universe] cannot be read until we

have learnt the language and become

familiar with the characters in which it

is written. It is written in mathematicallanguage, and the letters are triangles,

circles and other geometrical figures,

without which means it is humanly

impossible to comprehend a single

word.

Galileo, Il Saggiatore, p. 25.

18



C )*&+&"'? )*&+&"+ RGKINS? 6/ 7(22&(!+-# R+4 J0*&+*4S? <'/";

C )*&+&"+? )*&+&"' RGPVUS hGKNIi; 8&(/+20# 9+)*#-)&)2 :3+ ;#3 7*&#)*#'<

A1*4-&*0"$ 9> D"41> [1"@ *4$ \&5'4-' T" H*&%+'; c"@ X'1=? cXF T'%"1

`79&+2*0+'4- R((; GVNCGQLS; \%*+&*9&" '4&+4"F A@' c"@ H2+"42"-;

C J-0+070' " b7-"' $+ H0'1+* $"&&* H2+"4_*; RILGGS; 9(!(/+20# +, !"# =0'#+

>(/&/#+?' &)'!-05#)!' +) %&'4/(@< J0*&+*;

\((&"0- $"-+.4"$ @+06 ;$.;$"64 9> j7*4 [*1&'- `'42" [*/(7_*4'

600(FMM@@@;."'."91*079";'1.M7-"1M(1'3+&"M+$MQPVI

5

D$2$6$/%$1

19

Front page from the original version of Galileo’s book:Dialogue Concerning Two New Sciences.

Gallery 1.5

http://www.geogebratube.org/user/profile/id/8952

http://www.geogebratube.org/user/profile/id/8952

http://galileoandeinstein.physics.virginia.edu/tns_draft/index.html

http://galileoandeinstein.physics.virginia.edu/tns_draft/index.html



20

j'64 <+2"

D'4'1*1> `1'5"--'1H26''& '5 b*06"/*0+2- *4$ H0*0+-0+2-O4+%"1-+0> '5 H>$4">

k"&&> l; b*006"@-A"*26+4. *4$ l$72*0+'4*& T"%"&'(/"40 J4-0+070"O4+%"1-+0> '5 m7""4-&*4$

j7*4 [*1&'- ̀ '42" [*/(7_*4'H26''& '5 b*06"/*0+2- *4$ ̀ 6>-+2-O4+%"1-+0> '5 m7""4-&*4$

F9&-.61F

70$%'45 &-4/J1 &.F

`"0"1 \$*/-\--'2+*0" T"*4H26''& '5 b*06"/*0+2- *4$ ̀ 6>-+2-O4+%"1-+0> '5 m7""4-&*4$

Date post:	02-Jun-2018
Category:	Documents
Upload:	juancarlos-ponce
View:	222 times
Download:	0 times

A Geometrical Approach to Speed and Acceleration (complete)

Documents