TEMA 5 ASIE

8/9/2019 TEMA 5 ASIE

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Regula trapezului(lansarea unei rachete)

Miclaus Alexandru:322/2

Problema propusă:

O rachetă decolează pe vertical şi elimină combustibil cu o viteză de 2000

m/s cu un consum de combustibil de 200 !"/s# $ni%ial masa rachetei este de&0'000 !"# (acă racheta porneşte de la 0=t sec#' cum putem calcula

distan%a parcursă de rachetă )n perioada de la *=t p+nă la 30=t sec#,

Rezolvare:

(aca=

0m masa initiala a rachetei la momentul 0=t -!".

=q debitul cu care este expulzat combustibilul -!"/sec.

=u viteza cu care este expulzat combustibilul -m/s.(eoarece combustibilul este expulzat din racheta' masa rachetei va scadea in

timp' si devine la un moment dat de timp t:qt mm −= 0

ortele exercitate asupra rachetei in orice moment rezulta din aplicarea le"ii

a doua a miscarii' a lui e1tonaqt m g qt muqmamg uqma F .-.- 00 −=−−⇒=−⇒=∑

nde = g acceleratia "ravitationala -m/s2.

C gt qt mudt

dx g

qt m

uq

dt

xd g

qt m

uqa

e +−−−=⇒−−

=⇒−−

= .-lo" 00

2

2

0

Pornind de la aptul ca racheta pleaca de pe suport

.-lo".-lo"000 00 muC C mudt

dx

t La ee =⇒+=⇒=⇒=

(eci:

gt qt m

mu

dt

dxmu gt qt mu

dt

dxee −

−

=⇔+−−−=0

0

e00 lo".-lo".-lo"

4ezulta ca distanta parcursa de racheta de la 0t t = la t t = este'


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∫

−

−

=

0i

i

m

mln

t

t

dt gt qt

u x

$nlocuind cu datele precizate' se obtine:

In continuare rezolvam in MATLAB problema:

*programul in matlab este urmatorul:

clc

clf

clear all

% Purpose

% To illustrate the trapezoidal method as applied to a function

% of the user's choosing.

% Keyword

% Trapezoidal Method% Numerical Integration

% Inputs

% This is the only place in the program where the user makes the

changes

% ased on their wishes

% f!"#$ the function to integrate

f &!"# ()))*log!+,))))-+,)))).-(+))*"#./*" 0

% a$ the lower limit of integration

a/ 0

% $ the upper limit of integration

1) 0

% n$ the ma"imum numer of segments


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Miclaus Alexandru:322/2 n2) 0

%**********************************************************************

% This displays title information

disp!sprintf!'3n3n4imulation of the Trapezoidal Method'##

disp!sprintf!'5ni6ersity of 4outh 7lorida'##

disp!sprintf!'5nited 4tates of 8merica'##

disp!sprintf!'kaw&eng.usf.edu3n'##

disp!sprintf!'3n*******************************Introduction*************

********************'##

% 9isplays introduction te"t

disp!'Trapezoidal rule is ased on the Newton:otes formula that if one

appro"imates the'#

disp!'integrand y an nth order polynomial$ then the integral of the

function is appro"imated'#

disp!'y the integral of that nth order polynomial. Integrating

polynomials is simple and is'#

disp!'ased on calculus. Trapezoidal rule is the area under the cur6efor a first order'#

disp!'polynomial !straight line#.'#

% 9isplays what inputs are eing used

disp!sprintf!'3n3n********************************Input

9ata**********************************3n'##

disp!sprintf!' f!"#$ integrand function'##

disp!sprintf!' a %g$ lower limit of integration '$a##

disp!sprintf!' %g$ upper limit of integration '$##

disp!sprintf!' n %g$ numer of sudi6isions'$n##

format short g

% :alculate the spacing parameter

disp!sprintf!'3n7or this simulation$ the following parameter is

constant.3n'##

h!a#-n 0

disp!sprintf!' h ! a # - n '##

disp!sprintf!' ! %g %g # - %g '$$a$n##

disp!sprintf!' %g'$h##

% This egins the simulation of the method

disp!sprintf!'3n*******************************4imulation***************

******************3n'##

sum) 0

disp!'The appro"imation is e"pressed as'#

disp!' '#

disp!' appro" h * ! ).2*f!a# ; 4um !i+$n+# f!a;i*h# ;).2*f!# #'#

disp!' '#

% 4um all function 6alues not including e6alations at a and

disp!'+#


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Miclaus Alexandru:322/2disp!' 4um !i+$n+# f!a;i*h#'#

disp!' '#

for i+=n(

disp!sprintf!' f!%g#'$a;i*h##

end

disp!sprintf!' ; f!%g#'$a;!n+#*h##

disp!sprintf!' '##

for i+=n(

sumsum;f!a;i*h# 0

disp!sprintf!' %g'$f!a;i*h###

end

sumsum;f!a;!n+#*h# 0

disp!sprintf!' ; %g'$f!a;!n+#*h###

disp!sprintf!' '##

disp!sprintf!' %g3n'$sum##

% Now add half the end point e6aluations

disp!'(# 8dd to this ).2*!f!a# ; f!##'#

disp!' '#

disp!sprintf!' %g ; ).2*!%g ; %g# '$sum$f!a#$f!###

disp!sprintf!' %g ; %g %g'$sum$).2*!f!a#;f!##$sum;).2*!f!a#;f!####

sumsum;).2*!f!a#;f!## 0

disp!' '#

% 8nd finally multiply y h

disp!'1# Multiply this y h to get the appro"imation for the integral.'#

disp!' '#

disp!sprintf!' appro" h * %g'$sum##

disp!sprintf!' appro" %g * %g'$h$sum##

appro"h*sum 0

disp!sprintf!' appro" %g'$appro"##

% The following displays results

disp!sprintf!'3n3n**************************>esults*********************

*******'##

% The following finds what is called the '?"act' solution

e"act @uad!f$a$# 0

disp!sprintf!'3n 8ppro"imate %g'$appro"##

disp!sprintf!' ?"act %g'$e"act##

disp!sprintf!'3n True ?rror ?"act 8ppro"imate'##

disp!sprintf!' %g %g'$e"act$appro"##

disp!sprintf!' %g'$e"actappro"##

disp!sprintf!'3n 8solute >elati6e True ?rror Percentage'##

disp!sprintf!' A ! ?"act 8ppro"imate # - ?"act A *

+))'##disp!sprintf!' A %g - %g A * +))'$e"actappro"$e"act##

disp!sprintf!' %g'$as! !e"actappro"#-e"act #*+))##

disp!sprintf!'3nThe trapezoidal appro"imation can e more accurate if we

made our'##

disp!sprintf!'segment size smaller !that is$ increasing the numer of

segments#.3n3n'##


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Miclaus Alexandru:322/2% The following code is needed to produce the trapezoidal method

% 6isualization.

"!+#a 0

y!+#f!a# 0

hold on

for i+=n

"!i;+#a;i*h 0

y!i;+#f!"!i;+## 0

fill!B"!i# "!i# "!i;+# "!i;+#C$ B) y!i# y!i;+# )C$ 'y'#

end

"rangea=!a#-+)))=0

plot!"range$f!"range#$'k'$'Dinewidth'$(#

title!'Integrand function and Eraphical 9epiction of Trapezoidal

Method'#

5r ezultatele problemei sunt:55555555555555555555$nput (ata5555555555555555555555555

-x.' inte"rand unction a 6 *' lo1er limit o inte"ration

b 6 30' upper limit o inte"rationn 6 70' number o subdivisions

or this simulation' the ollo1in" parameter is constant#

h 6 - b 8 a . / n

6 - 30 8 * . / 706 0#&&

555555555555555555559imulation5555555555555555555555555

he approximation is expressed as


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approx 6 h 5 - 0#75-a. ; 9um -i6'n8. -a;i5h. ; 0#75-b. .

.


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rue rror 6 xact 8 Approximate

6 207>*? 8 207>?2 6 82#?7=3*

Absolute 4elative rue rror Percenta"e 6 B - xact 8 Approximate . / xact B 5 00 6 B 82#?7=3* / 207>*? B 5 00 6 0#00&3==?

*se obtine graficul urmator:

Concluzie:


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TEMA 5 ASIE

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