<Text-field style="Heading 1" layout="Heading 1">Introduction</Text-field>

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Deriving the Rocket Equation

\302\251 Maplesoft, a division of Waterloo Maple Inc., 2005

Maple 10 lets you create sophisticated live documents that seamlessly integrate the mathematical results with derivations, explanations, plots, and other forms of technical knowledge. This document develops a simple model of a rocket to illustrate the use of many of these new features.

The following Maple features are highlighted:

Document mode and document blocks

2-D math editor and palettes

Autoexecute regions

Plotting

Tables

Embedded components

Visit www.maplesoft.com to learn about other new features, to watch Maple 10 Feature Demos, and take the 10 Minute Tour.

From inside Maple 10, see Help>What's New, and Help>Take a Tour of Maple for more information.

Throughout this document, explanations will appear in this style of text, to distinguish the explanations from the text of the example.

This document was created in Maple 10's Document mode. Some computations are hidden from view for presentation reasons. Maple 10 uses markers to indicate the presence of hidden formatting information, including document blocks and bookmarks, in a column to the immediate left of this text. From the View menu, ensure that the Markers entry is selected so that you can see this information.

<Text-field style="Heading 1" layout="Heading 1">Initialization</Text-field>

Initialization steps are crucial to the proper functioning of the document, but generally do not need to be seen by the reader. In such cases, the initialization steps can be hidden inside a document block. As well, to ensure that the necessary initialization is done before the user starts to interact with the document, these steps are put into an autoexecute region. Even people who know nothing about Maple will be able to read, interact with and modify this document once it is opened, without the need to first execute anything. The marker to the left of the blue text below indicates the presence of an autoexecute region inside a document block.

To see inside a document block, place the cursor on the visible part of the region, and select View>Expand Document Block from the menu. Alternatively, you can right-click (control-click on Macintosh) on the marker bar beside the text, and select Document Block>Expand Document Block from the menu.

restart: interface( warnlevel = 0 ): with( DocumentTools ): printf ("Loading packages... done");

Loading packages... done

getParameters := proc() local m, f, b, v, g; m[r] := convert( parse ( GetProperty ( tfMassRocket, value ) ), unit_free) ; f[0] := convert( parse ( GetProperty ( tfMassFuel, value ) ), unit_free ); b := convert( parse ( GetProperty ( tfFuelRate, value ) ), unit_free ); v[e] := convert( parse ( GetProperty ( tfVelExhaust, value ) ), unit_free ); g := convert( parse ( GetProperty ( tfG, value ) ), unit_free ); (m[r], f[0], b, v[e], g) end proc: setDefaultValues:=proc(); SetProperty ( tfMassRocket, 'value' , "0.6e6" ); SetProperty ( 'tfMassFuel', 'value', "1.92e6" ); SetProperty ( 'tfFuelRate', 'value', "1.6e4" ); SetProperty ( 'tfVelExhaust', 'value', "-3000" ); SetProperty ( 'tfG', 'value', "-9.81" ); end proc: setPlot:= proc( plotName, plotValue, plotVar, minName, maxName, plotTitle ) local minValue, maxValue; minValue := GetProperty( minName, value ); maxValue := GetProperty( maxName, value ); SetProperty( plotName, value, plot(plotValue, plotVar=minValue..maxValue, title=plotTitle, labels=["Time (s)", "Velocity (m/s)"], labeldirections=[horizontal, vertical], gridlines ) ): end proc: updateSlider := proc (minMassName, maxMassName, massSlider) minMass := parse ( DocumentTools:-GetProperty( minMassName, value ) ); maxMass := parse ( DocumentTools:-GetProperty( maxMassName, value ) ); mass := DocumentTools:-GetProperty( massSlider, value ); if minMass >= maxMass then tmp := maxMass; maxMass := minMass; minMass := tmp; DocumentTools:-SetProperty( maxMassName, value, maxMass); DocumentTools:-SetProperty( minMassName, value, minMass); end if; if minMass > mass then mass := minMass; DocumentTools:-SetProperty( massSlider, value, mass); end if; if maxMass < mass then mass := maxMass; DocumentTools:-SetProperty( massSlider, value, mass); end if; DocumentTools:-SetProperty( massSlider, lower, minMass); DocumentTools:-SetProperty( massSlider, upper, maxMass); minorT := floor((maxMass-minMass)/50); DocumentTools:-SetProperty( massSlider, minorTicks, minorT); majorT := floor((maxMass-minMass)/2); DocumentTools:-SetProperty( massSlider, majorTicks, majorT); end proc: setPlot3d:= proc( plotName, plotValue, plotVar, minName, maxName, plotTitle, minMassName, maxMassName, massName ) local minValue, maxValue, minMass, maxMass; minValue := GetProperty( minName, value ); maxValue := GetProperty( maxName, value ); massX := GetProperty( massName, value ); minMass:= parse ( GetProperty( minMassName, value ) ); maxMass := parse ( GetProperty( maxMassName, value ) ); plt := plots[display] (plot3d(plotValue, plotVar=minValue..maxValue, mass=minMass..maxMass, axes=normal, title=plotTitle, labels=["Time", "Mass", "Velocity"], labeldirections=[HORIZONTAL, HORIZONTAL, VERTICAL], transparency=0.4, glossiness=0.9, lightmodel=light2, style=patchnogrid, orientation=[45, 135], shading=zhue), plots[spacecurve] ([plotVar, massX, Re(eval (plotValue, mass=massX))], plotVar=minValue..maxValue, color=black, thickness=2) ); SetProperty(plotName, value, plt ) end proc: # load units after becuase Units overrides normal which is used in setPlot3d with(Units[Standard]): printf( "Loading custom procedures... done");

Loading custom procedures... done

<Text-field style="Heading 1" layout="Heading 1">Problem statement</Text-field>

This section demonstrates the use of tables for organizing information and layout control. This table shows borders. Further below, you will see tables where the borders are hidden. To create a table in your document, select Insert>Table from the menu.

It also used embedded GUI controls. The user can change the values in the edit boxes, and press the button to restore the default values. Embedded GUI controls are added to a document through the use of the Components palette.

Here are the parameters for the problem, the values for the Saturn V launch are already filled in. You can change the values if you would like to experiment with how different values change the results. To see the result of your changes, press the execute all button (!!!) in the Maple toolbar.

Variable

Meaning

Value

Unit

To fill in the values for the Saturn V launch, click this button:

setDefaultValues()

JkkibUc2IjYjSSJyR0Yk

mass of an unfueled rocket

SSNLZ0c2Ig==

JkkiZkc2IjYjIiIh

mass of the initial fuel

SSNLZ0c2Ig==

SSJiRzYi

rocket's constant burn rate of fuel

KiZJI0tnRzYiIiIiSSJzR0YkISIi

Jkkidkc2IjYjSSJlR0Yk

constant velocity of exhaust (negative)

KiZJIm1HNiIiIiJJInNHRiQhIiI=

SSJnRzYi

acceleration of gravity (negative)

KiZJIm1HNiIiIiJJInNHRiQhIiM=

We will use the formula L0knVGhydXN0RzYiLCQqJkkiYkdGJCIiIiZJInZHRiQ2I0kiZUdGJEYoISIi, to find the velocity Jkkidkc2IjYjSSJyR0Yk of a rocket T seconds after lift off in KiZJIm1HNiIiIiJJInNHRiQhIiI=.

<Text-field style="Heading 1" layout="Heading 1">Model and solution</Text-field>

Plan:

(1) Compute the net upward force on the rocket as a function of time.

(2) Divide the net force by the mass to get the net acceleration.

(3) Integrate the net acceleration from L0kidEc2IjsiIiFJIlRHRiQ= to get velocity.

In the sentence below, note that the formula calculated by Maple (in blue) appears on the same line as the initial formula entered by the author (in black). Maple 10's document mode supports inline output, so the input and output appear on the same line. Inline output allows you to reduce unnecessary whitespace and maintain the flow of your text. The Maple commands used to calculate the answer are hidden inside a document block. The document block markers show the start and finish of the document block, and can be expanded through the View menu or by right-clicking between the markers, as before.

Enter the formula for the mass of the rocket as a function of time t, which is L0kiTUc2IiwoJkkibUdGJDYjSSJyR0YkIiIiJkkiZkdGJDYjIiIhRioqJkkiYkdGJEYqSSJ0R0YkRiohIiI=

PkkiTUc2Ii1JIitHNiQlKnByb3RlY3RlZEcvSSttb2R1bGVuYW1lR0YkSSlTdGFuZGFyZEc2JC9GKkkmVW5pdHNHNiRJKF9zeXNsaWJHRiRGKEYoNiQtRiY2JC1JIipHRic2JCZJIm1HRiQ2I0kickdGJC1JJVVuaXRHRiQ2I0kja2dHRiQtRjU2JCZJImZHRiQ2IyIiIUY7LUkiLUdGJzYjLUY1NiQtRjU2JC1GNTYkSSJiR0YkLUY8NiMtRjU2JEY+LUkiL0dGJzYjSSJzR0YkSSJ0R0YkLUY8RlU=

KiYsKCZJIm1HNiI2I0kickdGJiIiIiZJImZHRiY2IyIiIUYpKiZJImJHRiZGKUkidEdGJkYpISIiRiktSSVVbml0RzYkL0krbW9kdWxlbmFtZUdGJkkmVW5pdHNHNiRJKF9zeXNsaWJHRiYlKnByb3RlY3RlZEdGOjYjSSNrZ0dGJkYp

;

KiYsKCZJIm1HNiI2I0kickdGJiIiIiZJImZHRiY2IyIiIUYpKiZJImJHRiZGKUkidEdGJkYpISIiRiktSSVVbml0RzYkL0krbW9kdWxlbmFtZUdGJkkmVW5pdHNHNiRJKF9zeXNsaWJHRiYlKnByb3RlY3RlZEdGOjYjSSNrZ0dGJkYp

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJw==

The inline mathematics is still live, so if you change the initial formula, by say changing JkkibUc2IjYjSSJyR0Yk to LCQmSSJtRzYiNiNJInJHRiUiIiM=, highlight the entire line, and press the ! button on the toolbar, the line will re-execute and the output will be updated appropriately. Be sure to change it back again if you wish to use the application as intended.

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnNet upward force on the rocket is LCYqJkkiYkc2IiIiIiZJInZHRiU2I0kiZUdGJUYmISIiKiZJIk1HRiVGJkkiZ0dGJUYmRiY= (difference between thrust and gravity). Notice Maple automatically converts the units KihJI2tnRzYiIiIiSSJtR0YkRiVJInNHRiQhIiM= to Newtons (N).

PiZJIkZHNiI2I0kkbmV0R0YlLUkiK0c2JCUqcHJvdGVjdGVkRy9JK21vZHVsZW5hbWVHRiVJKVN0YW5kYXJkRzYkL0YtSSZVbml0c0c2JEkoX3N5c2xpYkdGJUYrRis2JC1JIi1HRio2Iy1JIipHRio2JC1GOTYkLUY5NiRJImJHRiUtSSVVbml0R0YzNiMtRjk2JEkja2dHRiUtSSIvR0YqNiNJInNHRiUmSSJ2R0YlNiNJImVHRiUtRkE2Iy1GOTYkSSJtR0YlRkYtRjk2JC1GOTYkSSJNR0YlSSJnR0YlLUZBNiMtRjk2JEZSLUZHNiMtSSJeR0YqNiRGSSIiIw==

KiYsJiomSSJiRzYiIiIiJkkidkdGJjYjSSJlR0YmRichIiIqJiwoJkkibUdGJjYjSSJyR0YmRicmSSJmR0YmNiMiIiFGJyomRiVGJ0kidEdGJkYnRixGJ0kiZ0dGJkYnRidGJy1JJVVuaXRHNiQvSSttb2R1bGVuYW1lR0YmSSZVbml0c0c2JEkoX3N5c2xpYkdGJiUqcHJvdGVjdGVkR0ZCNiNJIk5HRiZGJw==

;

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn

PkkiYUc2Ii1JIipHNiQlKnByb3RlY3RlZEcvSSttb2R1bGVuYW1lR0YkSSlTdGFuZGFyZEc2JC9GKkkmVW5pdHNHNiRJKF9zeXNsaWJHRiRGKEYoNiQmSSJGR0YkNiNJJG5ldEdGJC1JIi9HRic2I0kiTUdGJA==

KigsJiomSSJiRzYiIiIiJkkidkdGJjYjSSJlR0YmRichIiIqJiwoJkkibUdGJjYjSSJyR0YmRicmSSJmR0YmNiMiIiFGJyomRiVGJ0kidEdGJkYnRixGJ0kiZ0dGJkYnRidGJ0YuRiwtSSVVbml0RzYkL0krbW9kdWxlbmFtZUdGJkkmVW5pdHNHNiRJKF9zeXNsaWJHRiYlKnByb3RlY3RlZEdGQjYjKiZGMEYnKUkic0dGJiIiI0YsRic=

;

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn

Integrate a from t = 0 .. T to get v.

PiZJInZHNiI2I0kickdGJS1JIipHNiQlKnByb3RlY3RlZEcvSSttb2R1bGVuYW1lR0YlSSlTdGFuZGFyZEc2JC9GLUkmVW5pdHNHNiRJKF9zeXNsaWJHRiVGK0YrNiQtSSRpbnRHRiU2JEkiYUdGJS9JInRHRiU7IiIhSSJUR0YlLUklVW5pdEdGLzYjSSJzR0Yl

KiYtSSRpbnRHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2JComLCYqJkkiYkdGKCIiIiZJInZHRig2I0kiZUdGKEYuISIiKiYsKCZJIm1HRig2I0kickdGKEYuJkkiZkdGKDYjIiIhRi4qJkYtRi5JInRHRihGLkYzRi5JImdHRihGLkYuRi5GNUYzL0Y/O0Y9SSJUR0YoRi4tSSVVbml0RzYkL0krbW9kdWxlbmFtZUdGKEkmVW5pdHNHNiRGJ0YmRiY2IyomRjdGLkkic0dGKEYzRi4=

print();

All the input equations (in black) were created using the new 2-D math editor. Integrals, superscripts, subscripts, fractions, as well as many other layouts and over a thousand different symbols are available through the palettes and keyboard commands. The mathematics is formatted as you type, so you can easily make it look just the way it would in a technical paper or text book. You will find the list of math editor keyboard shortcuts for your platform from the Quick Help pop-up, which appears on start-up or can be invoked by selecting Help>Quick Help from the menu.

The quantities in LUklVW5pdEc2IjYjSSV1bml0R0Yknotation represent units. These units can be added to an equation using the new Units palettes, or through the new Units context menus. Using the context menus, you can also simplify expressions involving units, convert from one system of measurement to another, or replace the displayed unit with an equivalent unit. To see the context menu options, right-click on the LUklVW5pdEc2IjYjKiZJIm1HRiQiIiJJInNHRiQhIiI= unit in the output, and select Units>.

Maple does not evaluate this integral because it results in a logarithmic term that would be infinite if the denominator were nonpositive. Let's call int again using the continuous option. This tells Maple not to worry about discontinuities in the integrand.

v[r]:=int(a, t*Unit(s) = 0..T, continuous);

KiYsKComJkkidkc2IjYjSSJlR0YnIiIiLUkjbG5HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRic2IywmJkkibUdGJzYjSSJyR0YnISIiJkkiZkdGJzYjIiIhRjZGKkY2KiZJImdHRidGKkkiVEdGJ0YqRioqJkYlRiotRiw2IywoRjJGNkY3RjYqJkkiYkdGJ0YqRj1GKkYqRipGKkYqLUklVW5pdEc2JC9JK21vZHVsZW5hbWVHRidJJlVuaXRzRzYkRi9GLkYuNiMqJkYzRipJInNHRidGNkYq

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn

This time, when we ask for the value of Jkkidkc2IjYjSSJyR0Yk, Maple gives the classic rocket equation. Best of all, Maple carries the units through the integration, giving its symbolic answer in KiZJIm1HNiIiIiJJInNHRiQhIiI=.

We now take the numeric values for each of the parameters from the table at the beginning of the worksheet:

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn

PjYnJkkibUc2IjYjSSJyR0YmJkkiZkdGJjYjIiIhSSJiR0YmJkkidkdGJjYjSSJlR0YmSSJnR0YmLUkuZ2V0UGFyYW1ldGVyc0dGJkYm

NickIiInIiImJCIkIz4iIiUkIiM7IiIkLUkiLUc2JCUqcHJvdGVjdGVkRy9JK21vZHVsZW5hbWVHNiJJKVN0YW5kYXJkRzYkL0YxSSZVbml0c0c2JEkoX3N5c2xpYkdGMkYvRi82IyIlK0ktRi02IyQiJCIpKiEiIw==

;

NickIiInIiImJCIkIz4iIiUkIiM7IiIkISUrSSQhJCIpKiEiIw==

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn

JkkibUc2IjYjSSJyR0Yk

JCIiJyIiJg==

;

JCIiJyIiJg==

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn

JkkiZkc2IjYjIiIh

JCIkIz4iIiU=

;

JCIkIz4iIiU=

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn

SSJiRzYi

JCIjOyIiJA==

;

JCIjOyIiJA==

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn

Jkkidkc2IjYjSSJlR0Yk

ISUrSQ==

;

ISUrSQ==

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn

SSJnRzYi

JCEkIikqISIj

;

JCEkIikqISIj

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn

How long will the fuel last?

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn

PkkpZnVlbFRpbWVHNiItSSZzb2x2ZUdGJDYkLUkiK0c2JCUqcHJvdGVjdGVkRy9JK21vZHVsZW5hbWVHRiRJKVN0YW5kYXJkRzYkL0YtSSZVbml0c0c2JEkoX3N5c2xpYkdGJEYrRis2JCZJImZHRiQ2IyIiIS1JIi1HRio2Iy1JMGRlbGF5RG90UHJvZHVjdEc2JEYrL0YtSSxUeXBlc2V0dGluZ0c2JEYrRjM2JEkiYkdGJEkidEdGJEZE

JCIkPyIiIiE=

;

JCIkPyIiIiE=

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn

The next paragraph shows another example with inline output. In this case, the number of seconds is displayed twice seamlessly inside the same paragraph. If you expand the document block, you will see that the number comes from the value of the fuelTime variable. If that variable is changed and the document re-executed, the paragraph will automatically be updated to reflect the new value.

This provides about

fuelTime;

JCIkPyIiIiE=

seconds' worth of fuel. Now let's plot the velocity as T goes from 0 to

fuelTime;

JCIkPyIiIiE=

seconds. To plot an expression that has units, we must first strip off the units with the unit_free option to convert.

PkkndkZpbmFsRzYiLUkoY29udmVydEclKnByb3RlY3RlZEc2JCZJInZHRiQ2I0kickdGJEkqdW5pdF9mcmVlR0Yk

LCgsJiQiK1EzJD5VJSEiJiIiIiomJCIraXp4QyUqISInRideI0YnRidGJ0YnKiYkIiQiKSohIiNGJ0kiVEc2IkYnISIiKiYiJStJRictSSNsbkc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGMjYjLCYkIiRfIyIiJUYzKiYkIiM7IiIkRidGMUYnRidGJ0Yz

;

LCgsJiQiK1EzJD5VJSEiJiIiIiomJCIraXp4QyUqISInRideI0YnRidGJ0YnKiYkIiQiKSohIiNGJ0kiVEc2IkYnISIiKiYiJStJRictSSNsbkc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGMjYjLCYkIiRfIyIiJUYzKiYkIiM7IiIkRidGMUYnRidGJ0Yz

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn

In Figure 1, we see that the rocket will have a velocity of over

maxVelocity:=eval(vFinal, T=fuelTime)*Unit(m/s); convert ( floor ( convert(maxVelocity, units, km/s) ), unit_free);

KiYsJiQiKmZgIUdKISImIiIiKiYkIiIhRipGJ14jRidGJ0YnRictSSVVbml0RzYkL0krbW9kdWxlbmFtZUc2IkkmVW5pdHNHNiRJKF9zeXNsaWJHRjElKnByb3RlY3RlZEdGNTYjKiZJIm1HRjFGJ0kic0dGMSEiIkYn

IiIk

KiZJI2ttRzYiIiIiSSJzR0YkISIi when its fuel runs out.

plot(vFinal,T=0..fuelTime,title="Figure 1", labels=["Time (s)", "Velocity (m/s)"], labeldirections=[horizontal, vertical], gridlines );

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

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn

Note that the plot command itself, which is frequently of little interest to the reader, is hidden.

<Text-field style="Heading 1" layout="Heading 1">Velocity after fuel runs out</Text-field>

There's a problem with the above. We very politely asked our model for the rocket's velocity up until the time at which it runs out of fuel,

fuelTime;

JCIkPyIiIiE=

seconds. But try redrawing the plot above with a maximum time greater than

fuelTime;

JCIkPyIiIiE=

It predicts the velocity of the rocket will become infinite!

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn

SetProperty( Fig2Min, upper, round( fuelTime) ); SetProperty( Fig2Max, lower, round( fuelTime) ); SetProperty( Fig2Max, upper, round( fuelTime)*2); setPlot(Figure2, vFinal, T, Fig2Min, Fig2Max, "Figure 2");

Function:

vFinal;

LCgsJiQiK1EzJD5VJSEiJiIiIiomJCIraXp4QyUqISInRideI0YnRidGJ0YnKiYkIiQiKSohIiNGJ0kiVEc2IkYnISIiKiYiJStJRictSSNsbkc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGMjYjLCYkIiRfIyIiJUYzKiYkIiM7IiIkRidGMUYnRidGJ0Yz

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You can observe the velocity of the rocket extended beyond the point when fuel has run out by adjusting the slider bar below. Move the slider to pick the maximum time for plotting.

Minimum Time: setPlot(Figure2, vFinal, T, Fig2Min, Fig2Max, "Figure 2")

Maximum Time: setPlot(Figure2, vFinal, T, Fig2Min, Fig2Max, "Figure 2")

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This plot is controlled by a slider. When the sliders are moved, the curve is updated. Along with sliders, edit boxes, buttons and plot regions, you can also embed toggle buttons, check boxes, list boxes, and math edit boxes in which 2-D math can be entered and displayed.

The plot has been placed to the right of the sliders that are used to control it. This layout has been done by using a table. If you move your mouse over the table, you will see where the hidden borders lie. Tables can be embedded inside cells of other tables, and you can set the visibility of each table's exterior and interior borders. In this way, you can control the layout, placement and appearance of text, plots and other objects in your documents.

The problem is with our mathematical model. We defined M to be LCgmSSJtRzYiNiNJInJHRiUiIiImSSJmR0YlNiMiIiFGKComSSJiR0YlRihJInRHRiVGKCEiIg== without any bounds on t. But as t increases, this formula says that M shrinks to zero and then becomes negative! We can plug this hole in our model using Maple's handy piecewise command.

M := ( m[r]+piecewise(t<f[0]/b, f[0]-b*t,0) ) * Unit(kg);

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But the mass formula is not the only difficulty. The net acceleration on the rocket becomes just g when the fuel runs out. So we need to modify our force equation as well. Rewrite the formula for the velocity of the exhaust using the piecewise command, assuming that the exhaust velocity is still v[e] for as long as the fuel holds out.

velExhaust := piecewise( t<f[0]/b, v[e],0 );

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F[net] := -b*Unit(kg/s) * velExhaust * Unit(m/s) + M*g*Unit(m/s^2);

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;

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Integrating the modified expression of the acceleration to obtain the velocity:

v[r] := int( a, t*Unit(s) = 0..T, continuous );

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;

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;

JCIkIz4iIiU=

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;

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;

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;

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vFinalFixed:=convert(v[r],unit_free);

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This is more realistic. After the fuel runs out, the velocity decreases linearly with time.

SetProperty( Fig3Min, upper, round( fuelTime) ); SetProperty( Fig3Max, lower, round( fuelTime) ); SetProperty( Fig3Max, upper, round( fuelTime)*2); setPlot(Figure3, vFinalFixed, T, Fig3Min, Fig3Max, "Figure 3");

Function:

vFinalFixed;

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Move the slider to adjust the maximum time shown in the plot.

Minimum Time: setPlot(Figure3, vFinalFixed, T, Fig3Min, Fig3Max, "Figure 3");

Maximum Time: setPlot(Figure3, vFinalFixed, T, Fig3Min, Fig3Max, "Figure 3");

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Maple 10 allows you to add grid lines to your 2-D plots. You can add grid lines through the plot command, the plot builder, the context menus, or simply by clicking on the TUZOV3RLVWI8b2I8Uj1NRExDZE5SPmI+S2I6YkZxZmJBOmFUTENUSmNEWG94UnNnVk1JPUFSXFw7VjtLVkdjRT07RnNNdjtJQ0VjQkU7Rl1lckxnREtDSEs7RGM9aVpLSD1HRGJ1c0FrZVlbRlVZYmBNRDtVWVI/RUJ1QlpzaFc9aWpVYl51aDtdVTxVWXFnQ2BBWFVjWFVDWWhVRk91Qzo/UkdnTW9oeUVpQjxqXlBOYUxOUUVOakRCOjo1OlwiXHtcfQ== button on the toolbar, which acts as a grid line toggle.

<Text-field style="Heading 1" layout="Heading 1">Investigate the relationship between the mass and velocity of the rocket</Text-field>

Here we show how the velocity of a rocket changes from its launch time into its time in space. We also show the differences in velocity caused by having a lighter or heavier initial mass.

We use the following formula for the velocity:

vFinal3D:=3000*ln(-1*mass-1.92*10^6)+piecewise(T<=120,-9.81*T-3000* ln(-1*mass-1.92*10^6+16000*T),-9.81*T-3000*ln(-1*mass) ): setPlot3d(Fig4, vFinal3D, T, Fig4MinTime, Fig4MaxTime, "Figure 4", Fig4MinMass, Fig4MaxMass, Fig4Mass); vFinal3D;

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You can use the below sliders to pick the time interval to show in the plot. Enter the range of masses for the rocket in the text boxes.

Minimum Time:

setPlot3d(Fig4, vFinal3D, T, Fig4MinTime, Fig4MaxTime, "Figure 4", Fig4MinMass, Fig4MaxMass, Fig4Mass);

Maximum Time:

setPlot3d(Fig4, vFinal3D, T, Fig4MinTime, Fig4MaxTime, "Figure 4", Fig4MinMass, Fig4MaxMass, Fig4Mass);

Minimum Mass:

use DocumentTools in # Enter Maple commands to be executed when the specified # action is carried out on the component. # Use: # GetProperty( component_name, attribute_name ) # and # SetProperty( component_name, attribute_name, value ) # to access any of the attributes of the component. # See ?CustomizingComponents for more information. updateSlider (Fig4MinMass, Fig4MaxMass, Fig4Mass); setPlot3d(Fig4, vFinal3D, T, Fig4MinTime, Fig4MaxTime, "Figure 4", Fig4MinMass, Fig4MaxMass, Fig4Mass); end use;

Maximum Mass:

Mass:setPlot3d(Fig4, vFinal3D, T, Fig4MinTime, Fig4MaxTime, "Figure 4", Fig4MinMass, Fig4MaxMass, Fig4Mass);

As the mass increases from

minMass:=parse(GetProperty(Fig4MinMass, value))*Unit(kg): maxMass:=parse(GetProperty(Fig4MaxMass, value))*Unit(kg): minTime := GetProperty(Fig4MinTime, value): maxTime := GetProperty(Fig4MaxTime, value): minMass;

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maxMass;

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, the maximum velocity drops down from

vel := eval ( vFinal3D, mass=convert(minMass, unit_free) ): max( seq( Re(vel), T=minTime..maxTime ) )*Unit(m/s);

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vel := eval ( vFinal3D, mass=convert (maxMass, unit_free) ): max( seq( Re(vel), T=minTime..maxTime ) )*Unit(m/s);

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. The maximum velocity for a lighter rocket is higher than that for a heavier one (as expected).

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Other new plotting features include logarithmic axes and glossiness of 3-D plots, as seen in the plot above. To add glossiness to a 3-D plot, first add a lighting model. Like other plot options, you can set these options programmatically or interactively using the context menus. To set glossiness using the context menu, right-click on the plot, and then choose Lighting> to select a lighting model. Right-click again and choose Glossiness... The amount of glossiness is controlled by a slider.

More information about Maple 10 can be found at www.maplesoft.com

Legal Notice: The copyright for this application is owned by Maplesoft. The application is intended to demonstrate the use of Maple to solve a particular problem. It has been made available for product evaluation purposes only and may not be used in any other context without the express permission of Maplesoft.

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