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Narrative
In this, the second of four aerospace-based S.T.E.M. project, students will calculate the payload capacity of the Skylon space plane.

Students will use the launch site latitude as well as the orbital inclination to determine the amount of weight that the Skylon can carry.

Time Frame

Mathematics Used
Linear Equations

Material List
Connection to the Internet

Science Topics
Physics, Aerospace

11th and 12th

Essential Questions
• Who are the pioneers of spaceplane technology?
• What is the Orbital Inclination of a spacecraft?
• Where is the payload bay of the R.E.L. Skylon located?
• When will be the first flight of the Skylon spaceplane?
• Why do people want to fly payload into orbit on a spaceplane?
• How does the latitude of the Launch Site effect an orbital payload?
• Wait. I have to do science and technology and engineering and mathematics, all at the same time? Woah.

Activating Previous Learning
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Lesson Overview
Note: This website incorporates spreadsheets and slide-show presentations that are provided to teachers for use in the classroom.
• Students first learn the basics of spaceflight launch payload using pencil, paper, and scientific calculator.
• Students then use what they have learned to create an Aerospace Mission Design App (AMDA), designed according to the Engineering Design Process, that will be used for real-world spacecraft. They will use spreadsheet software to create the app.
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Learning Objectives
Evaluation
• Interpret data related to aerospace, and rocketry.
• Select an optimum design from many design options to solve technological problems.
Synthesis
• Explain the principles of an launching a payload into space in mathematical and physical terms.
• Integrate mathematics and aerospace in the engineering design process.
Analysis
• Analyze the physical principles of a an launching a payload into space, and relate these to a space mission design.
• Use mathematics to calculate the cargo capacity of the Skylon space plane.
Application
• Use the Engineering Design Process to construct a real-world space mission app that is constrained by certain aerospace factors.
Comprehension
• Define constraints to the real-world model.
• Explain how solutions to the problem address the specific requirement.
Knowledge
• Explain the relationships of the principles of aerospace to the concept of payload and orbital inclination of the Skylon.
• Demonstrate how their space mission design app addresses the requirements of payload and orbital inclination of the Skylon.
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Science As Inquiry
• Identify questions and concepts that guide scientific investigations.
• Design and conduct scientific investigations.
• Use technology and mathematics to improve investigations and communications.
• Formulate and revise scientific explanations and models using logic and evidence.
• Communicate and defend a scientific argument.
Physical Science
• Use mathematics and logic to explain scientific principles.
• Look up and use aeronautical constants.
Science and Technology
• Identify a problem or design an opportunity.
• Propose designs and choose between alternative solutions.
• Implement a proposed solution.
• Evaluate a solution and its consequences.
• Communicate the problem, process, and solution.
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Visual Learning
Here is a medium length (7 min) video of the Reaction Engines, Ltd. (R.E.L.) Skylon spaceplane (the accompanying music is excellent, so make sure the volume is slightly elevated). It shows how the vehicle is loaded with payload and propellant, then takes off like an ordinary airplane, except this one can go all the way into space!

R.E.L. Skylon taking a typical flight (0:04:43)

The Skylon can operate out of any airport, even (especially) from Spaceport America.

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Constants
• (none)
Input
• Launch Site Latitude (deg)
• Orbital Altitude (mi)
Output
• At Latitude (km)
• To I.S.S. (km)
• To Polar Orbit (km)
• Orbital Altitude (km)
• At Latitude (kg)
• To I.S.S. (kg)
• To Polar Orbit (kg)
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Vocabulary
• International Space Station (I.S.S): The space station currently orbiting the earth; it is at an orbital altitude of 230 mi with an Orbital Inclination of 52 degrees
• Latitude: The number of degrees above (or below) the equator
• Launch Site: The spaceport where the Skylon spaceplane launches and recovers
• Launch Site Latitude: The latitude (measured in degrees) of the launch site
• Orbital Altitude: The height above Mean Sea Level (MSL) of a spacecraft
• Orbital Inclination: The number of degrees that an orbit subtends relative to the equator
• Polar Orbit: An orbit that flies above the North and South poles; it has an Orbital Inclination of 98 degrees
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Analysis
To determine the weight and orbital altitude of a spacecraft climbing into earth orbit, we need information on the space plane's capabilities at various launch latitudes. Fortunately, REL provides us with exactly what we need (3.1 MB pdf):

Here are the screenshots taken from the pdf file above of the information that we need:

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Here are the same graphs as an animated GIF:

Using these graphs, we can determine the general formula for each of these lines. So, let's make a couple of tables, shall we?

We will concentrate on the Launch Site Latitude equal to the Orbital Inclination. So for the Launch Site Latitude of 0 degrees, we will look for the graph at 250 km for the Orbital Inclination of 0 degrees. For the 15 degree graph, we will look at the 250 km point on the 15 degree Orbital Inclination line. This process continues for each graph:

Now we can analyse the tables to see what kind of equation that we have. We can use the old trick of subtracting the dependent variables (absolute value) to determine the degree of the polynomial equation. If the all the subtractions keep coming up with the same number it is a linear (degree 1) polynomial. If after 2 subtractions we get a constant, then it is a quadratic (degree 2) polynomial. If 3, then a cubic (degree 3), etc. Let's see what we get for the first table:

15.50 - 15.25 = 0.25
15.25 - 14.50 = 0.75 => 0.75 - 0.25 = 0.50
14.50 - 13.25 = 1.25 => 1.25 - 0.75 = 0.50
13.25 - 11.75 = 1.75 => 1.75 - 1.25 = 0.50

So we get a constant after two iterations. Therefore, we are dealing with a degree 2 polynomial, or a quadratic equation.

Since the initial weight of the payload is irrelevant, we can zero out b:
We can calculate c by using the table and plugging in x = 0 and y = 15.5:

We can then calculate a by using the table (again) and plugging x = 15 and y = 15.25:

We now have the equation that we need to determine the payload capability (y) depending on the latitude of the launch site (x) for a 250 km Orbital Altitude.

The polynomial equation can be determined using the same technique on the second table:

We now have the equation that we need to determine the payload capability (y) depending on the latitude of the launch site (x), this time for an 800 km Orbital Altitude.

We can now determine the two points needed to draw the linear equation for the payload.

The same technique described above can be used to determine the equations to reach the I.S.S. and for a polar orbit.

The app can now be constructed and the numbers entered.

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The Space Mission App
The NMSTARG/Google S.T.E.M. Orbital Spacecraft app is broken into four parts:
1. Input/Output
2. Graph
3. Constants
4. Calculations

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Teacher Lesson Plan
Use the slide-show presentation below to give a lesson about basic payload profiles and calculating the altitude and weight of a Skylon payload.

In this lesson, students will identify the various aspects of a payload profile diagram, matching them with the terms and definitions.

Students then practice the calculations using pencil, paper, and scientific calculator.

Students then learn about the Engineering Design Process, and begin the process of laying the groundwork for the app built with a spreadsheet. Sample Open Source computer code is provided to aid students with their spreadsheet formulas.

TEACHER PRESENTATION HERE

Some screenshots of the Teacher Presentation:

TEACHER PRESENTATION SCREENSHOTS HERE

The slides will guide the students as they run through the lesson powered by E^8:
1. Engage
• Lesson Objectives
• Lesson Goals
• Lesson Organization
2. Explore
• ...
• The ... and its Components and Definitions
3. Explain
• ...
4. Elaborate
• Other Spaceplane Examples
5. Exercise
6. Engineer
• The Engineering Design Process
• AMDC Spaceflight Plan
• Designing a Prototype
• AMDC Software
7. Express
• Displaying the AMDC
• Progress Report
8. Evaluate
• Post Engineering Assessment
This lesson can be delivered in one or two class periods, with students working on the project after school. It is recommended that a few minutes of a few class periods be set aside for student help.

The Student Workbook (below) accompanies the Teacher Lesson Plan.

STUDENT WORKBOOK HERE

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Missions
The input values of the app can be varied to create different spaceflight landing scenarios. These scenarios will add a sense of realism to the student project. Students will be placed into groups and asked to determine the payload or the orbital altitude based on the latitude of the launch Site:

SPACECRAFT LANDING PARAMETERS HERE

Encourage students to design their own scenarios by using different input values. This project is very flexible in that regard.

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Presentation
Students will be asked to present their findings to the rest of the class. Parents are, of course, encouraged to attend (it is suggested that a pot luck would make things more festive).

Each presentation will have slides that introduces the group, describes the spaceflight, and displays the calculations. A short biography (Alan Bond, Eugen Sanger, etc.) of the pioneers of space planes will also be included.

Students that know how to use presentation software should be encouraged to create their own presentations (remember, Google Apps are free!)

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Rubric
Students will also create a website and embed their slide presentation and their S.T.E.M. app in a webpage. Their journal will be kept on the webpage as well. If the class presentation is recorded to video, it can be uploaded to YouTube, then embedded in the webpage.

Therefore, each webpage (one for each project) should have the following items:
1. Embedded Slide Presentation
2. Embedded view of S.T.E.M. app
4. Link to working prototype of S.T.E.M. app
5. Journal Entries
6. (optional) Embedded Youtube video of the presentation
EXAMPLE STUDENT WEBSITE

This is not a scoring rubric; rather it is a guide of what is expected for the project.

The presentation should take between 5 and 10 minutes, unless there are a lot of questions from the audience. For a class with 6 groups, this comes out to between 30 and 60 minutes.

Students should be encouraged to dress professionally, and to practice their presentations beforehand.

Scoring and grading these projects is left up to the professionalism of the teacher.

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Conclusion
The linear and quadratic equations used in this project should be easy enough for the average high school Algebra 2 student. The teacher may need to guide students through the setup of the equations and the calculations. As the quarter progress, the concepts and the mathematics will become more challenging.

Using technology to do a high school math project should be easy and fun.

But above all, it should be free.

Both the spreadsheet and the presentation were built using Google Docs, a free application when you sign up for gmail through Google. Therefore, any student with an Internet access can use these tools for free, whether at the school, or at the library, or at the coffee shop, or at home, etc.

Students that do well on this project will learn many important skills that will help to succeed in whatever field they desire to choose.

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