In this, the fourth and final aerospace-based S.T.E.M. chapter, students will track the position and speed of a spacecraft that is landing at Spaceport America, which is located at 33 degrees North Latitude.

Students will also determine the spacecraft descent rate, ground speed, and time until touchdown.

**Start Date**

Fourth Quarter

Fall Semester

**Time Frame**

About 4.5 weeks

**Mathematics Used**

Trigonometry

Vectors

**Essential Questions**

- Who are the pioneers of spaceports?
- What is the Complement of an angle?
- Where can a spaceport be located?
- When was Spaceport America open for business?
- Why would people prefer to land at a spaceport as opposed to an airport?
- How do I use Trigonometry to calculate the distance and altitude of a spacecraft?
- Wait. I have to do science
technology__and__engineering__and__mathematics, all at the same time? Woah.__and__

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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 unpowered glide landing 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 aeronautics, aerospace, and rocketry.
- Select an optimum design from many design options to solve technological problems.

*Synthesis*

- Explain the principles of an unpowered spacecraft landing in mathematical and physical terms.
- Integrate mathematics and aeronautics in the engineering design process.

*Analysis*

- Analyze the physical principles of a an unpowered spacecraft landing, and relate these to a space mission design.
- Use mathematics to calculate the height and distance of a landing spacecraft.

*Application*

- Use the Engineering Design Process to construct a real-world space mission app that is constrained by certain aeronautical 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 aeronautics to the concept of height and distance.
- Demonstrate how their space mission design app addresses the requirements of height and distance.

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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 short (3 min) video of the Spaceport America complex, and a glimpse of what tourists will see when they visit the facility.

Animated Tour of Spaceport America (0:03:05)

The spaceport should be completed in time for the Virgin Galactic flights (see Project #1).

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**Constants**

- (none)

**Input**

- Glide Angle (deg)
- Glide Distance 1 (ft)
- Glide Distance 2 (ft)

**Output**

- Altitude (m AGL)
- Distance from Spaceport (m)
- Glide Slope (deg)
- Glide Speed (mps)
- Descent Rate (mps)
- Ground Speed (mps)
- Time to Touchdown (min)

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**Vocabulary**

*Adjacent Side of a Right Triangle:*The side next to the given angle (*not*the Hypotenuse)*Altitude:*The distance a spacecraft is above a given point*Above Ground Level (AGL):*The distance a spacecraft is above the ground*Descent Rate:*The distance a spacecraft descends over a certain period of time*Distance From Spaceport:*The ground distance form the edge of the runway to the spacecraft*Glide Angle:*The angle from the vertical that a Landing Laser points*Glide Distance:*The distance the Landing Laser measures*Glide Slope:*The angle a spacecraft makes to the horizontal*Glide Speed:*The speed of the spacecraft during the unpowered glide landing*Ground Speed:*The speed of the spacecraft as related to the ground*Hypotenuse of a Right Triangle:*The longest side of a right triangle*Landing Laser:*The laser used to determine the Glide Distance to a spacecraft*Landing Profile:*The graph of a landing spacecraft*Line-Of-Sight Distance:*The Glide Distance converted to S.I. units*Opposite Side of a Right Triangle:*The side opposite the given angle*Right Triangle:*A triangle with one of the angles equal to exactly 90 degrees*Time Until Touchdown:*The time the spacecraft will take to glide to a landing*Touchdown:*The moment the spacecraft makes contact with the runway during a landing

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**Analysis**

We will be tracking a hypothetical spacecraft returning from space (such as the Virgin Galactic SpaceShipTwo) as the pilots on board perform an unpowered glide landing back at the Spaceport.

A Landing Laser located at the edge of the Spaceport runway will be used to track the landing spacecraft. The laser will determine the Glide Angle and the Glide Distance:

Landing Laser determining Glide Distance and Glide Angle |

This laser will measure the Glide Angle from the vertical, since the ground may or may not be level. The laser itself when triggered will perform two bursts over a one second period. This gives us Glide Distance 1 and Glide Distance 2.

*Note: Ideally, the laser would be firing every second so that a more accurate plot of the spacecraft can be made as it comes in for the landing. This constraint to the project means that we are basically taking a snapshot of the position and speed of the spacecraft with each laser firing.*

The resulting Landing Profile can be represented as a Right Triangle, and can then be labeled appropriately. The Glide Slope is simply the Complement of the Glide Angle.

Glide Slope = Complement(Glide Angle) = 90 - Glide Angle

A Right Triangle can be formed representing the landing profile |

The Right Triangle can be solved by using the Trigonometric functions of Sine and Cosine. Of course, the angles will have to first be converted to radians.

cos(Angle) = Adjacent Side / Hypotenusesin(Angle) = Opposite Side / Hypotenuse

or

cos(Glide Slope) = Distance to Spaceport / Glide Distance 2sin(Glide Slope) = Altitude / Glide Distance 2

Rearranging, we get,

Distance to Spaceport = Glide Distance 2 * cos(Glide Slope)Altitude = Glide Distance 2 * sin(Glide Slope)

To graph the Landing Profile, simply graph the two points:

(0, 0) & (Distance to Spaceport, Altitude)

Graph of the Landing Profile |

The two laser bursts one second apart gives us two distances with t=1. Using

*d = rt*and rearranging, we get,Glide Speed = Glide Distance 2 - Glide Distance 1

Using the same trigonometric functions as before, the other rates can be calculated.

Ground Speed = Glide Speed * cos(Glide Slope)Descent Rate = Glide Speed * sin(Glide Slope)

and

Time To Touchdown = Glide Distance 2 / Glide Speed

We now have all the information that we need to build our app.

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**The Space Mission App**

The NMSTARG/Google S.T.E.M. Spacecraft Landing app is broken into four parts:

- Input/Output
- Graph
- Constants
- Calculations

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**Teacher Lesson Plan**

Use the slide-show presentation below to give a lesson about basic landing profiles and calculating the altitude and distance from the Spaceport.

In this lesson, students will identify the various aspects of a landing 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 ground work for the app built with a spreadsheet. Sample Open Source computer code is provided to aid students with their spreadsheet formulas.

INSERT TEACHER PRESENTATION HERE

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The slides will guide the students as they run through the lesson powered by E^8:

- Engage
- Lesson Objectives
- Lesson Goals
- Lesson Organization
- Explore
- The Cosine Function
- The Sine Function
- The Glide Slope and its Components and Definitions
- Additional Terms and Definitions
- Explain
- The Glide Slope
- Using Cosine for Altitude and Descent Rate
- Using Sine for Distance to the Spaceport and Ground Speed
- Glide Speed Equation
- Elaborate
- Other Spaceport Examples
- Exercise
- Spacecraft Landing Parameters
- Spacecraft Landing Scenario 1
- Spacecraft Landing Scenario 2
- Engineer
- The Engineering Design Process
- AMDC Spaceflight Plan
- Designing a Prototype
- AMDC Software
- Express
- Displaying the AMDC
- Progress Report
- 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.

INSERT 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 Altitude, Distance, Descent Rate, Ground Speed, and Glide Speed for a spacecraft returning from space:

INSERT 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 (Sir Richard Branson, Burt Rutan, etc.) of the pioneers of parabolic spaceflight 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:

- Embedded Slide Presentation
- Embedded view of S.T.E.M. app
- Link to S.T.E.M. app
- Link to working prototype of S.T.E.M. app
- Journal Entries
- (optional) Embedded Youtube video of the presentation

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 trigonometric 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.

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