Note to readers: This lesson plan, as well as all the other similar articles we've written, may appear, to some folks, like the next thing to Greek language (the math stuff), but bear in mind these S.T.E.M. lessons are really all about the students. Ergo, preparing high school students for the real rocketry world using real math. That being said, we invite you into their world, their minds, and let's support them in this heroic endeavor, because these students are truly going where no high school student has ever gone before.
::
4.1 Narrative
In this, the fourth and final aerospace-based S.T.E.M. project, students will track the position and speed of a spacecraft that is landing at Spaceport America.
Students will also determine the spacecraft descent rate, ground speed, and time until touchdown.
Time Frame
About 4 weeks (22 days)
Aerospace Problems
Glide Slope
Altitude
Distance from Spaceport
Descent Rate
Ground Speed
Mathematics Used
Trigonometry
Vector Analysis
Material List
A connection to the Internet
Google GMail account
Science Topics
Physics, Aerospace
Activating Previous Learning
Basic Mathematics
Scientific Calculator
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 and technology and engineering and mathematics, all at the same time?
::
This lesson is powered by E^8:
1. Engage
Lesson Objectives
Lesson Goals
Lesson Organization
2. Explore
The Right Triangle
The Trigonometric Identities
Basic Vector Analysis
Additional Terms and Definitions
3. Explain
Glide Slope
4. Elaborate
Other Spaceport Examples
5. Exercise
Spacecraft landing Parameters
Spacecraft landing Scenario
6. Engineer
The Engineering Design Process
SMDA Spacecraft Landing Plan
Designing a Prototype
SMDA Software
7. Express
Displaying the SMDA
Progress Report
8. Evaluate
Post Engineering Assessment
::
Lesson Overview
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 a Space Mission Design App (SMDA), designed according to the Engineering Design Process, that will be used for real-world spacecraft.
We will be using Spaceport America (33o N, 107o W), located just north of Las Cruces, NM. Typical spacecraft that could land at this facility are the Virgin Galactic SpaceShipTwo and the R.E.L. Skylon.
Students will use spreadsheet software to create the app, and will use slide-show software for their presentations. They will also create a document of their experiences engineering the SMDA and presenting their findings to the rest of the class.
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)
::
Visual Learning
Spaceport America, located in the high desert of New Mexico, is ideally suited for spacecraft launch and recovery activities.
::
Continued...
4.2 Vocabulary
Adjacent Side Altitude Above Ground Level (AGL) Descent Rate
Distance From Spaceport Glide Angle Glide Distance
Glide Slope Glide Speed Ground Speed
Hypotenuse Landing Laser Landing Profile
Line-Of-Sight Opposite Side Right Triangle
Time To Touchdown Touchdown
::
Spaceport America
4.3 Analysis
Any spacecraft returning from space is always out of propellant. This is because all the propellant is used during the trip into space. Consequently, there is none available for the return trip. All machines that have wings can glide, that is, fly with the engine turned off. Some glide better than others, but still, they all glide.
This is why spacecraft come in with their nose down; they are maintaining required airspeed. As they cross over the edge of the runway, the nose is pulled up and the spacecraft flattens out its glide as air is packed underneath the wings. It’s then just a simple matter of letting the spacecraft sink to a gentle touchdown.
Once on the ground the nose is kept in the air “wheelie” fashion, so that speed can be bled off without using brakes, because they can get very hot extremely quickly. After the nose comes down on its own the brakes can then be (sparingly) applied. Eventually, the spacecraft rolls to a full stop.
Back home once again!
Graph of the Glide Path of a Returning Spacecrat
::
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 to 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:
Diagram of Landing Laser Configuration
::
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 degrees - Glide Angle
Landing Laser Diagram as a Pythagorean Triangle
::
The Right Triangle can be solved by using the Trigonometric functions of Sine and Cosine. For the purposes of this exercise, the angles will not have to first be converted to radians.
- cos(angle) = Adjacent Side / Hypotenuse
- sin(angle) = Opposite Side / Hypotenuse
Writing the trigonometry in aerospace form, the general equation becomes:
- cos(Glide Slope) = Distance To SpaceportLine-Of-Sight Distance
- sin(Glide Slope) = AltitudeLine-Of-Sight- Distance
Rearranging the equation, we get,
- Distance To Spaceport = Line-Of-Sight Distance * cos(Glide Slope)
- Altitude (AGL) = Line-Of-Sight Distance * sin(Glide Slope)
To graph the Landing Profile, simply plot the two points and connect them:
- (0, 0) & (Distance to Spaceport, Altitude)
The linear equation can easily be derived from these two points.
::
The two laser bursts one second apart gives us two distances with t=1. Thus we get two different distances, Line-Of-Sight Distance 1 and Line-Of-Sight Distance 2. Using d = rt and rearranging, we get,
- Glide Speed = Line-Of-Sight Distance 2 - Line-Of-Sight 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 = Line-Of-Sight DistanceGlide Speed
Example
An R.E.L. Skylon is returning to Spaceport America from space after dropping off some passengers and picking up more that are homeward bound. The Landing Laser bounces a laser off of the spaceplane to determining the following information:
- Glide Angle = 55 degrees
- Glide Distance 1 = 19.80 mi
- Glide Distance 2 = 19.75 mi
Find the following information about the landing spacecraft at this moment in time.
- Altitude AGL
- Distance to Spaceport
- Glide Slope
- Glide Speed
- Descent Rate
- Ground Speed
- Time To Touchdown
First, we must change our inputs to S.I. units:
Line-Of-Sight Distance 1 = Glide Distance 11609 = 31,865 m
Line-Of-Sight Distance 2 = Glide Distance 21609 = 31,785 m
So,
Glide Slope = 90 deg -55 deg
= 35 deg
Altitude (AGL) = Line-Of-Sight Distance 2 sin(Glide Slope)
= 31785(0.57)
= 18,277 m
Distance To Spaceport = Line-Of-Sight Distance 2 cos(Glide Slope)
= 31785(0.82)
= 26,102 m
Glide Speed = Line-Of-Sight Distance 1 - Line-Of-Sight Distance 2
= 31865 - 31785
= 81 mps
Descent Rate = Glide Speed * sin(Glide Slope)
= 81(0.57)
= 46 mps
Ground Speed = Glide Speed * cos(Glide Slope)
= 81(0.82)
= 46 mps
Time To Touchdown = Line-Of-Sight Distance 2 / Glide Speed
= 690 s
= 11.5 min
S.T.E.M. Education. Don’t come home without it...
::
R.A.F.T. Writing
Role: Teacher
Audience: Middle School students
Format: Five paragraph essay
Topic: The Kennedy Space Center (KSC). What spacecraft were launched from there? Did any of the space launches go to the Moon? What was unique about their missions? What was in common with all the missions? How does KSC differ from the spaceport presented in this textbook? How are they the same? Why even bother to build a spaceport anyway?
::
4.4 Unpowered Glide Landing App
Given the above information, we can use a spreadsheet to enter equations and data to create a Space Mission Design App (SMDA).
The S.T.E.M. for the Classroom/Google App is broken down into four (4) parts:
1. Input/Output Interface
2. Graph
3. Constants
4. Calculations
The App can now be developed.
Sample Open Source Code
Once the cells have been named referencing cells is easy.
CALCULATIONS
LOSDist1=GlideDist1/1609
LOSDist2=GlideDist2/1609
GlideSlope=90-GlideAngle
Alt=LOSDist2*sin(GlideSlope)
DistToSpaceport=LOSDist2*cos(GlideSlope)
Spaceport America Spacecraft Landing App
::
4.5 Chapter Test
I. VOCABULARY
Match the aerospace term with its definition.
1. Adjacent Side of a Right Triangle
2. Descent Rate
3. Glide Slope
4. Hypotenuse
5. Right Triangle
A. A triangle with one of the angles equal to exactly 90 degrees.
B. The side next to the given angle (not the Hypotenuse).
C. The distance a spacecraft descends over a certain period of time.
D. The angle a spacecraft makes to the horizontal.
E. The longest side of a right triangle.
::
II. MULTIPLE CHOICE
Circle the correct answer.
6. A spacecraft is returning from space for a landing back at Spaceport America. It is required for the spacecraft to have the engines turned on in order to land safely.
A. TRUE B. FALSE
7. Given a Right Triangle, the sine of an angle (that is not the Right Angle) is defined as the Hypotenuse divided by the Adjacent side.
A. TRUE B. FALSE
8. What is the Glide Slope of a landing spacecraft if the Glide Angle is 60 degrees?
A. 60 deg B. 30 deg C. 15 deg D. Cannot be determined
9. The Landing Laser is malfunctioning and is giving the Glide Angle of a landing spacecraft. What is the distance to the Spaceport of the spacecraft?
A. 8.3 km B. 16.6 km C. 33.2 km D. Cannot be determined
10. If the Adjacent Side of an angle is increased, then the measurement of that angle ____.
A. Increases B. Decreases C. Unchanged D. Cannot be determined
::
III. CALCULATIONS
An R.E.L. Skylon is in an unpowered glide landing returning to Spaceport America. The Landing Laser paints the spacecraft with a 61 degree Glide Angle and a 19.70 miles Glide Distance. Exactly one second later, the spacecraft is holding steady at 61 degrees, but is now at 19.65 miles.
11. What is Glide Distance 1? Glide Distance 2?
12. What is Line-of-Sight Distance 1?
13. What is Line-of-Sight Distance 2?
14. What is the Glide Slope?
15. What is the Distance to the Spaceport?
16. What is the Altitude (AGL)?
17. What is the Glide Speed?
18. What is the Ground Speed?
19. What is the Descent Rate?
20. What is the Time To Touchdown?
::
IV. WRITING
Write a one paragraph essay on the topics below.
21. Explain why a triangle can never have two right angles.
22. Explain why if in a Right Triangle, increasing the Opposite Side of an angle increases the measurement of that angle.
23. Explain how to convert Glide Distance into Line-Of-Sight Distance, that is, convert Glide Distance to S.I. units.
24. Explain how to find the Descent Rate of a spacecraft returning from space.
25. Write a short story about what it would be like to feel the adrenaline after an unpowered glide back to Spaceport America from Low Earth Orbit aboard any spacecraft that you desire.
::
<<>>
CLICK HERE TO OPERATE THE SPACECRAFT UNPOWERED GLIDE LANDING APP
CLICK HERE FOR THE TEACHER SLIDE SHOW
(coming soon)
CLICK HERE FOR THE STUDENT HANDOUT
(coming soon)
CLICK HERE FOR THE ORBITAL PAYLOAD SPACE MISSION DESIGN PARAMETERS HANDOUT
(coming soon)
CLICK HERE TO GO TO THE EXAMPLE RUBRIC STUDENT WEBSITE
(coming soon)
::
END OF DIARY
::
A (partial) list of future topics in the series:
- S.T.E.M. Education For the 21st Century and Beyond
An Introduction to S.T.E.M. For the Classroom
- Go Where No Student Has Gone Before
A more indepth discussion of what we’re trying to accomplish.
- Suborbital Spaceflight - Quadratic Equations
Students calculate the height that SpaceShipTwo reaches space.
- Orbital Payload - Quadratic and Linear Equations
Students calculate the payload that the R.E.L. Skylon can place into Low Earth Orbit (LEO).
- A City in the Sky - Matrices
Students design a space station, and find the cost to place it into orbit. They also find the total volume and the number of crew that can safely occupy the station.
- Landing is the Hardest Thing to Do - Trigonometry
Students calculate the ground speed and altitude of a spacecraft returning from space.
- Delta V and Transfer Time - Square Root Equations
Students calculate the change in orbital velocity needed to go from a lower orbital altitude to a higher orbital altitude and find the time it takes for the maneuver.
- Spacecraft Weight Analysis - Linear Equations
Students find the weight of a real crew capsule that was designed in 1971 and determine the mission duration and the number of crew that can fly the mission.
- The Rocket Equation - Exponential Equations
Students determine the amount of cryogenic propellant needed to fly a space mission using an engine module designed in 1971.
- Fly Me to the Moon - Finance
Students calculate the amount of cryogenic propellant needed to land on the Moon and find the amount of profit you can make by selling moon rocks.
- Delta V and the Gravity of the Situation - Square Root Equations
Where we ask the question: does the mathematics add up to what the astronauts are depicted doing?
- The Thrill(e) in the Rille - Trigonometry
Students calculate the amount of rope needed for Apollo astronauts to safely descend into a lunar canyon.
- The Bone of Contention - Proportions
Students determine the identities of fictitious astronauts who have perished on a lunar landing mission using their recovered femur bones.
- TBA - Mathematics Topic is also TBA
Lesson plans that are still in the works...
::