Note: This article was initially published some time ago. However, recent tragic events warrant mentioning the loss of SpaceShipTwo, and one of the pilots. Hence, today's article is dedicated to space pilot Mike Alsbury and the men and women of Virgin Galactic and their endeavors to continue with suborbital spaceflight.
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Narrative
In this, the first of four aerospace-based S.T.E.M. lessons, students will calculate various Virgin Galactic SpaceShipTwo (SS2) spaceflight parameters and milestones, create an app, and write a report about it.
Time Frame
About 4.5 weeks (22 days)
Aerospace Problems
Maximum Altitude
Time Weightless
Time in Space
Spaceflight Duration
Mathematics Used
Quadratic Equations
The Quadratic Formula
Material List
A connection to the Internet
Google GMail account
Science Topics
Physics, Aerospace
Activating Previous Learning
Basic Algebra
Scientific Calculator
Essential Questions
- Who are the pioneers of parabolic spaceflight?
- What is the altitude at rocket burnout of a parabolic spacecraft?
- What is the maximum height of a parabolic spacecraft?
- Where does space actually begin?
- When was the first COMMERCIAL parabolic spaceflight?
- Why do people want to go on a parabolic spaceflight?
- How can I be weightless if I am still increasing my altitude?
- Wait. I have to do science and technology and engineering and mathematics, all at the same time?
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This lesson is powered by E^8:
1. Engage
Lesson Objectives
Lesson Goals
Lesson Organization
2. Explore
The Quadratic Equation
The Quadratic Formula
The Parabola and its Components and Definitions
Additional Terms and Definitions
3. Explain
The Vertex
The Vertex as a Maximum
Mission Duration Equation
4. Elaborate
Other Suborbital Spacecraft Examples
5. Exercise
Suborbital Space Mission Parameters
Suborbital Space Mission Design Scenario
6. Engineer
The Engineering Design Process
SMDA Spaceflight Plan
Designing a Prototype
SMDA Software
7. Express
Displaying the SMDC
Progress Report
8. Evaluate
Post Engineering Assessment
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Lesson Overview
Students first learn the basics of parabolic spaceflight 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.
Students will use spreadsheet software to create the app. The app will accept input on the spacecraft flight data and display the space flight profile.
Constants
Standard Gravity (m/s2)
Input
Rocket Burnout Time (min MET)
Rocket Burnout Altitude (ft MSL)
Rocket Burnout Velocity (mph)
Output
Maximum Altitude (m MSL)
Time Spent Weightless (min)
Time Spent In Space (min)
Spaceflight Duration (min)
Weightless Phase
Begin Weightlessness
Begin Spaceflight
Maximum Altitude
End Spaceflight
End Weightlessness
Spaceflight Duration
Carrier Phase
Boost Phase
Weightless Phase
Reentry Phase
Glide Phase
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Visual Learning
Here is a short (3 min) video of the first Virgin Galactic SS2 rocket engine flight test. The video explains the flight testing sequence, and how it relates to eventual commercial spaceflight profiles.
The successful test occurred 05Sep2013 at the Mojave Air and Space Port. This means that the final phase of flight testing has begun.
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Continued...
Continued...
1.2 Vocabulary
Boost Phase Begin Spaceflight Begin Weightlessness
Carrier Phase Duration Drop
End Spaceflight End Weightlessness Glide Phase
Maximum Altitude Mean Sea Level (MSL) Mission Elapsed Time (MET)
Reentry Interface Reentry Phase Rocket Burnout
Space Interface SpaceShipTwo (SS2) Suborbital Spaceflight
White Knight 2 Weightless Phase
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1.3 Analysis
A suborbital spacecraft, such as SS2, after a Drop from the White Knight 2 carrier aircraft, follows a flight profile that takes the shape of a parabola.. A parabola can be described with a quadratic equation, so that is what we will use.
The SS2 follows a similar trajectory that a baseball thrown to another baseball player follows. As all baseball players are aware, a baseball is never thrown in a straight line; rather it is thrown slightly upward. As a result, the path the baseball follows is curved (parabolic).
If the same baseball is thrown straight into the air, it will continue moving upward after it leaves the ball player's hand. The baseball at that point has an initial thrust to it, and the moment the ball is released it immediately begins to slow down. The ball will eventually reach a maximum height, where the speed becomes zero, and then drop back down to Earth (in this case, into the player's glove). The ball increased speed on the way down and arrived at the glove with the same energy that it left with.
Note: the moment the player releases the ball, the ball is weightless. Weight returns when the ball is catched.
That baseball (spacecraft) follows a nice parabolic curve and can be described by a quadratic equation.
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Virgin Galactic’s SpaceShipTwo is poised to go into space in the near future. Many people have already paid for their ticket, and will be flying into space as soon as the spaceship is ready. You are asked to build the prototype space mission app for your company.
Given the initial conditions as input, the app should display the following:
- Rocket Burnout Time (TimeRBO)
- Initial Velocity (InitVel)
- Initial Height (InitHt)
- How high into space you will go [Maximum Altitude (m)]
- How long you will be weightless [Time Spent Weightless (min)]
- How long you will be in space [Time Spent In Space (min)]
- How long your space flight will be [Spaceflight Duration (hr:min)]
- A graph of the weightless period [Time (min) vs. Altitude (m)]
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Flight Path of a Suborbital Spacecraft
The graph represents a typical weightless period spaceflight profile for SpaceShipTwo. The horizontal axis of the graph represents time (in seconds), and the vertical axis represents height (in meters). Space is defined as beginning at 100,000 m.
As you can see from the graph, the spaceflight takes the shape of an inverted parabola. This means that we can use a Quadratic Equation.
To find the maximum height and the time spent weightless, first determine the time component of the vertex, a.k.a, the axis of symmetry.
To find the time spent in space, find the time that the spacecraft entered space.
But first, the Quadratic Equation is:
- h(t) = -0.5*g0*t^2 + v0*t + h0
where,
g0 = Standard Gravity (9.80665 m/s2)
t = time (s)
v0 = initial velocity of the spacecraft (m/s)
h0 = initial height of the spacecraft (m MSL)
h(t) = height of the spacecraft at time t (m MSL)
Therefore the time component of the vertex is:
- vertex-t = -v0/[2*(-0.5*g0)] = -v0/-g0 = v0/g0
The time spent weightless is twice vertex-t, taking advantage of the symmetry of a parabola.
- Time-weightless = 2*vertex-t
To find the maximum height (vertex-h), just plug vertex-t into the Quadratic Equation.
- vertex-h = h(vertex-t) = -0.5*g0*vertex-t^2 + v0*vertex-t + h0
To find the time the spacecraft enters space, let h = 100,000, make the Quadratic Equation equal to zero, then use the Quadratic Formula:
100,000 = -0.5*g0*t^2 + v0*t + h0
0 = -0.5*g0*t^2 + v0*t + h0- 100,000
0 = -0.5*g0*t^2 + v0*t + h1
where,
h1 = height at space = h0 - 100,000
Now we use the Quadratic Formula to solve for the time you enter space:
space-t = [-v0 + SQRT(v0^2 - 4(-0.5*g0)(h1)] / 2(-0.5*g0)
= [-v0 + SQRT(v0^2 + 2*g0*h1)] / -g0
= [v0 - SQRT(v0^2 + 2*g0*h1)] / g0
Once you have the time you enter space, subtract it from the vertex-t, double it (again, because of symmetry), and we have the Time Spent in Space.
- Time-space = 2(vertex-t - space-t)
To find the Mission Elapsed Time for the five milestones of the Weightless Phase use the following equations:
- WeightlessBegin = TimeRBO
- SpaceBegin = TimeRBO + space-t
- Altitudemax = TimeRBO + vertex-t
- SpaceEnd = TimeRBO + 2*space-t
- WeightlessEnd = TimeRBO + 2*vertex-t
We can now tackle the spaceflight duration calculations.
Given:
- Boost Phase = 70 s = 1.17 min
- Reentry Phase = 3.50 min
- Glide Phase = 25.00 min
Then
- CarrierPhase = TimeRBO - Boost Phase
and
- DurationSpaceflight = Carrier + Boost + TimeWeightless + Reentry + Glide
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Example
Suppose we have an input of
- TimeRBO = 110 min
- InitHt = 135,000 ft MSL
- InitVel = 2,600 mph
What is the Time Spent Weightless, the Maximum Height achieved, the Time Spent In Space, and the Time Spent on Spaceflight aboard SpaceShipTwo?
First, let’s convert the input into S.I. Units:
- v0= 2,600 mi/hr * 1,609 m/mi * 1 hr/3,600 s = 1,162 mps
- h0= 135,000 ft * 1 m/3.28 ft = 41,148 m MSL
So,
- vertex-t= v0/g0 = 1162/9.80665 = 118.49 s
The Time Spent Weightless is:
Time-weightless = 2 * vertex-t
= 2 * 118.49
= 236.98 s
= 3.95 min
and the maximum altitude is:
vertex-h = f(vertex-t) = -0.5*g0*vertext^2 + v0*vertex-t + h0
= -0.5*9.80665*118.49^2 + 1162*118.49 + 41148
= 110,027 m MSL
Finding h1,
h1= h0 - 100000 = -58,852 m
we can use it to find the time you enter space:
space-t = [v0 - SQRT(v0^2 + 2*g0*h1)] / g0
= [1162 - SQRT(1162^2 + 2*9.80665*-58852] / 9.80665
= 73 s
Once we know when we enter space we can calculate the Time Spent in Space:
Time-space = 2(vertex-t - space-t)
= 2(118.49 - 73) s
= 90.98 s
= 1.52 min
We now calculate the other mission milestones and the total mission duration:
Carrier Phase = TimeRBO - Boost Phase
= 110 min - 1.17 min
= 108.83 min
and
DurationSpaceflight = Carrier + Boost + TimeWeightless + Reentry + Glide
= (108.83 + 1.17 + 3.95 + 3.50 + 25.00) min
= 142.45 min
= 2 hrs 22.45 min
So, given an initial velocity at Rocket Burnout Time of 110 min going 2,600 mph, with an initial height of 135,000 ft MSL, we can make the following conclusions:
- Time Spent Weightless is 3.95 min
- Maximum Height achieved is 110,027 m MSL
- Time Spent In Space is 1.51 min
- Time Spent on Spaceflight aboard SpaceShipTwo is 2 hr 22.45 min
Therefore, out of an almost two and a half hour flight, the passengers spend less than four minutes weightless, and less than two minutes in space. If each ticket costs $250,000, that comes out to about $2,748 for each second spent in space!
The time spent in space is irrelevant anyway; that the passengers went into space is the real story.
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R.A.F.T. Writing
Role: Teacher
Audience: Middle School students
Format: Five paragraph essay
Topic: The X-15. Who were some of the astronauts that flew the missions? Did any of the pilots fly into space? What was unique about their missions? What was in common with all the missions? How does an X-15 suborbital space mission differ from the space mission presented in this textbook? How are they the same? Why even bother to fly a suborbital spaceflight anyway?
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1.4 Suborbital Space Mission Design 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
- VertexTime=v0/g0
- TimeWeightless=2*VertexTime
- MaxAlt=-0.5*g0*VertexTime^2+v0*VertexTime+h0
- h1=h0-Space
- SpaceTime=(v0-SQRT(v0^2+2*g0*h1))/g0
- TimeInSpace=2*(VertexTime - SpaceTime)
- GRAPHING
- BeginWt=RBO
- BeginSpace=RBO+TimeSpace
- MaxAlt=RBO+VertexTime
- EndSpace=RBO+2*TimeSpace
- EndWt=RBO+2*VertexTime
Suborbital Space Mission App
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1.5 Chapter Test
I. VOCABULARY
Match the aerospace term with its definition.
1. End Spaceflight
2. Mission Elapsed Time
3. Rocket Burnout
4. SpaceShipTwo
5. Weightless Phase
A. The moment a rocket engine shuts itself off, where the spacecraft continues upward on its own momentum.
B. The spacecraft that is dropped from White Knight 2. After rocket burnout, the spacecraft coasts up to space and back.
C. The third of six phases in a parabolic spaceflight, where the spacecraft and its occupants experience weightlessness.
D. The moment a spacecraft exits from space. The spacecraft returns to the atmospheric environment.
E. Time since the beginning of the spaceflight.
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II. MULTIPLE CHOICE
Circle the correct answer.
6. The Quadratic Equation describing the parabolic flight profile of a Virgin Galactic suborbital spaceflight has a leading coefficient that is less than zero.
A. True B. False
7. After reaching maximum altitude, weight returns and the Virgin Galactic passengers are no longer weightless.
A. True B. False
8. If Vertex-t = 1.75 min, then the Time Spent Weightless is
A. 1.75 min B. 3.50 min C. 5.25 min D. None
9. You reach the maximum altitude 30 seconds after crossing into space. How long will you be in space?
A. 30 sec B. 60 sec C. 90 sec D. None.
10. A Virgin Galactic suborbital spacecraft is going 1,100 mps at the moment of Rocket Burnout. How long will it take to coast up to the maximum altitude?
A. 56.12 sec B. 112.24 sec C. 224.48 sec D. 336.72 sec
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III. CALCULATIONS
A Virgin Galactic suborbital spacecraft has an initial velocity at Rocket Burnout Time of 108 minutes, going 2,550 mph, with an initial height of 140,000 feet MSL.
11. How long does it take to reach maximum altitude after rocket burnout?
12. What is the maximum altitude of this spaceflight?
13. How long does it take to reach space after rocket burnout?
14. How long were the Virgin Galactic passengers weightless?
15. How long were the Virgin Galactic passengers in space?
16. How long was the Carrier Phase of the suborbital spaceflight?
17. How long was the spaceflight?
18. What percent of the suborbital spaceflight was spent during the Weightless Phase?
19. What percent of the suborbital spaceflight was spent in space?
20. What percent of the suborbital spaceflight was not spent in space or during the Weightless Phase?
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IV. WRITING
Write a one paragraph essay on the topics below.
21. Explain why the leading coefficient of the quadratic equation describing the parabolic spaceflight profile of a Virgin Galactic suborbital spacecraft is negative.
22. Explain why the total time weightless can be calculated by doubling the time it takes to reach the vertex of the parabola of a Virgin Galactic suborbital spaceflight.
23. Explain why passengers feel weightless even though the spacecraft is coasting to a maximum altitude in the UP direction.
24. Describe the step-by-step procedure to calculating the maximum altitude reached by a Virgin Galactic suborbital spacecraft given the time that the maximum altitude occurred.
25. Write a short story about what it would feel like to float weightlessly inside of SpaceShipTwo while gazing at the curvature of the Earth as it flies a parabolic spaceflight profile.
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CLICK HERE TO OPERATE THE SUBORBITAL SPACEFLIGHT APP
CLICK HERE FOR THE TEACHER SLIDE SHOW
CLICK HERE FOR THE STUDENT HANDOUT
CLICK HERE FOR THE SUBORBITAL SPACE MISSION DESIGN PARAMETERS HANDOUT
CLICK HERE TO GO TO THE EXAMPLE RUBRIC STUDENT WEBSITE
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END OF DIARY
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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...
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