AP Physics C: Lunar Lander Project
Last modification September 25, 2009 11:10 AM by Byron Philhour
Mission
Over the course of the 1st semester, students in AP Physics C will investigate and simulate aspects of a lunar landing and return mission. The purpose is to provide a unifying story for the different theoretical and experimental expectations of the course, and to give us a chance to have some fun. This project is significantly more challenging, complex, and time-consuming than most, and necessarily involves a very high level of student engagement and self-direction.
Objectives [skip to the current objective]
Objective 1: Familiarize yourself with the basic terminology of the human spaceflight program and the main tools we'll need to study the topic
- Set the historical context
- Read this background material on the Apollo program [Step 1]
- Watch an Apollo 11 launch video montage [Step 1]
- Check out the mission transcripts [Step 1]
- Watch documentary footage (In the Shadow of the Moon) regarding the Apollo program [Step 1]
- Optional: Watch the From the Earth to the Moon miniseries (a copy will be available for showings outside of classtime upon request from BJP) [Step 1]
- Set the current scene
- Read this background material on Project Constellation [Step 1]
- Learn to use the LabPro Data Acquisition System (DAQ)
- Visit the website documenting the Vernier LabPro specs [Step 1]
- Make use of the LabPro User Manual
[Step 2]
- You are responsible for pages 1 - 6 and Appendices A - C
- Read about the large variety of LabPro probes (many of these can be found in the green cabinet on the northern wall of the lab room (Room 312) [Step 1]
- Learn to use the Excel spreadsheet program
- Work this Excel tutorial and the quiz that follows [Step 1]
- Write a projectile motion simulation that accomplishes the following:
[Step 2]
- given an intial position (x,y), initial speed, initial angle with respect to horizontal, and gravitational field g, calculate:
- the x and y components of initial velocity
- the time of flight
- the range
- data representing the position (x,y), velocity (v_x,v_y), speed, and angle with respect to horizontal at 0.1 s intervals along its trajectory
- a plot of the trajectory of the projectile at appropriate time intervals (by trajectory, we mean a plot of y vs. x ... so each data point will correspond to a different point in time)
- given an intial position (x,y), initial speed, initial angle with respect to horizontal, and gravitational field g, calculate:
- Re-familiarize yourself with the Scientific Method
- Read the rubric/guide to the Scientific Method for SI students: (PowerPoint format, color .pdf, grayscale .pdf) [Step 1]
Objective 2: Analyze the forces and accelerations involved in a launch to near Earth orbit
- Verify the underlying theory
- Work the following laboratory activities using provided class time AND your own out-of-class time (as homework) -- do a complete write-up and (for each) attach the Level III Rubric I'll use for grading
- Expand your Excel spreadsheet to include Newton's 2nd Law
- Given an initial two-dimensional position, initial two-dimensional velocity, constant two-dimensional net force, and an object's mass, calculate the two-dimensional position, velocity, and acceleration for 100 future points in time separated by 1 s intervals; plot the trajectory (x vs. y positions) of the object [Step 2]
- Simulate a Launch
- Vertical launch [Step 2]
- Begin with the following assumptions:
- flat Earth
- g = 9.8 m/s^2 at all altitudes (this is a better approximation than you might think: suborbital and orbital spaceflight takes place very close to the Earth, relatively speaking)
- no rotation of the Earth
- no rotation of the rocket about its center of mass (i.e., no pencil-tip balance problem)
- no orbit
- one-dimensional motion (vertical only)
- Write a launch simulation in Excel that accomplishes the following:
- simulation inputs
- payload mass (determined by project needs; 47000 kg to lunar orbit for Saturn V / Apollo)
- rocket diameter (unconstrained; 10.1 m for Saturn V)
- air density profile for the Earth (NRLMSISE-00) -- you'll notice that the vertical scale is logarithmic, so that'll take some thinking to see how it works // also, if you need a value between two data points, you can estimate it from the graph. The better your air density profile (that is, the more detailed) the less 'jerky' your simulated launch will turn out.
- You can also use the less accurate formula for air density p = p_0*e^(-h/z), where h is the altitude and z is the 'scale height' of the atmosphere, usually 10 km, and p_0 is the density of air at sea level. This will not produce as realistic a simulation but might be a good starting point.
- thrust, fuel mass, stage mass, fuel volume, and burn time for each stage (these are not all independent - determine the missing values from the provided values)
- cross-sectional area ("profile") of rocket -- also not independent of the values above
- coefficient of drag (C_d) for air resistance (you can leave at 0.5 for now, but have it be editable)
- simulation outputs
- maximum rocket height (constrained by fuel volume and rocket diameter) -- does your rocket "escape" or must it come back to Earth eventually?
- data & plots representing the following values at 10 s (or other appropriate) intervals:
- vertical acceleration
- vertical velocity
- vertical position
- force of air resistance (drag force / drag equation)
- force of gravity (weight)
- thrust force
- time and vertical height of "max-q"
- simulation inputs
- Here is a .pdf printout of BJP's "Flat Earth" launch simulation for your reference -- yellow painted objects are inputs, the rest are outputs
- From the flight journal above: you can use this to judge your acceleration vs. time graph
- Begin with the following assumptions:
- Equatorial Orbit launch [Step 3]
- Check out this Apollo 8 flight journal
- Check out my notes on an equatorial coordinate system -- use if you'd like [.doc, .pdf]
- Check out my Excel spreadsheet for this exercise [.pdf]
- This more complicated launch and orbital insertion operates under the following assumptions:
- round, rotating Earth
- launch from the equator (note that you'll need to calculate the tangential velocity of the Earth at the equator)
- Newtonian gravity
- no rotation of the rocket about its center of mass (i.e.,no pencil-tip balance problem)
- orbit!!!
- simulation inputs as for vertical launch, but also add:
- attitude of rocket (angle with respect to the horizontal immediately after launch)
- pitch program for rocket -- how the attitude changes with time
- simulation outputs as for vertical launch, but also add:
- horizontal and vertical (x and y) positions measured with respect to the center of the Earth
- horizontal and vertical velocities (x and y)
- forces and acceleration expressed as vectors (x and y components)
- a plot of the horizontal and vertical positions of the rocket from launch to stable orbit (see launch profile below)
- a plot of the trajectory of the spacecraft (x & y)
- a plot of the attitude angle (compared to vertical) of the rocket vs. time
- From the flight journal above: you can use this image to determine the nominal attitude of the rocket as a function of position
- Optional: launch from Kennedy Space Center in Florida (use z to represent positions 'above' or 'below' the equator)
- Vertical launch [Step 2]
Objective 3: Analyze the energy and angular momentum considerations involved in a journey to, landing on, and return from the Moon
- Explore the basic concepts
- Have fun with a (cartoonish) simulation of a moon landing using the Simple Java Lunar Lander [Step 1]
- Download and watch this simulated landing [.wmv] created using Eagle Lander 3-d -- you'll notice there's an object already waiting for them in the crater near the landing spot. What is that? Find a close-up photo of it (a real photo) and include it in your binder with this project [Step 1]
- Verify the underlying theory
- Explore the theory of conservation of (vector) momentum and (scalar) energy
- Simulate an Earth orbit to Moon orbit trajectory
[Step 3]
- Check out my Excel spreadsheet for this exercise [.pdf]
- Check out my notes on an Lunar Transfer Orbit coordinate system -- use if you'd like [.doc, .pdf]
- Write an Excel-based simulation of a transfer from Earth orbit to Moon orbit with the following parameters
and assumptions
- Start in a circular orbit of Earth with nominal orbital velocity
- Use an x-y coordinate system centered at the center of the Earth
- The only force applicable is gravity -- we will model thrust as small bursts known as delta-v's(literally the change in velocity of the craft as a result of the trust, aka the impulse divided by the mass)
- Inputs to the model include
- the time step (begin at about 100 s time steps -- note this is much longer than in our prior simulation - this leads to some slight errors in energy conservation)
- starting altitude above the surface of Earth
- starting orbital velocity above the surface of the Earth
- the mass of the spacecraft (can start at 47000 kg)
- assume the mass doesn't change appreciably when the rockets fire (note change from prior sim)
- a set of 'delta-v's which consist of
- the magnitude of the change in speed delta-v in km/sec as a result of the thrust impulse
- the moment in time when this delta-v occured measured from start time (for the purposes of this simulation, we'll assume the delta-v is instantaneous)
- the orientation of the craft as measured in the absolute coordinate system centered on the earth at t = 0.
- assume the moon is in circular orbit with (sidereal) period 27.3 days
- Outputs of the model include
- at each time step
- the kinetic energy of the craft
- the potential energy of the craft measured with respect to 0 J at infinite distance
- the total energy of the craft measured with respect to 0 J at infinite distance
- the momentum of the craft (two-dimensional)
- the angular momentum of the craft measured with respect to the Earth's center
- the angular momentum of the craft measured with respect to the Moon's center
- the gravitational acceleration due to the Earth and due to the moon
- evidence that
- the gravitational torques acting on the spacecraft (r x F) lead to equivalent changes in angular momentum (dL/dt)
- the total mechanical energy of the craft only changes significantly during the delta-v maneuvers
- the position of the 'cross-over' point where the spacecraft transitions from dominance by the Earth's gravitational influence to dominance by the Moon's gravitational influence
- a plot of the trajectory including an Earth orbit, a set of two or more delta-vs, and finally several Moon orbits. The trajectory plot should show the position of the moon at the time of launch and at the time of reaching lunar orbit, but need not be displayed at intermediate times
- at each time step
- Experience a Moon landing
- This is an optional Step 2 or Step 3 exercise for students who are interested.
- Learn to use the free, downloadable portion of the Eagle Lander 3-d
- Demonstrate to me your understanding of the program and lunar landings in general (in person)
- Show me something cool
- This is an optional Step 2 or Step 3 exercise for students who are interested.
- Investigate the energetics of an Earth-atmosphere re-entry and splashdown
- Write an Excel-based simulation of an atmospheric re-entry
- You'll probably want to use, as a starting point, your launch simulation from Objective 2. Remove all reference to thrust, but do include weight and drag force. Don't worry about rotation of the Earth -- use the absolute velocity as the wind velocity when calculating drag.
- Inputs to the model include
- mass of the return vehicle (find online)
- initial trajectory of the return vehicle
- start outside the atmosphere with a certain speed and trajectory angle with respect to tangential (this is what we called 'pitch' in the first assignment)
- use the drag to slow the spacecraft
- deploy a parachute at the appropriate time to significantly increase the drag area -- bear in mind that these parachutes do not hold up well at high speeds/temperatures -- try to model your simulation on the actual Apollo landings
- outputs from the model should include
- total, kinetic and gravitational energy at each time step
- dissipated power at each time step (in Watts)
- landing speed
- acceleration at each point (should not exceed fatal values)