SI Prep -- Physics -- Lunar Lander Project

AP Physics C: Lunar Lander Project

Last modification December 29, 2009 8:39 PM by Byron Philhour

Mission

Over the course of the school year, 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

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
- Watch an Apollo 11 launch video montage
- Check out the mission transcripts
- Watch documentary footage (In the Shadow of the Moon) regarding the Apollo program
- Optional: Watch the From the Earth to the Moon miniseries
Set the current scene
- Read this background material on Project Constellation
Learn to use the Excel spreadsheet program
- Work this Excel tutorial and the quiz that follows
- Write a projectile motion simulation that accomplishes the following:
  - 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)

Objective 2: Analyze the forces and accelerations involved in a launch to near Earth orbit

Simulate a Launch
- Vertical launch
  - 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)
        
        to get these properties, you can scale from the Saturn V numbers, presuming that the fuel quality hasn't changed =0
        
        Stage I
        
        Stage II
        
        Stage III
      - 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"
  - 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
- Equatorial Orbit launch

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)
Simulate an Earth orbit to Moon orbit trajectory
- Check out my a sample spreadsheet for this exercise [.pdf] (thanks Daine D. '10) -- be sure to zoom in to the .pdf!
- Check out my notes on an Lunar Transfer Orbit coordinate system -- use these 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
  - Launch to a near-circular orbit of Earth
  - Use an x-y coordinate system centered at the center of the Earth, as before
  - Once in space, 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) - this means we'll be including a conditional (IF) statement that adds a certain x and y component of velocity in the direction of motion at a provided time. Make the model flexible enough to allow for multiple 'delta-v's
  - Inputs to the model include
    - all the inputs from our launch to equatorial orbit of Earth
    - the time step (once in orbit, shift to much longer time steps -- even 100 seconds -- you should make this vary using a conditional (IF) statement so that various parts of launch & orbit can have different time steps)
    - assume the mass doesn't change appreciably when the rockets fire (note change from prior sim -- we aren't using significant fuel during these thrusts - so the max of the vessel is effectively fixed)
    - a table 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 - or at least shorter than the time step)
      - the orientation of the craft at each delta-v
    - assume the moon is in circular orbit around the Earth with (sidereal) period 27.3 days (if you'd like you can make it elliptical, but you aren't expected to)
  - Outputs of the model include
    - at each time step
      - all of the parameters from the initial simulation (equatorial launch)
      - 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 vector)
      - the angular momentum of the craft measured with respect to the Earth's center as pivot point (assume everything happens in the same plane, so no need to create a vector)
      - the angular momentum of the craft measured with respect to the Moon's center as pivot point
      - the gravitational acceleration acting on the spacecraft due to the Earth and, separately, due to the moon
      - evidence that
        
        the gravitational torques acting on the spacecraft (r x F) about an Earth-centered pivot lead to equivalent changes in angular momentum (dL/dt) about the same pivot (that is, verify the rotational version of the force-impulse relationship) -- (see Reher '10 results for Earth and Moon as pivot)
        
        the total mechanical energy of the craft only changes significantly during the delta-v maneuvers (that is, verify the conservation of energy) -- (see Reher '10 result)
    - find 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 a close up of launch and trip to Earth orbit, the set of two or more delta-vs, and finally close-ups of 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 the moon need not be displayed at intermediate times.
Experience a Moon landing [under construction]
- 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
Investigate the energetics of an Earth-atmosphere re-entry and splashdown [under construction]
- 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)

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