CK-12 21st Century Physics: A Compilation of Contemporary and Emerging Technologies
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Mass is the amount of stuff in an object. The more atoms, molecules, or whatever there is in an object, the more mass it has. Mass used to be called inertia because it is hard to change the motion of a large mass. Mass likes to continue moving like it is moving and it takes a force to change its motion.
Force
Force is a push or a pull. When an elevator accelerates upward, the motor exerts a force on the cable attached to the elevator to pull it up. Even when the elevator is going down, the motor must exert a force on the cable to keep the elevator from falling too fast. When you stand on the floor, the floor exerts an upward force on you to keep you from falling through the floor. When you stand on a scale, the floor exerts an upward force on the scale and the scale in turn exerts an upward force on you, which is read on the scale. Complicated - isn't it?
Gravity
Now that you understand the basics of motion, we can discuss an important force in many cases involving motion. All masses attract each other. We call that gravity. The Earth is a very large mass equal to about a trillion billion elephants. The Earth attracts the rocket towards its center and this often causes the rocket to return back to Earth. The acceleration of a freely falling body near the surface of the Earth is given the symbol . How would things change on the Moon or other planets? You can explore that later.
Weight
When you stand at rest on the floor, the floor pushes up on you to keep you from falling through. The value of this force is your mass times and we call the value of this force your weight . This is what you actually feel because of Earth's gravity.
Newton’s Laws: Putting It All Together
Isaac Newton (1643-1727), one of the greatest minds ever, discovered that force was directly related to acceleration. The more total force you exert on an object, the greater its acceleration (, where stands for the total force on the mass in question). In many cases there can be many objects acting on one another, so using Newton's idea can be complicated. In our case the rocket motor and gravity will be acting on the rocket in the simplest case. We will also consider the air drag acting on the motion (wind can also act on the rocket, but we won’t model that). Thanks to Newton, we now have the reason for acceleration and thus motion. See Laboratory Activities in this book for experiments on Newton’s laws.
Modeling and Simulation
In the simplest case where the rocket motor provides a constant upward acceleration and we neglect air drag, wind, and the rotation of the Earth (not a realistic assumption for an actual rocket), the motion can be solved in closed form with relatively simple algebra. However, as we add air drag and model the rocket engine more realistically, we have to resort to solving differential equations. Ultimately, a numerical simulation using a computer is called for. Leonhard Euler (1707-1783) developed a conceptually simple method for numerically solving equations like those we will deal with. The method involves rates of change and equations that can’t be solved with algebra. Although there are more sophisticated and efficient numerical methods, modern computers are fast enough that we will be able to get by with Euler’s conceptually simpler method. It is very powerful in its simplicity.
Approach to Learning
Students can learn a lot by looking at examples of how others solved a particular problem and then modifying the solution to adapt the problem to less stringent assumptions. It is in the modification process that the student’s understanding of the solution is tested and where valuable learning takes place. We will follow that process in this chapter. Using Etoys, a simulation of the rocket, with simplifying assumptions will be available for the learner to modify to a more realistic solution. As more complexity is included along the way, the learner will be rewarded with a sense of accomplishment and a deeper understanding of modeling, simulation and motion.
General Assumptions
We start by making several assumptions to simplify the problem of the LAS and relax several of them as we approach a more realistic model.
The acceleration of gravity is constant at . This is a reasonable assumption near the surface of the earth, but at the International Space Station, that acceleration has already dropped to around 0.9 the value that it has at the Earth’s surface. We can relax this assumption without too much effort.
We will neglect air resistance. This isn’t a reasonable assumption and it is one of the first we will attack with Euler’s method.
The rocket engine will be modeled as providing a constant upward acceleration. This isn’t realistic and we will later model the rocket engine with a constant gas exhaust velocity relative to the rocket and a constant rate of mass decrease of the rocket fuel. However, we won’t be able to model the actual engine used by NASA, as that is proprietary information – proprietary information is information or intellectual property owned by a company and protected from unauthorized distribution for the purpose of allowing the company to make money.
We will neglect the fact that Earth is rotating. Newton’s laws apply in a coordinate system that isn’t accelerating. Because the Earth rotates about its axis, a coordinate system attached to the Earth from which we observe the motion, is accelerating. Anything moving in a circle accelerates because the direction of the velocity is always changing. Velocity is a vector and has both magnitude and direction. If either the direction or magnitude changes, the velocity changes and the object is said to accelerate. The acceleration of a point on the equator is around . Although this is a small fraction of , the effect is very noticeable for the range of motion of a typical rocket. In addition there is a smaller acceleration because the Earth revolves around the Sun and an even smaller one because the Sun revolves about the center of our Milky Way galaxy , and so on... It isn’t easy to do physics calculations in a rotating coordinate system and we will leave that as an independent investigation for the learner. Read about the Coriolis effect (Gaspard-Gustave Coriolis 1792-1843) if this topic interests you.
We will neglect forces from winds.
We will neglect departures of the Earth from a uniform sphere.
We will neglect the buoyant force of the Earth’s atmosphere on the rocket.
Model 1: Constant Acceleration in One Dimension
We could solve some of the models using algebra and others would require solving differential equations, which requires a greater mathematical sophistication. Although we could use closed form techniques, we will use the Euler method, which is conceptually simple, powerful and can be used to solve nearly any problem given enough computing power and time. Learning to solve problems numerically is a valuable skill to have. We will start modeling the rocket engine as providing a constant upward acceleration of the rocket, which moves upward in one dimension until the burn is completed. Once the burn is finished, the rocket will be in freefall as we will neglect air resistance and winds. We will ignore the fact that we are in a rotating coordinate system.
Description
You can access this model after downloading and installing Etoys at squeakland.org. The URL for the model is http://www.pcs.cnu.edu/~rcaton/flexbook/flexbook.html. If you can’t find a computer where you can install Etoys, it is possible to run the simulation off a memory stick. You can download a copy of Etoys-To-Go at squeakland.org and the simulation at http://www.pcs.cnu.edu/~rcaton/flexbook/flexbook.html. The simulation is an Etoys project with an extension .pr. If the downloading process puts a .txt extension on the end, remove the .txt extension. You need to unzip the downloaded file and put Etoys-To-Go and the simulation on the same memory stick. The simulation .pr file should go in the Etoys directory.
Experimental Observation and Understanding
Scientists and engineers find it useful to plot data on a graph to visualize what is happening and you will too. Before modifying the simulation, it is instructive to take data on the simple model in the Etoys project.
Exploration 1: To start collecting data, set your values for burn time and acceleration in the Control and Data Center, click the yellow reset button, record the time a
nd altitude, click the red launch button, click the yellow pause button approximately every second, record your new data, and click the red launch button to resume. Take data until 10 seconds after the burn time is over. Plot the position on the vertical axis against the time on the horizontal axis. Work in groups and brainstorm how you should best collect and record the data. Have each member choose different burn times and accelerations for the rocket. Discuss your results in the group and compare with others' data and plots.
How does the burn time affect your graph's shape?
How does the acceleration affect your graph's shape?
Exploration 2: Record data in your notebook on the velocity of the rocket approximately every second until 10 seconds after the rocket's burn time is over. Plot the velocity on the vertical axis against the time on the horizontal axis. Work in groups and brainstorm how you should best collect and record the data. Have each member choose different burn times and accelerations for the rocket. Discuss your results in the group and compare with others' data and plots.
What is special about the shape of your graph? Remember, there is always error in real data, so your graph may not be perfect. Try to visualize what the ideal graph would look like.
How does the burn time affect your graph's shape?
How does the acceleration affect your graph's shape?
Exploration 3: What would happen if you launched the rocket on the Moon or a planet other than Earth? Work in groups and have each member find a value for the acceleration of gravity at the surface of the Moon or planet. Use books and the Internet. Be sure the units are meters per second per second so you can compare with the value given for Earth. If the values are in different units, look up how to convert the values to the needed units. Enter the new gravity values in the second from bottom box in the Control and Data Center. Take data on the new motions and plot it. Discuss your results in the group and compare with others' data and plots.
Model 2: Air Resistance
We will relax some of the simplifying assumptions one at a time starting with air resistance. For large objects like the CEV, the air resistance can be modeled as producing a force proportional to the square of the speed and opposing the motion of the object. Your assignment is to add this to the previous model by modifying the force expressions for the two regions: the constant acceleration upward while the rocket engines are burning and the subsequent freefall. It would be wise to include a proportionality constant in the air resistance term so you can vary the strength of the force of the air on the rocket. Although Euler’s method is used in the simulation, it would be good to explicitly present the notation here.
Because and are constantly changing with time, they need to be continually updated. Take a small step in time and change and . Calculate the new values for and using
For example, , with positive when the rocket is freely falling toward Earth. You need to figure out the other cases during the motion. The green values are current values of and and the blue are the new values. The amazing thing is that for small enough the result approaches the correct solution with arbitrary precision for well behaved systems.
Model 3: A More Realistic Rocket
Rockets aren’t usually designed to provide a constant upward acceleration. Although we can’t divulge the proprietary propulsion scheme, a first step in modeling a more realistic rocket is to assume a constant burn rate for the fuel and a constant ejection speed of gas relative to the rocket. The force on a rocket taking off from Earth has two components: one from the gravitational force of the Earth and the other from the expulsion of gases from the rocket engine (thrust). We will continue to neglect the rotation of the Earth and small variations in the acceleration of gravity from height above the Earth and departures of the Earth from a perfect uniform sphere. The discussion below leaves out air resistance. You can add it if you wish.
Call the mass of the rocket. The force of gravity is if we take upward as positive and assume we are near the Earth's surface where is fairly constant. The rocket force can be found from the impulse momentum theorem: because is negative. The d's stand for small changes and is the velocity of the ejected gas relative to the rocket. We assume that and are constant to make the problem easier. The total force can be written as
We get the following finite differentials:
Because , and are constantly changing with time, they need to be continually updated. Take a small step in time and change , and . Calculate the new values for , and using
The green values are current values of , and and the blue are the new values. The amazing thing is that for small enough the result approaches the correct solution with arbitrary precision for well--behaved systems.
Your assignment is to modify the previous model to use the more realistic rocket engine described above. You can include air resistance if you wish, but it may be best to leave it out at first until you have the rocket engine model working properly. You can get some numbers to test your model from Physics by Alonso and Finn.
Model 4: Accounting for the Change in Gravity as the Distance from the Earth Changes
Earth’s force of gravity reduces as the rocket’s distance from Earth increases. To model this, assume an Earth with spherically symmetrically distributed matter and then you can replace the Earth with a point mass at its center. Newton held up publication his Principia until he could prove this was true. For the LAS at the launch pad, the variation of the force of gravity isn’t a big effect because the CEV doesn’t get that far from the Earth’s surface, but it is important for the journey to the moon and valuable for your education to model this behavior.
Model 5: Simplified LAS
Figure 2: Scope of LAS Operating Environment. (Attribution: NASA)
A simplified 2D model of the LAS implemented in Etoys can be downloaded at http://www.pcs.cnu.edu/~rcaton/flexbook/flexbook.html. Explore this model to learn how it works. A simulation with changing mass would be a much better representation. The LAS + CM loses about a 7th of its mass during the abort motor burn. We used a simplified thrust curve for the abort motor because the actual thrust curve is “NASA sensitive.”
Extensions
Adapt the 2D LAS model to employ a rocket with constant exhaust velocity and constant rate of mass loss. Model a two-stage 1D rocket.
Contact Information
If you have questions about the simulations, feel free to contact Randall Caton at rcaton@cnu.edu.
This chapter has been reviewed by John Stadler.
References
1. Orion Crew Exploration Vehicle Launch Abort System Guidance and Control Analysis Overview, John B. Davidson, Sungwan Kim, David L. Raney, Vanessa V. Aubuchon, Dean W. Sparks, and Ronald C. Busan, Ryan W. Proud and Deborah S. Merritt
TABLE OF CONTENTS
CK-12 License
Chapter 1: VA Introduction
Chapter 2: Toward Understanding Gravitation.
Chapter 3: Nuclear Energy
Chapter 4: The Standard Model of Particle Physics
Chapter 5: The Standard Model and Beyond
Chapter 6: A Brief Synopsis of Modern Physics
Chapter 7: Nanoscience
Chapter 8: Biophysics (Medical Imaging)
Chapter 9: Kinematics: Motion, Work, and Energy
Chapter 10: Laboratory Activities
Chapter 11: Statitical Physics and Random Walks
Chapter 12: Modeling and Simulation in the Physics Classroom
Chapter 13: Modeling and Simulating NASA's Launch Abort System
CK-12 21st Century Physics Flexbook: A Compilation of Contemporary and Emerging Technologies
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