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.
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 9.8 m/s2. 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 0.034 m/s2. Although this is a small fraction of g, 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 0.006 m/s2 and an even smaller one because the Sun revolves about the center of our Milky Way galaxy 2.4 x 10 -10 m/s2, 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.
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 and 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 y and vy are constantly changing with time, they need to be continually updated. Take a small step in time dt to change y and vy. Calculate the new values for y and vy using
For example, , with k 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 y and vy and the red are the new values. The amazing thing is that for small enough dt 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 M the mass of the rocket. The force of gravity is -Mg if we take upward as positive and assume we are near the Earth's surface where g is fairly constant. The rocket force can be found from the impulse momentum theorem: Frdt = -vedM because dM (the small change in M) is negative. ve is the velocity of the ejected gas relative to the rocket. We assume that dM/dt ( = -C) and ve are constant to make the problem easier. The total force can be written as
We get the following finite differentials:
Because y, vy and M are constantly changing with time, they need to be continually updated. Take a small step in time dt to change M, y and vy. Calculate the new values for M, y and vy using
The green values are current values of M, y and vy and the red are the new values. Again for small enough dt 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 (p. 136).
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 valuable for your education to model this behavior.
Model 5: Simplified LAS
Figure 3: 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.”
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.