ENGAGE: Introduction to Model Rocketry (80 minutes)
Watch the NASA eClip video, "Launchpad: Designing a Capsule for Space" to set the stage both for the engineering design process and the complexities of launching a rocket upward.
Ask students if they have ever built and launched a model rocket. Briefly discuss their experiences with model rocketry. Explain that they will be completing an engineering project (which differs from a "science" project) focused on designing a model rocket and predicting its motion. Provide teams of three students with a simple standard model rocket kit. Allow students approximately 45–60 minutes to build the rocket, according to the instructions provided in the kits, asking them to take note of the various parts of the rocket as they build the model.
When students have completed their models, pass out or project the student handout "The Alpha Model Rocket Nomenclature" (page 2 of the Estes Reproduction Masters). Review the basic structure of a model rocket and ask students to brainstorm which parts of the rocket could be changed in order to improve its performance, where "performance" refers to how high and how straight the rocket flies. Offer no opinion as to the efficacy of their suggestions.
Explain to the students that it is important to allow the glue used to construct the rockets time to dry; therefore, it is not safe to launch on the same day as constructing the model.
EXPLORE: The Flight Fidelity Challenge (80 minutes)
Pass out or project the handout, "Model Rocket Engine Functions" (page 6 of the Estes Reproduction Masters). Ask students to examine the diagram of the engine. Correlate the burn of the engine to the flight of the rocket: the burn of the solid propellant creates an upward thrust and, therefore, an upward acceleration.
Ask students to draw a free body diagram of the rocket while the engine is burning propellant (see diagram below). Ask students, "What happens to the rocket’s forces when the propellant is exhausted?" Ask them to draw a free body diagram.
Ask students, "What happens to the rocket’s motion when the propellant is exhausted?" Students may assume the rocket will immediately begin plummeting towards Earth; if so, remind them that the rocket still has upward momentum, so the rocket will continue to travel upward, slowing down due to the downward forces of gravity and drag.
Eventually, the rocket’s velocity will slow to zero, at which point it will turn around and begin accelerating towards Earth. Point out to students that during this time, the engine is burning through its "Delay and Smoke Tracking" layer. The last digit in the engine’s designation number defines the length of time of the delay: for instance, the A8-3 engine that is recommended for this project has a three-second delay. (For more information on how to interpret the letter-number designation on the engines, please refer to the "Model Rocket Engines" handout, page 5 of the Estes Reproduction Masters.) Have students draw a free body diagram of the rocket during the delay (before parachute deployment).
Once the "Delay and Smoke Tracking" layer has burned through, the "Ejection Charge" layer sends a burst of force up through the rocket body, pushing the nose cone out, which allows the parachute to deploy. Have students draw a free body diagram of the rocket immediately after parachute deployment. Be sure students understand that the increase in area created by the parachute will cause an increase in the drag force that opposes the rocket’s motion. According to Newton’s Second Law, the direction of an object’s net force must be the same as the direction of the object’s acceleration. Since the net force acting on the rocket is upward, the rocket must be experiencing an upward acceleration (slowing down as it travels downward). Because the rocket is slowing down, the drag force, which is dependent on the square of the velocity, will simultaneously and continuously decrease until both the drag force and the rocket’s weight are equal. Ask students what happens at that point. (Forces are equal, so speed remains constant; this is where the rocket reaches its terminal velocity.)
Explain to the students that, ideally, you choose an engine that will allow the rocket to attain maximum altitude before ejecting the parachute.
At this point in the activity, refrain from deriving the full, expanded equation for the rocket’s motion (see the Teacher Resources section for "Rocketry: Equation Derivation"). Students can be encouraged to attempt to independently derive the equation; however, this part of the lesson does not require the use of the equation, only a conceptual understanding of which rocket characteristics might influence which forces (for instance, increasing the mass of the rocket will increase the rocket’s weight, and increasing the cross-sectional area will increase the drag force).
It is very important to review the safety hazards of launching a rocket (see page 4 of the Estes Reproduction Masters). If time and weather conditions allow, take students outside with one of the rockets students constructed on the first day to demonstrate a launch. Ensure the launch pad is not on dry grass or dry underbrush, as it could easily start burning by coming into contact with the flames from the engine ignition. Load the rocket with an A8-3 engine. Before launching, ask students to observe the motion of the rocket to verify the points made in the class discussion. Launch and retrieve the rocket.
Return to the classroom. Pass out the "Flight Fidelity Challenge—Phase 1" handout (provided in the "Student Handouts" section). Review the challenge, criteria, and constraints with the students. Show them the materials that will be made available to them.
Pass out the "Rocket Stability" handout (provided in the "Student Handouts" section) and review how to measure centers of mass and pressure. You should tell students only that there must be at least two body diameters of separation between the rocket’s centers of pressure and mass. At this level, students should independently determine that the center of mass needs to be above (more towards the nose of the rocket) the center of pressure. Students will be able to logically deduce that it is advantageous to have the payload in the nose cone towards the top of the rocket as opposed to near the engine on the bottom.
Calculate center of pressure (included in the "Mathematics Lesson Extensions" section). In this extension, students will learn the process of using geometry (and, for more advanced students, calculus) to mathematically determine the centroid of a geometric object. They will then use that process to calculate the centroid of their model rocket, which corresponds to their rocket’s center of pressure. The position they calculate can be compared to the position they obtain by experimentally finding the center of mass of rocket’s shadow, as described in "Rocket Stability" handout.
Pass out the "Engineering Design Brainstorming Worksheet" (provided in the "Student Handouts" section), which steps students through the first four phases of the engineering design process. For the remainder of the class period and for homework, students should individually complete this worksheet.
EXPLORE: Engineering Design Process (160+ minutes)
Have the "Engineering Design Process" projected as students enter the room; alternatively, copy the page as a student handout. If necessary, review the engineering design process with the class. Ask students what this project’s problem is, and what the problem’s criteria and constraints are.
Tell students that once they build their rocket prototype, they will need to evaluate its performance. Once they evaluate the performance of their rocket, they will be able to refine and improve the design to improve performance. Ask students how their rocket’s performance might be evaluated (observing the direction of flight, measuring the maximum height). Remind students that one of the criteria was that they be able to fully describe the physical characteristics of the rocket so that they can predict its motion under other conditions. Ask students to brainstorm constructs that could be measured within their launching trials (initial mass, final mass of the engine cartridge, approximate angle of launch takeoff, and maximum height of flight). Ask students what tool they would use to take each measurement. The masses are measured on a balance, the degree to which the rocket flight is vertical is approximated using a protractor (or, students can just eye the flight path and estimate). Ask students how they would measure the maximum height. If the flight of the rocket is close to vertical, students can calculate the maximum height by measuring the distance from the observer to the launch controller and using a handheld altimeter to measure the angle of sight to the maximum height of the rocket; with these two values, simple trigonometry can be used to calculate the maximum height of the rocket. (For increased accuracy, remember to add the height of the observer!)
Determine Maximum Height (included in the "Mathematics Lesson Extensions" section). In this extension, students will develop the geometry of a maximum height model that accounts for errors associated with rockets that don’t fly in a perfectly vertical trajectory.
As a verification of this calculation, students can include as part of their payload a PICO altimeter, which is small enough to fit inside a 10mm-diameter rocket body, and is capable of digitally measuring the exact altitude attained by the rocket. Do a Google search of "pico altimeter" for purchasing options.
For the remainder of the EXPLORE phase, students should collaborate, design, build, and modify a model rocket according to the engineering design process. Stress to the students the value of the iterative process: even with the best simulations available, "real-life" engineers always build, test, evaluate, and modify their ideas; they never give themselves "only one chance." "Getting it wrong" is expected and should be thought of as a positive step on the path to getting it right!
Each group should be provided with three engines, which will allow them to perform three iterations of the engineering design process as they attempt to idealize their rocket’s performance. All students should be using the same engine power. A8-3 engines work well for the parameters defined in this lesson plan. Before each launch, students should be required to perform a swing test to verify the rocket’s stability: use a couple of pieces of masking tape to secure a length of string around the rocket’s center of mass. Stand in a clear area and slowly start the rocket swinging in a circle. If the rocket is stable, it will swing with its nose forward and the tail to the back.
The most common alterations to the rocket design include changing the diameter of the rocket body, the shape of the nose cone, and/or the shape, size or number of fins. Students will also need to design a way to secure the payload in a way that does not interfere with the rocket’s parachute. Encourage students to research their ideas before building their rockets. There is a lot of information that is easily accessible online (for instance, http://library.thinkquest.org/10568/ has some good general tips, or do a Google search of "model rocket plans" for more specific ideas). It is also a good idea to require students to use glue on their rockets either at the end of a class period or at home, so that class time can be spent testing, evaluating, and collaborating, instead of class time being wasted waiting for the glue to dry.
EXPLAIN: Analyzing the Motion (60 minutes)
By the end of the EXPLORE phase, students should have launched their rocket three times and should have recorded the maximum height attained by their final rocket design.
Show the NASA video "Launchpad: Newton’s Laws On-Board the International Space Station" or "Launchpad: Liftoff with Solid Rocket Boosters" to review how Newton’s Laws relate to a rocket.
Say to the students, "In order to successfully launch a shuttle, NASA engineers need to calculate the fuel necessary to reach a certain altitude, the altitude of the shuttle’s orbit. I’d like to model this situation for you, but we don’t have a shuttle available. What we do have are these rockets you’ve designed. These rockets are powered by small engines, so they clearly can’t make it into orbit. Therefore, instead of calculating the fuel necessary to get to a specific height, you’re going to use the amount of fuel you have (or in other words, your engine) to predict exactly how high your rocket will go. And, by the way, I’m going to give you a completely different engine, one with a completely different power than the type of engine you’ve used so far. So, listen carefully and ask questions if you get confused; otherwise, you may find yourselves unable to apply this process to the new engine."
Write the new challenge on the board. Ask students to brainstorm what values they would need to make this prediction and what physics concepts/equations they will need to apply. After giving time to brainstorm individually, write students’ ideas on the board. Values that are needed to make the prediction include the rocket’s mass, the mass of the propellant, the thrust provided by the propellant, and the drag coefficient of the rocket. The primary concepts/equations involved are Newton’s Second Law of Motion and kinematics equations.
Have students derive an appropriate Newton’s 2nd Law equation for a rocket as it flies upward to its maximum height, and algebraically solve for acceleration in that equation (see the Teacher Resources document "Rocketry: Equation Derivation").
Tell students to consider each variable in this equation individually:
Thrust Force: What causes this force? (the engine) How long does that force last? (until the engine burns up, partway up to the rocket’s maximum height) During the burn, is the thrust produced a constant thrust or a varying thrust? (a varying thrust) Do we know how that thrust changes over time? Luckily, the answer to this question is "yes" because the engine manufacturers provide this information in graphical form, called a thrust curve. Pass out or project the handout "Typical Time/Thrust Curve" (page 7 of the Estes Reproduction Masters) and review its characteristics.
Analyze a Thrust Curve (included in the "Mathematics Lesson Extensions" section). Prior to this extension, students estimate thrust values by interpolating a provided thrust curve. In this extension, students will use piecewise linear functions and sequences, as applied to the same thrust curve, to refine and improve their estimates.
CD, coefficient of drag. Review with students (or introduce the concept, if they haven’t yet been exposed to it) that this is a function of the shape of an object moving through a fluid. Coefficient of drag is normally measured experimentally using wind tunnels or calculated using computational methods. However, in this lesson, it will be estimated by adjusting the drag coefficient in the simulation until the simulation matches the maximum height achieved when the rocket was launched.
A, area, the cross-sectional area of the rocket, in the unit m2. What shape is your rocket? (a tube) What shape is the cross-sectional area of a tube, in the direction that it travels through the air? (a circle) What is the formula for the area of a circle? (πr2) It’s easier to measure the diameter of a tube. So what’s the relationship between the diameter and the radius? (d=2r)
ρ, air density, in the unit kg/m3. If you live in cities that are close to sea level, a standard value of 1.225 kg/m3 can be used. However, if you live at higher altitudes, it is recommended that you use average air density in your area. For example, average air density in Denver is 1.047 kg/m3. Denver’s altitude is approximately a mile above mean sea level. For a more precise value, students can perform an Internet search to find the exact air density at or close to the launch location.
v2, velocity squared, in the unit (m/s)2. Does the velocity of the rocket change as it is moving upward? (yes) Can you measure the velocity of the rocket in flight? (no, not easily, at least) Is there some other construct that is directly related to the rocket’s velocity that we can measure? (yes, maximum height, using kinematics equations) This will be tricky because this is a complex, real life problem; the drag force changes when velocity changes, which means that the acceleration is also changing. There is a trick that we can use to overcome that complexity though. We’ll talk about it after we’re done looking at every variable in this equation.
m, total mass of the rocket, payload, and engine in the unit kg. Is this variable changing and why? (yes, because the thrust is produced by burning the propellant. When the propellant is burned, its mass is expelled, causing an overall decrease in mass of the rocket.) Another changing variable!
g, the Earth’s standard gravity constant, 9.81 m/s2
a, acceleration, in the unit m/s2. Clearly, since FT, v, and m are all changing, acceleration is changing as well.
So, we have an equation that describes the motion of a rocket traveling straight up, but there appear to be many variables that change along the way. We need a method of accounting for those changes. To begin, let’s idealize the situation. Let’s say that the rocket’s thrust was a constant value, that somehow the mass was not changing, and that you were traveling in space where there was no air resistance. How could you figure out how far you would travel in 10 minutes? (Use FT=ma to find acceleration, then use x=x0+vt+12at2). You are able to do that because acceleration is constant — remember, this kinematics equation holds true only in constant acceleration situations. That is the exact idea we’re going to use to solve our much more complex problem.
Draw a "Z-t" graph on the board similar to the figure below.
Tell students that this graph represents some variable (Z) whose value is changing over a specified time interval. Suppose you need to use Z in a simulation and its value has to be constant within each time interval. This graph clearly indicates a large difference between the Z values at the beginning and end of the time period shown. However, a constant value of Z can be approximated by chopping the function into multiple segments occurring over shorter time intervals. The change in Z within any one of these segments is much less than the change in Z over the original time span, so each segment can more appropriately be assumed to have a constant value for Z (see figure below).
This approximation can be improved further by segmenting the time interval into smaller and smaller pieces. When the time intervals are very small (for instance, breaking Z into 1000 time intervals), the difference between the beginning and end Z values within each time interval will be very small. In other words, within a very small time interval, Z approaches being a constant value and that value of Z can be used in equations that require no change over time. This overall method is called the Euler method of integration, and it is a basic method used to carry out integration on a computer.
Let’s apply Euler’s method to a rocket’s upward motion. Consider the equation we derived for the upward motion of the rocket, assuming the rocket has no (or very little) horizontal motion, and, unlike "real" rockets such as the Space Shuttle, the air density and gravitational field remain constant:
The constants (such as area, the Earth standard gravity constant, and air density) are known values or are easily measured. A constant acceleration is needed so that it can be used in the constant acceleration kinematics equations. To this end, consider very small time intervals of the flight and assume that, within those very small time intervals, all forces and the rocket’s mass remain constant. Because all values to the left of the equal sign are estimated to be constant during a single time interval, then the construct to the right of the equal sign (acceleration) will also be constant. As long as the initial conditions for the rocket are known (y = 0 m; v = 0 m/s), that acceleration can be used in the kinematics equations to calculate a final velocity and final position of the "first time interval." Those final values then become the beginning conditions for the second time interval. The Euler method of integration moves through each time interval sequentially, ending with the final conditions of the final time interval.
Going through the Euler method manually could be very time consuming, especially if the time increments are small. However, the Euler method involves doing the exact same thing over and over again. We have a tool available to us — a computer — that will follow directions and perform repetitive tasks very well. So, if we tell the computer the specific steps and instruct it to keep repeating those steps until the end, we will have the work done for us. Essentially, we will be creating a simulation of the motion of the rocket. The easiest and most readily available tool for this task is a spreadsheet, but it could also be done using a simple computer programming language such as Octave or Matlab.
Ask students to consider the effect of the size of the time interval that is used (the smaller the time interval, the more accurate the results).
Each constant will be defined within the spreadsheet, and a dynamic equation for each variable will become a column in the spreadsheet. Pass out the "Model Rocket Engine: Type ½ A6 Specifications" and the "Rocketry Spreadsheet: Student Guide" (both included in the "Student Handouts" section). Lead students through the process of defining dynamic spreadsheet equations for each variable (see "Numerical Simulators Using Euler’s Method of Integration," included in the "Teacher Resources" section).
Some important notes about using the spreadsheet:
- It may take one, two, or more time increments for the engine thrust force to exceed the weight of the rocket and produce a positive acceleration in the simulation. In real life, rockets rest on the ground and will not lift off until positive acceleration is achieved. Positive acceleration is achieved when the thrust force is greater than the weight of the rocket. In the spreadsheet, acceleration should manually be set to zero until the acceleration is positive.
- When the rocket reaches maximum height, it turns around and begins traveling downward. The only difference between the upward and downward motion of the rocket (prior to parachute deployment) is the direction of the drag force. Therefore, after the rocket reaches maximum height, the sign of the drag force needs to change. This is accomplished using the SIGN function. Drag force is always opposite to the direction of velocity. The argument to the SIGN function is velocity. The SIGN function returns +1 if velocity is positive (moving up) and returns -1 if velocity is negative (moving down). The SIGN function saves the student the trouble of examining velocity in the spreadsheet and manually changing the sign on drag force calculations. Note that the spreadsheet is valid until parachute deployment. Coefficient of drag changes drastically after parachute deployment.
- The spreadsheet accounts for the mass that is ejected during the engine firing. A constant mass flow rate is used in the spreadsheet, where mass is calculated as a function of time. After engine burnout, mass remains constant. Students should manually set the mass values in the spreadsheet to a constant value after burnout. Forgetting this step will produce erroneous results.
- Time increment is the amount time between one integration time step to the next. Choosing a proper time increment is very important in a numerical simulation. A large integration time step leads to large integration errors, and a time increment that is too small results in a simulation that is too computation-intensive and takes up too many computer resources. In any numerical integration problem, the engineer needs to do a convergence analysis. In a convergence study, the engineer chooses an initial time increment, runs the simulation and records a few important output parameters, such as the altitude in this case. The engineer then reduces the time increment, runs the simulation again, and records how much the output parameters change, repeating the process until the answers don’t change very much. At this point, the simulation has converged. Reducing the time increment any further would just be waste of computer resources. For this simulation, an integration time increment of 0.01 seconds has been chosen. A convergence study has not been performed on this problem. Students are encouraged to perform a convergence study to find the most appropriate time increment.
ELABORATE: Calculating a Drag Constant (20 minutes)
Once students understand the process, provide them with a digital copy of the spreadsheet, appropriate for the A8 engine they used. Have students use the spreadsheet to determine the drag coefficient. To do this, students first input all relevant constants for their rocket (mass and diameter). They should then guess what their coefficient of drag is (0.5 is a good starting number) to see what maximum height the spreadsheet calculates. If the maximum height shown in the spreadsheet is too low, students should deduce that they need to decrease the drag coefficient. Conversely, if the maximum height shown in the spreadsheet is too high, students should deduce that they need to increase the drag coefficient. Students should continue to manipulate the value for drag coefficient until the maximum height shown by the spreadsheet matches their experimentally measured maximum height.
Have students attempt to sketch the y-t, v-t, and a-t graphs of the rocket’s motion, from launch to just prior to parachute deployment. Then, have students use the graph function of their completed spreadsheets to check their predictions. It is interesting to graph all three curves on a single axis system, as shown in the "Teacher Resources" section. Discuss the striking characteristics of the graphs, such as: (a) the drastic change in acceleration; (b) the similarity in shape between the thrust curve and the beginning of the acceleration curve; (c) the correlation between of the velocity curve’s inflection point and the acceleration changing from a positive value to a negative value; (d) the correlation of maximum height and zero velocity; and (e) the acceleration’s approach to -9.81 m/s2.
Project the example spreadsheet and discuss the structure of the spreadsheet in the detail appropriate for your particular students. Then, instead of providing the spreadsheet to the students, have them "self-create" the spreadsheet to simulate their rocket’s motion.
EVALUATE: Making Predictions (60 minutes)
When most students have calculated the drag coefficient of their rocket, present the final challenge to them (pass out "Flight Fidelity Challenge: Part 2," and "Model Rocket Engine: Type ½ A6 Specifications," both provided in the "Student Resource" section). Begin by stressing that their drag coefficient is a constant. It will not change as long as the size or shape of the rocket structure doesn’t change. (Changing mass does not change an object’s drag coefficient.) The challenge, then, is for students to use their calculated drag coefficient and their understanding of the spreadsheet to predict the maximum height attained by their rocket when it is launched with a ½ A6 engine. Manipulating the spreadsheet to make a prediction can be assigned as homework if students have access to a computer.
To successfully predict a maximum height, students will need to make several changes to their spreadsheet:
- Change the engine type designation to ½ A6.
- Correct the initial mass of the rocket, which will be 1.2 g less than the mass of their previous trials because of the difference in mass between the two engine types. Mass should be converted to the unit kg before being entered on the spreadsheet.
- Determine a new burn rate by dividing the mass of the propellant by the burn time (as read from the thrust curve). For the ½ A6 engine, the burn rate would be .00156 kg/.325 s = .0048 kg/s.
- Values for thrust should be interpolated from the provided ½ A6 engine thrust curve.
Ask students to print their spreadsheet when they have made all of their changes. They should circle the maximum height shown on the spreadsheet.
Students should launch their rocket with an ½ A6 engine, measuring the maximum height using the method they used earlier. If time and resources allow, repeat the launch and have students average their two heights. Students should calculate the percent error associated with their prediction and discuss the errors inherent in the overall process they used to make their prediction.
Calculate a More Accurate Propellant Mass (included in the "Mathematics Lesson Extensions" section). Up to this point, students have assumed a linear degradation of mass. In this extension, students will use numerical extensions to solve the differential equations that govern the actual change in propellant mass as it burns and is expelled. In addition, they account for the change in mass that occurs during the delay and tracking stage. Students then complete an error analysis to compare the various methods of assigning an instantaneous mass within the spreadsheet.
The acceleration, velocity, and position of a rocket, as calculated by the spreadsheet simulator, can be verified through use of a video camera and video-based motion analysis software such as VideoPoint (http://www.lsw.com/videopoint
). Position the video camera such that you will be able to see the rocket in the frame for approximately the first second of motion. Position a meter stick (for reference) immediately beside the launch pole; be sure, though, that the meter stick will not interfere with the launch of the rocket. Film the launch of the rocket and upload the clip to your computer. The motion analysis software will allow a frame-by-frame identification of the rocket’s position within the clip. Based on those positions, the program generates graphs and tables of the rocket’s changing position, velocity, and acceleration. These values can be compared to the values calculated by the spreadsheet simulator. Any differences can be discussed, with emphasis on the errors inherent to each method.
Computer Programming through Octave (included in the "Mathematics Lesson Extensions" section). A final extension to this lesson is presented. In that extension, students are led through how a computer programming language can be used to perform a numerical simulation similar to the one they created with the spreadsheet.