## Introduction

Model rockets come in a variety of shapes and sizes (Figure 1). A basic model rocket consists of the rocket’s body, the **nose cone**, a **recovery system** (like a parachute), the **fins**, and the **engine** (also called the motor or propellant), along with some additional components, as shown in Figure 2. You can safely launch low-power model rockets from a field using a launch system like the one shown in Figure 3.

**Figure 1.**Three different types of model rockets. From top to bottom: the Estes Loadstar II, the Apogee Research Express, and the Apogee Apprentice.

**Figure 2.**Cross-sectional diagram of a basic model rocket. Note: This diagram does not show every single part of the rocket. Read the instructions that come with your rocket for assembly details.

**Figure 3.**A typical low-power model rocket launch pad.

Model rocket flight is broken up into three phases (Figure 4). During the **boost** phase, the rocket’s engine burns propellant, which generates **thrust** and pushes the rocket upward. During the **coast** phase, the engine has burned out, but the rocket continues to fly upward due to the high velocity it achieved during the boost phase. The highest point of the rocket’s flight is called the **apogee**. Finally, during the **recovery** phase, the rocket slowly falls back to the ground. The descent is typically slowed by something like a parachute or streamers, although some rockets may be designed to tumble or glide back to the ground.

**Figure 4.**The three phases of a model rocket’s flight.

Electronic sensors can help you measure different aspects of your rocket’s flight. An **accelerometer** is an electronic sensor that measures acceleration. An **altimeter** is an electronic sensor that measures the rocket’s altitude by measuring **barometric pressure**. If your rocket has a **payload bay** (Figure 5), you can mount a sensor inside your rocket and it will record information about the flight. Altimeters require **static ports**, small holes that are drilled in the payload bay so air pressure inside the bay will equal the outside air pressure (Figure 6).

**Figure 5.**A rocket with a payload bay, which can be used to carry cargo like an altimeter.

**Figure 6.**A close-up view of the payload bay. One of the static ports is visible just below the Science Buddies logo. There are two more ports on the other side of the rocket (not visible).

Different electronic sensors record different information. For example, some sensors record certain values that characterize a flight, like the apogee, along with maximum and/or average accelerations, the flight duration, or the amount of time between different points in the flight, like the **burn time** (how long the engine burns before running out of propellant) (Figure 7). Other sensors may continuously log data throughout the entire flight, producing graphs of altitude and acceleration vs. time when the data is downloaded to a computer or displayed on a smartphone (Figure 8). Either type of sensor should work for measuring the maximum altitude in this project.

**Figure 7.**Screenshot from the Jolly Logic Altimeter Two user manual showing all the information that is recorded during a flight. Photo courtesy of Jolly Logic.

**Figure 8.**The Jolly Logic Altimeter Three software showing recorded flight data. Photo courtesy of Jolly Logic.

We have briefly covered how you can *measure* rocket flight, but how can you *predict* a rocket’s motion? You are probably familiar with **Newton’s second law of motion**, which is commonly expressed in equation form as:

**Equation 1:** F=maF=ma

where

*F*is force, in newtons (N)*m*is mass, in kilograms (kg)*a*is acceleration, in meters per second squared (m/s^{2})

In high school physics problems, Newton’s second law is usually applied to objects with constant mass (for instance, the classic example of a force pushing a block on an inclined plane). However, that presents a problem when applied to rockets: *their mass is not constant!* Rockets expel high-velocity exhaust in one direction, which pushes the rocket in the other direction due to **conservation of momentum**. As the engine burns and expels fuel, its mass decreases; therefore, rocket motion is modeled mathematically with the **Tsiolkovsky rocket equation**, which takes the rocket’s changing mass into account. This equation is *extremely* important for real rockets that fly into space. Reaching orbit requires so much fuel that the majority—nearly or sometimes exceeding 90%—of a rocket’s mass at launch is fuel (this ratio is called the **propellant mass fraction**).

However, the propellant mass fraction is much lower for model rockets. For example, the model rocket at the bottom of Figure 1 (the Apogee Apprentice) has a mass of 60.8 g when loaded with a new A8-3 model rocket engine, and a mass of 54.9 g with a used engine (we will discuss model rocket engines and what the letters and numbers mean below), a difference of 5.9 g. Less than 10% of the rocket’s weight at launch is propellant. This means that you can roughly predict a rocket’s maximum height using a simplified equation that assumes the rocket’s mass is constant. This height is the sum of the height the rocket reaches at the end of its boost phase plus the additional height it gains during the coast phase.

**Equation 2:** hmax=12aboostt2boost+v2max2ghmax=12aboosttboost2+vmax22g

where

*h*is the rocket’s maximum height, in meters (m)._{max}*a*is the rocket’s acceleration during the boost phase, in meters per second squared (m/s_{boost}^{2}).*v*is the rocket’s maximum velocity, which is achieved at the end of the boost phase, in meters per second (m/s)._{max}*g*is acceleration due to gravity, which is 9.81 meters per second squared (m/s^{2}). Note: Do not plug*negative*9.81 into the equation, the minus sign is already accounted for.

To solve for the maximum height in this equation, first you need to find the boost phase acceleration, *a _{boost}*, and the maximum velocity,

*v*. The average acceleration during the boost phase can be calculated using Newton’s second law (see technical note):

_{max}**Equation 3:** aboost=Tavgmavg−gaboost=Tavgmavg−g

where

*a*is the rocket’s acceleration during the boost phase in meters per second squared (m/s_{boost}^{2})*T*is the engine’s average thrust in newtons (N)_{avg}*m*is the rocket’s average mass in kilograms (kg)_{avg}*g*is acceleration due to gravity, which is 9.81 meters per second squared (m/s^{2}). Note: do not plug*negative*9.81 into the equation, the minus sign is already accounted for.

The maximum velocity can be calculated using the equations of motion for constant acceleration (see Technical Note):

**Equation 4:** vmax=aboosttburnvmax=aboosttburn

where

*v*is the rocket’s maximum velocity, in meters per second (m/s)._{max}*a*is the rocket’s acceleration during the boost phase, in meters per second squared (m/s_{boost}^{2}).*t*is the engine’s burn time, in seconds (s)._{burn}

You can measure the mass of your rocket yourself, but where do you find out the engine thrust and burn time? You will need to look up that information from the manufacturer (note: information from the manufacturer may not always be 100% reliable, but it is a good place to start. See the Technical Note about rocket engines for more details). For example, look at this page for the popular Estes A8-3 engine and click on “Technical Specifications.” You will see that it lists a “Total Impulse” of 2.50 N-sec and a “Thrust Duration” of 0.50 sec (thrust duration is another way of saying “burn time”). An **impulse** is a force that is applied for a certain amount of time. If you know the total impulse and amount of time, you can calculate the average force using this equation:

**Equation 5:** J=FavgΔtJ=FavgΔt

where

*J*is impulse, in newton-seconds (N-s).*F*is average force, in newtons (N)._{avg}*Δt*is change in time, in seconds (s).

See the Technical Note for the full derivation of these equations. Even if you do not understand all the physics to derive the equations yourself, you should understand that it relies on a few assumptions:

- As discussed above, the physics assumes the rocket’s mass is constant. It ignores the change in mass due to the engine burning propellant.
- It ignores air resistance.
- It assumes the rocket moves straight up in a vertical line. It ignores any horizontal motion due to wind or launching the rocket at an angle.

That is a lot of information to absorb! Take your time to review this introduction, the references in the Bibliography, and the terms and questions before you proceed. Once you have done so, you should be prepared to do a science project with independent and dependent variables involving a model rocket. For example, you can test different rocket engines with different total impulses. You can predict your rocket’s maximum height using Equation 2, measure its maximum height using an altimeter, then compare your theoretical and experimental results. For other ideas, see the Variations section. Now, get ready to do some rocket science!

**Technical Note: Derivation of Equation 2**You might be wondering where Equations 2–4 come from. They are derived from analyzing the rocket’s vertical motion using the **constant acceleration equations**, which you can find in most physics textbooks. These equations allow you to calculate the position and velocity of an object that has a constant acceleration (note: deriving these equations requires the use of calculus—if you have not taken calculus yet, you can just accept the information for now). You may also hear them referred to as the “one-dimensional (1D) kinematic equations,” meaning they describe the motion of an object moving in a straight line. The four main equations are:

**Equation 6:** d=vit+12at2d=vit+12at2**Equation 7:** vf=vi+atvf=vi+at

**Equation 8:** v2f=v2i+2advf2=vi2+2ad

**Equation 9:** d=vi+vf2td=vi+vf2t

where

*d*is the object’s displacement (distance traveled), in meters (m).*t*is the elapsed time, in seconds (s).*v*is the initial velocity, in meters per second (m/s)._{i}*v*is the final velocity, in meters per second (m/s)._{f}*a*is the constant acceleration, in meters per second squared (m/s^{2}).

Using these equations, along with Newton’s second law (Equation 1), you can calculate a rocket’s maximum height by adding up the distances traveled during its two upward phases of flight (boost and coast). Figure 9 shows a diagram with information about the rocket’s flight, including known and unknown variables.

- At point 1, the rocket launches from the ground. It starts with zero velocity and a height of zero.
- During the boost phase, thrust from the engine (
*T*) pushes the rocket upward, and gravity pulls it down. The net upward force on the rocket is the engine thrust minus the rocket’s weight (*W*). The acceleration during this phase can be calculated with Newton’s second law. - At point 2, the engine burns out and the rocket transitions to the coast phase. The rocket’s height and velocity at this point are unknown, but can be solved for using the constant acceleration equations.
- During the coast phase, there is no longer any thrust acting on the rocket. The only force acting on the rocket is its own weight pulling it down (ignoring air resistance). The acceleration during this phase is equal to the acceleration due to gravity, -9.81 m/s
^{2}(assuming we have defined up as the positive direction). - At point 3 (the apogee), the rocket has reached its maximum height. The height is unknown, but importantly, the
*velocity*at this point will be zero. At the apogee, the rocket’s velocity briefly becomes zero before it becomes negative and the rocket starts to fall back to the ground.

**Figure 9.**Diagram of the boost and coast phases of the rocket’s flight, with known and unknown variables labeled.

Calculating the acceleration during the boost phase comes from writing Newton’s second law, where the net upward force on the rocket is the thrust minus the weight:

**Equation 10:** ΣF=T−W=T−mg=maΣF=T−W=T−mg=maRearranging Equation 10 to solve for the acceleration gives the acceleration during the boost phase (the same as Equation 3 above):

**Equation 11:** aboost=Tavgmavg−gaboost=Tavgmavg−gThe distance and velocity at point 2 can be calculated using Equations 6 and 7, respectively, filling in the known information from Figure 9 (you know the initial velocity at point 1 is zero, so the *v _{i}* term in Equations 6 and 7 cancels out):

**Equation 12:** d2=12aboostt2boostd2=12aboosttboost2

**Equation 13: ** v2=aboosttboostv2=aboosttboost

Note that the velocity at point 2 will be the rocket’s maximum velocity. When the engine burns out, it will stop accelerating and start decelerating. This means that

**Equation 14:** v2=vmaxv2=vmax

Now you can calculate the distance traveled during the coast phase using Equation 8, because you know the final velocity at point 3 is zero. Rearranging Equation 8 to solve for this distance gives the following (remember that the acceleration during the coast phase is equal to -g)

**Equation 15:** d=v222gd=v222g

However, this is the distance traveled *from point 2*, not from the ground. To get the total distance traveled (the distance at point 3), you need to add the distances from Equations 12 and 13. This gives you

**Equation 16:** d3=12aboostt2boost+V2max2gd3=12aboosttboost2+Vmax22gwhich is the same as Equation 2, (except there we used “h” for “height” instead of “d” for “displacement”).

**Technical Note: Model Rocket Naming Convention and Specifications**Model rocket engines are labeled with a letter-and-number scheme, such as A8-3, where:

- The letter indicates the possible
*range*for the total impulse of the engine in newton-seconds (N-s). See the “All About Rocket Engines” reference in the Bibliography for a table with the letters and ranges of impulse values. - The first number is the average thrust of the engine, in newtons (N).
- The second number is the
**delay time**, in seconds (s). This is the amount of time after the propellant finishes burning until the**ejection charge**ignites, which helps deploy the parachute.

However, different manufacturers might not be consistent about their use of this naming scheme, so you should not take it at face value. For example, the technical specifications page for the popular Estes A8-3 engine lists a total impulse of 2.5 N-s and a thrust duration of 0.5 s. According to Equation 3, rearranging to solve for *F _{avg}* gives an average force of 2.5/0.5 = 5 N. However, according to the name A8-3, the average force should be 8N! Which value should you believe? The National Association of Rocketry performs their own tests on rocket motors, and their list of NAR Certified Motors is a good source to verify motor specifications. Their tests for the Estes A8 and Quest A8 motors show average thrusts of 4.03 N and 4.84 N, respectively—neither of which matches the value you would expect from the name! Depending on which values you use, you will get different results from Equation 2.

### Terms and Concepts

- Nose cone
- Recovery system
- Fins
- Engine
- Boost
- Thrust
- Coast
- Apogee
- Recovery
- Accelerometer
- Altimeter
- Payload bay
- Static ports
- Burn time
- Newton’s second law of motion
- Conservation of momentum
- Tsiolkovsky rocket equation
- Propellent mass fraction
- Impulse
- Constant acceleration equations
- Delay time
- Ejection charge

### Questions

- Which forces act on a model rocket during flight?
- What factors influence the maximum height that a rocket will reach?
- How can you use math and physics to
*predict*a rocket’s maximum height? - How can you
*measure*a rocket’s maximum height? - What do the letter-and-number labels on a model rocket engine mean?

## Materials and Equipment

You will need to decide exactly what to purchase for this project based on your budget, how many trials you want to do, etc. There are multiple options, so we did not write the instructions for a specific model of rocket or altimeter. Model rocket supplies are available at many hobby stores. Supplies are also available from Estes Rockets and Apogee Components .

- Model rocket with payload bay
- Altimeter
- You will need mounting hardware or tape to attach the altimeter inside your rocket’s payload bay (check manufacturer recommendations).
- Make sure you buy an altimeter that is intended for use with model rockets and will fit in your rocket’s payload bay. Such altimeters are typically only a few centimeters long and may only weigh a few grams.
- Most model rocket altimeters will record maximum altitude. Some will record additional data. Make sure you understand what data your altimeter will record and how you will access it. For example, does the altimeter display the data on a built-in screen, connect to your phone, or do you need to download the data to a computer?

- Launch controller and launch pad
- Engines (at least 3 of each type you plan to test). A-sized engines are best if this is your first model rocket project. Your rocket will not go as high, but it will be easier to recover. For example, you could compare Estes 1/2A6-2 engines to Estes A8-3 engines. B-sized engines and above are best for students who already have some experience with model rockets and access to a larger launch site.
- Recovery wadding
- Tools and supplies for assembling your rocket. You will need to read the manufacturer’s instructions for your rocket, but these typically include a hobby knife, ruler, pencil, wood glue, and plastic cement or superglue.
- Materials to finish, paint, and/or decorate your rocket, such as sandpaper, spray paint, and stickers
- Kitchen scale to weigh your rocket, with at least 0.1 g sensitivity
- Large, open field free of obstructions like trees and power lines in which to launch your rocket. Refer to the Model Rocket Safety Code for required launch site dimensions.
- Helpers can be useful when launching the rocket, such as a spotter to carefully watch where the rocket lands or to take photos/video from a distance while you operate the launch controller
- Lab notebook

## Experimental Procedure

- Plan out your experiment. Review the Background section if you are not sure how to answer the following questions.
- What independent variable will you change on your rocket?
- What dependent variable will you measure?
- How will you calculate a theoretical value for the dependent variable?
- How will you measure the dependent variable, experimentally? How will you access the data?
- Does your altimeter store data for multiple flights? Will you need to download the data to a computer in between each flight?
- When and where will you launch your rocket? How many trials will you do? What is your backup plan if you are unable to recover your rocket? For example, is there a risk of your rocket getting stuck in a tree? Will you have the time and money to buy another rocket in time to do your science project? How would this affect your results?
- Will you be able to perform all your launches on the same day, in a relatively short period of time? If not, how could variable weather affect your experiment?

- Prepare a data table for your results. Table 1 shows an example data table. Remember that your table may be different depending on your independent and dependent variables. For model rocket projects, it is also a good idea to record observations about the weather and launch conditions. You could record these in a separate data table (Table 2) or include them in your main table.

Engine | Maximum Height (m) | ||||||||
---|---|---|---|---|---|---|---|---|---|

Name | Total Impulse (N-s) |
Burn time (s) |
Average thrust (N) |
Trial 1 | Trial 2 | Trial 3 | Average | Theoretical | Difference (Theoretical – Average) |

**Table 1.** An example data table to compare the rocket’s theoretical and experimental maximum height for three different engine types. Your data table could include additional columns; for example, the standard deviation of the experimental results or the percent error between the theoretical and experimental results. If you have enough engines, you can also do more than three trials per engine.

Engine | Trial | Wind Speed (km/h) |
Wind Direction | Temperature (°C) |
Other notes |
---|---|---|---|---|---|

1 | |||||

2 | |||||

3 |

**Table 2.** An additional data table for recording weather conditions at launch. You can easily look up weather information for your location on a smartphone weather app.

- Perform your theoretical calculations for your rocket’s maximum height (or, if you are too excited to go launch your rocket, you can do this step later!). You may want to create a separate data table to record values as you do your calculations (Table 2), or you could do the calculations in a spreadsheet program.
- Look up the total impulse and burn time for your motor.
- Use Equation 5 from the Background to calculate your motor’s average thrust.
- Calculate your rocket’s average mass (the average between new and used engines). You will either need to launch a rocket first so you can measure a used engine, or look up the mass of a used engine online.
- Use Equation 3 from the Background to calculate your rocket’s average acceleration during the boost phase.
- Use Equation 4 from the Background to calculate your rocket’s maximum velocity at the end of the boost phase.
- Use Equation 2 from the Background to calculate your rocket’s maximum height.

Motor Name | Total Impulse (N-s) |
Burn Time (s) |
Average Thrust (N) |
Rocket Average Mass (kg) |
a_{boost} (m/s^{2}) |
v_{max} (m/s) |
h_{max} (m) |
---|---|---|---|---|---|---|---|

**Table 3.** An example data table to help with doing your theoretical calculations.

- Follow the manufacturer’s instructions to assemble your rocket.

**Figure 10.**An assembled Apogee Research Express rocket.

- Follow the manufacturer’s instructions to set up your altimeter. These may include (but not be limited to):
- How to charge the altimeter’s battery.
- How to download and use software to access the altimeter’s recorded data.
- How to drill static ports in your rocket’s payload bay for the altimeter.
- How to install the altimeter in your rocket’s payload bay. This step is important so the altimeter does not fall out when the ejection charge fires to deploy the parachute (as this author learned the hard way)!

- Make sure you have everything you need to launch your rocket! This might not be a big deal if you can launch your rocket from your backyard, but if you need an adult to drive you somewhere (like a nearby park or soccer field), it will be frustrating if you forget a small part like recovery wadding or motor igniters. Bring everything you will need to conduct multiple launches.
- Follow the instructions for your rocket, launch pad/controller, and altimeter, along with the National Association of Rocketry’s Model Rocket Safety Code to launch your rocket.

**Figure 12.**A model rocket on the launch pad.

- Recover your rocket. If needed, download or record the data from your altimeter.
- Repeat steps 7–8 as many times as needed in order to collect all your data.
- Analyze your experimental results. Graphing the data may be helpful.
- Are the results what you expect?
- Is there a trend or relationship between your independent and dependent variables?
- How repeatable were your results? Are there any outliers or measurements that seem erroneous? For example, did your altimeter report that your rocket only went 10 ft. in the air when you could clearly see that it went over 100 ft? If so, it is possible that your altimeter was accidentally triggered before launch—for example, by bumping the rocket on the launch pad? You may need to repeat the trial if this occurs.
- Are there any other potential sources of error or variation in your experimental results? For example, how variable were the weather conditions? Did you perform some launches in calm weather and others in a steady breeze or a gusty wind?

- Compare your experimental results and theoretical predictions. Showing the results on the same graph may be helpful.
- Do the theoretical predictions show the same trend as your experimental results?
- How close are they to the experimental results?
- Make sure you review the assumptions used for Equation 2 in the Background section. How might these assumptions affect the accuracy of your theoretical results?

## Variations

- Other ideas for your independent variable could include:
- Use engines with the same total impulse, but different average thrust, for instance, a B4-2 versus a B6-2.
- Compare engines with the same rating from different manufacturers.
- Use the same type of engine for all trials, but put additional mass in the rocket’s payload bay.

- There are other aspects of your rocket’s flight that you can calculate using the equations of motion presented in the Background section. You can also measure some of these parameters with an altimeter (you will need to check the manufacturer’s instructions to see exactly what your altimeter can measure). How do theoretical and experimental results compare for these variables? For example:
- What is the rocket’s maximum velocity?
- At what altitude does the rocket achieve maximum velocity?
- How long does it take the rocket to reach its apogee?
- Based on the engine’s delay time, at what altitude will the rocket eject its parachute?
- How fast will the rocket be going when it ejects its parachute?

- As discussed in the Technical Note in the Background, the manufacturer ratings for rocket engine performance may not always be consistent with the engine naming convention, and testing by the National Association of Rocketry might give different values as well. That gives you three different, potentially conflicting, sources of information about the engine! How much do your theoretical results vary when you use information from the different sources in Equation 2? Which source seems to be the most accurate?
- You can analyze your rocket’s motion using a video of the launch. Set up a camera on a tripod aimed at the launch pad and film the launch. Using an object of known size in the video frame, in the same plane as the launch pad, will give you a scale for the video (for example, a meterstick or a person). You can then use video analysis software to track the rocket’s location in the video, and convert this location in pixels to real-world distance in meters. If you know the camera’s frame rate (the number of still images taken per second), you can also calculate velocity and acceleration. Many cameras film at 30 or 60 frames per second (fps), but some smartphones offer slow-motion video modes that capture at 120 fps, 240 fps, or even higher! Note that this approach might work to film the beginning of your rocket’s flight, but it can be difficult to place a camera far enough away to film the entire flight while fixed on a tripod.
- You can use computer simulation software to predict your rocket’s performance. This software makes it easier to account for calculations that are much more difficult to do by hand, like the effects of wind gusts on rocket stability. RockSim is a simulation program available from Apogee Components.
- Testing different fin shapes is a popular topic for a model rocket science project, but be careful! Fin shape can have a very subtle and hard-to-measure effect on rocket performance. For example, the average thrust and burn time of individual motors can vary by about +/-5% (see NAR motor testing data linked in the Background section). Unless you do many, many trials, it can be hard to tell if something like a 3% difference in altitude between two flights is due to different fins or simply variations in motor performance (or other factors like wind).