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# 8.11: Teacher Resources

Difficulty Level: At Grade Created by: CK-12

## Rocketry: Equation Derivation

Variables:

• Fg = force of gravity (weight)
• FD = drag force
• FT = thrust
• m = mass
• g = gravitational acceleration (9.81 m/s2)
• CD = drag coefficient
• A = cross sectional area
• ρ\begin{align*}\rho\end{align*} = air density
• v = velocity
• a = acceleration

Gravity: Fg=ma\begin{align*}F_g = ma\end{align*}

Drag Force: FD=12CDAρv2\begin{align*}F_D = \frac{1}{2} C_D A \rho v^2\end{align*}

Newton’s 2nd Law:

FFTFDFgFT12CDAρv2mgFT12CDAρv2mg=ma=ma=ma=a\begin{align*}\sum F &= ma\\ F_T - F_D - F_g &= ma\\ F_T - \frac{1}{2} C_DA \rho v^2 - mg &= ma\\ \frac{F_T - \frac{1}{2} C_DA \rho v^2}{m} - g &= a\end{align*}

## Numerical Simulators Using Euler’s Method of Integration

### Background Related to Rocketry

An elementary understanding of spreadsheets is required to be able to use and/or create the spreadsheet necessary for calculating the rocket’s drag coefficient. To use the spreadsheet, students must know how to change values in a cell. To self-create the spreadsheet (advanced physics extension), students must know how to format cells, how to define cells as functions, and how to copy those equations down the length of a column. Many spreadsheet tutorials can be found on the Internet. For example, see http://people.usd.edu/~bwjames/tut/excel/

The spreadsheet necessary for this project requires a number of constants to be defined, as well as six columns of varying constructs that are defined through functions. All values are recorded in MKS units.

The completed spreadsheet (truncated at the 36th row) is shown below:

### CONSTANTS

group name
defined by students
engine type
the lesson plan as written calls for Estes engine A8-3. If a different Estes engine is used, the thrust curve and mass of propellant can be accessed on the Estes web site (see citations below).
initial mass of rocket, including engine and payload
measure the mass of the rocket when both the payload and the engine are secured in it; convert this value to the unit kilograms.
body diameter
measure of the diameter of the tube which makes up the main cylindrical part of the rocket; convert this value to the unit meters.
cross-sectional area
Because the rocket’s cross sectional area is a circle, the equation for the area is A=πr2\begin{align*}A = \pi r^2\end{align*}. The prior cell can be accessed to create the function; using the example spreadsheet shown on the previous page, the function to calculate the area in cell D7 is as follows:
=((D6/2)^2)*3.14
air density
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.
burn rate
Burn rate defines how the mass of the engine decreases over time due to the combustion of the propellant; for the purposes of this project, the burn rate is assumed to be constant over the time of the burn.

For an A8 Estes engine, .00312 kg of propellant are burned in .7 seconds. Students can either determine the rate mathematically:

Burn Rate = Total Propellant Mass/Time of Burn

or they can graph the data (mass vs. time) and interpret the slope of the resulting line as the burn rate.

Propellant masses of other engine types can be found at:

drag coefficient
This is the value for which students will be "solving." The exact procedure is outlined in the lesson plan.

### VARIABLES

Set all initial values (except initial mass) as zero, as is shown on line 15 of the provided spreadsheet. This should be reflected as the first line in the spreadsheet. Initial mass of the rocket can be found in cell D5 in the spreadsheet shown.

time (unit s)
Choosing a proper time increment is very important for a good simulation. Refer to the procedure part of the lesson about choosing a proper time increment.

In the spreadsheet shown, the time increment is 0.01 seconds. Cell B15 was set to the function

=B14+0.01

This equation should be copied down the row to the end of the simulation.

thrust (unit N)
Estes website provides thrust curves for all of their engines in form of thrust-versus-time plots. However, in order to use thrust data in the simulation, the data has to be digitized into a tabular format and entered into the simulation (column C in the spreadsheet). Digitizing thrust curves is discussed in detail in the mathematics extensions.

Once the propellant has completely burned, the value of thrust should remain at zero.

Propellant masses of other engine types can be found at:

total mass (unit kg)
This column accounts for the loss of mass due to propellant combustion. For the spreadsheet shown, cell D15 was set to the function
=D14-$D$9*(B15-B14)

A constant mass flow rate model is being used to account for the propellant mass ejected from the engine. This constant mass flow rate is stored in spreadsheet cell D9. If an Excel cell contains an equation consisting of values of parameters stored in other cells, and this cell is copied to the cell below it, Excel automatically increments all cell locations referenced within the equation. In the equation above, we want all the cell locations to increment as cell D15 is copied down to the end of the simulation, except for D9. To prevent cell D9 from incrementing, you would have to place a dollar sign before D and before 9 ($D$9). A variable mass flow rate model will be discussed later, in which the mass flow rate is different within each time increment, with values stored in a separate column. To implement the variable mass flow rate model, simply replace $D$9 by the column containing the mass flow rate column.

This equation should be copied down the row until where thrust goes to zero. After engine burnout, the mass remains constant. Copy rocket mass at engine burnout down the row all the way down to the end of the simulation. Forgetting this step will result in errors.

acceleration (unit m/s2)
Acceleration is determined by using the previously derived equation
FT12CDAρv2mg=a\begin{align*}\frac{F_T-\frac{1}{2}C_D A \rho v^2}{m}-g=a\end{align*}

where v refers to the velocity on the previous line of data (since it is a known value), and F and m refer to the thrust and total mass on the current line of data.

For the spreadsheet shown, the above equation is written in cell E15 as follows:

=((C15–SIGN(F14)*(0.5*$D$10*$D$7*$D$8*F14^2))/D15)–9.81

This equation should be copied down the row. Note the SIGN function in the above equation. SIGN is a spreadsheet function that returns +1 if the argument is a positive number, and returns -1 if the argument is negative. The direction of the drag force changes when the rocket reaches maximum height and begins to move down. The argument in the SIGN function here is the velocity. On the way up, the direction of velocity is positive, therefore drag force must be negative in the equation above, and vice versa. The SIGN function accounts for the change in the direction of drag force when the rocket reaches maximum height.

After the acceleration equation is copied all the way down the spreadsheet to the end of the simulation, you will notice that the first few lines show negative acceleration. As shown in the thrust profiles of ESTES rockets, thrust force starts at zero and rapidly goes up to a near maximum value. Real life rockets act in a similar fashion at ignition. In real-life, a rocket that is ready to launch, does not get off the pad until enough thrust is built up to overcome the weight of the rocket. We need to mimic this behavior in the simulation. If this behavior is not accounted for in the simulation, the rocket would move in the downward direction for the first few time steps. To keep the rocket on the launch pad in the simulation at liftoff, manually zero out all the negative acceleration terms at the start of the simulation, as shown in the spreadsheet printout above. In this example, rows E15 to E18 were manually set to zero. As instructed previously, make sure the equation above is copied down the row to end of simulation before zeroing out acceleration cells at liftoff.

velocity(unit m/s)
Velocity is determined by using the basic kinematics equation
\begin{align*}v = at + v_0\end{align*}

The velocity equation above is represented in the spreadsheet by setting cell F15 to the function:

=E15*(B15-B14)+F14

This equation should be copied down the row to the end of simulation.

height (unit m)
Height is determined by the basic kinematics equation
\begin{align*}y = y_o + v_ot + \frac{1}{2} at^2\end{align*}

The height equation above is represented in the spreadsheet by setting cell G15 to the function:

=G14+(F15*(B15–B14))+(0.5*E15*((B15–B14)^2))

This equation should be copied down the row to the end of simulation.

kinematics graphs
Although the kinematic plots shown below are generated using a separate plotting package, all the data used to generate these plots can be extracted from the spreadsheet. The figure below shows position, velocity, and acceleration plotted on the same time scale. There are a few interesting features in these plots. The acceleration plot looks similar to the A8-3 engine thrust curve while it is firing. The velocity curve reaches maximum value at the end of the engine burn. This point also coincides with an inflection point on the position plot. Acceleration changes from positive to negative at engine burnout, because thrust is the only force in the positive (up) direction. After burnout, direction of acceleration changes from positive to negative, because gravity and drag both act in the negative (down) direction. The velocity decreases continually after burnout and crosses the x-axis at about 3.8 seconds. The height plot is at its maximum value at this point. Subsequent to reaching maximum height, the velocity becomes negative and position starts decreasing. Discuss with students how these plots relate to calculus concepts they learned in class. Velocity is derivative of position, and acceleration is derivative of velocity. In other words, the velocity plot is the slope of the height plot, and the acceleration plot is the slope of the velocity plot. Point out to the students how these calculus concepts are illustrated in a real life simulation of a rocket trajectory.

Use the figure below to get into a discussion with students about the kinematics of a rocket flight. The figure below shows accelerations due to all the forces acting on the rocket. The forces acting on the rocket are thrust, drag, and the gravity force. While the engine is ignited, acceleration due to thrust dominates all other forces. Note that the positive direction is up. While the engine is on, the direction of acceleration due to thrust is positive, and acceleration due to drag and gravity are both negative, opposing the thrust. But because acceleration due to thrust is so much greater than all other forces, the overall acceleration is positive during the engine burn. After engine burnout, gravity and drag take over. Gravity is always pulling the rocket down with an acceleration of 9.81 m/s2, and the aerodynamic drag force adds an additional deceleration component on the way up. The drag force is maximum at engine burnout, because the rocket is at maximum speed at this point. The drag force gradually decreases as the rocket reaches maximum height approximately 3.8 seconds into flight. Plots such as the one below are used by rocket engineers to analyze the performance of rockets, and lets them know what changes need to be made to improve performance.

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