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Computing the 1D Motion of a V2 Rocket
Developed by David Marasco  Published February 3, 2019
This exercise set requires the students to generate a computational model for rocket motion in one dimension in the absence of outside forces, and to compare the resulting velocity to the rocket equation ($ \begin{equation*}
\Delta v = u \ln(\frac{m_i}{m})
\end{equation*}
$). From there, students calculate position and velocity data for both constant and Newtonian gravity, and implement models with both altitudeindependent drag
($ \begin{equation*}
D = \frac{1}{2} \rho v^2 C_d A
\end{equation*}
$),
and one that considers the air density as a function of altitude.
________________________________________
One may ask why we pick a rocket with such a sordid past, that was built by slave labor in Nazioccupied Europe, and killed thousands of civilians. We made this choice because the V2 is a onestage rocket which makes the simulation cleaner, and in addition, a lot of the technical data can be found in open sources.
Subject Area  Mechanics 

Levels  First Year and Beyond the First Year 
Available Implementations  Spreadsheet, Glowscript, and MATLAB 
Learning Objectives 
Learning Objectives: Students who complete these exercises will be able to
• Model the 1D velocity and position of a rocket without gravity using the EulerCromer method, and verify the Rocket Equation (Exercise 1).
• Produce graphs the 1D velocity and position of a rocket in under different assumptions (Exercises 24,6).
• Compare the results from two models of gravity (Exercises 23).
• Compare the results from two models of drag (Exercises 4&6).
• Fit a function to the Earth’s atmospheric density data (Exercise 5).

These exercises are not tied to a specific programming language. Example implementations are provided under the Code tab, but the Exercises can be implemented in whatever platform you wish to use (e.g., Excel, Python, MATLAB, etc.).
**EXERCISE 1 – Rocket in the Absence of Outside Forces**
A V2 rocket has an empty mass of 4000kg and was loaded with 8800kg of fuel. It burned fuel at 129.4kg/s and had an exhaust velocity of 2050m/s. Use the last two pieces of information to calculate an effective thrust, and use the EulerCromer method to solve for the velocity of the rocket in one dimension in the absence of gravity, noting that it will be changing its mass at each step. Compare the computed velocity at each time to the results predicted by the Rocket Equation.
**EXERCISE 2 – Rocket in a Constant Gravitational Field**
Modify the model from Exercise 1 by adding in a constant gravity. Note that while g will not change with time, the weight will. Write code so that when the rocket runs out of fuel, its mass stops changing, and the thrust goes to zero. Graph the resulting 1D position, velocity and acceleration as a function of time and comment.
**Exercise 3  Rocket in a Newtonian Gravitational Field**
Modify the results from Exercise 2 to account for the weakening of gravity as a function of distance from the center of the Earth. Graph the resulting 1D position and velocity as a function of time.
**Exercise 4 – Drag on a Rocket with Constant Air Density**
Using the results of either Exercise 2 or 3, add in the effects of drag. Assume that a V2 has a diameter of 1.65m, a drag coefficient of 0.125 and that the density of air is 1.22kg/m^3. Graph the resulting 1D position and velocity as a function of time. Calculate a maximum height and compare that result to previous numbers.
**Exercise 5 – Profile of Atmospheric Density**
Given the following data, fit a function to the density of air as a function of altitude. Note that a piecewise continuous function may be the best fit. Since this exercise is already in Excel, use of the Logest function is suggested.
Altitude (m)  $\rho$ (kg/m^3) 
:
 0000 1.225
1000 1.112 
2000 1.007 
3000 0.9093 
4000 0.8194 
5000 0.7364 
6000 0.6601
7000 0.59 
8000 0.5258 
9000 0.4671 
10000 0.4135 
15000 0.1948 
20000 0.08891 
25000 0.04008 
30000 0.01841 
40000 0.003996 
50000 0.001027 
60000 0.0003097 
70000 0.00008283 
80000 0.00001846
(Source:https://www.engineeringtoolbox.com/standardatmosphered_604.html)
**Exercise 6  Drag on a Rocket with AltitudeDependent Air Density**
Using results from Exercise 5, repeat Exercise 4 with a more realistic model for drag. Graph the resulting 1D position and velocity and drag as a function of time. Comment on the velocity graph. Calculate a maximum height and compare that result to previous numbers.
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Credits and Licensing
David Marasco, "Computing the 1D Motion of a V2 Rocket," Published in the PICUP Collection, February 2019.
The instructor materials are ©2019 David Marasco.
The exercises are released under a Creative Commons AttributionNonCommercialShareAlike 4.0 license