+
Computing the 1-D 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 altitude-independent 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 Nazi-occupied Europe, and killed thousands of civilians. We made this choice because the V2 is a one-stage rocket which makes the simulation cleaner, and in addition, a lot of the technical data can be found in open sources.
Subject Area Mechanics First Year and Beyond the First Year Spreadsheet Learning Objectives: Students who complete these exercises will be able to • Model the 1-D velocity and position of a rocket without gravity using the Euler-Cromer method, and verify the Rocket Equation (Exercise 1). • Produce graphs the 1-D velocity and position of a rocket in under different assumptions (Exercises 2-4,6). • Compare the results from two models of gravity (Exercises 2-3). • 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 Euler-Cromer 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 1-D 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 1-D 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 1-D 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/standard-atmosphere-d_604.html) **Exercise 6 - Drag on a Rocket with Altitude-Dependent Air Density** Using results from Exercise 5, repeat Exercise 4 with a more realistic model for drag. Graph the resulting 1-D 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.