due 11/20/2024 before midnight via Learning Suite 15 possible points
This problem uses the same rocket from the previous homework. In this case we are interested in the flight trajectory. The Saturn V does not fly at a straight angle during the first stage and so it would be difficult to provide an accurate estimate using the closed-form rocket equation. Instead, we need to use numerical integration. Though we will still ignore drag in this analysis.
As mentioned the heading angle changes significantly throughout the flight. I fit a curve to postflight trajectory data and computed the heading angle as a function of time during stage 1. Note that \(\theta = 0\) corresponds to vertical flight:
\(\theta = p_1 \arctan\left(p_2 t^{p_3}\right)\) where \(p_1 = 0.866, p_2 = 2.665 \times 10^{-5}, p_3 = 2.378\)
Other parameters you will need:
| thrust | 35.1 MN |
| propellant mass | \(2.1 \times 10^6\) kg |
| total rocket mass | \(2.97 \times 10^6\) kg |
| specific impulse | 283 s |
You could solve this using any ODE solver (e.g. scipy.integrate.solve_ivp in python), or you could write a basic forward Euler method. This means that you setup a time vector, and a starting point for \(V, m, z, x\), and then execute a for loop. At iteration (\(i\)) you update those four values using data from the previous iteration (\(i - 1\)). For example, using the last ODE (and setting \(\Delta t = t^{(i)} - t^{(i-1)}\)):
Report the following. Be sure to clearly show your work and assumptions.