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.