In 2017, I simulated an orbital launch of an Atlas V 500 Series rocket using Wolfram Mathematica.
Standing far enough away, this is exactly what one would see when viewing an Atlas V 500 Series launch to orbit.
See the highlights below or download the resources directly:
Setting up the launch
The Atlas V rockets are HUGE, and propel both people and satellites into space. The first stage (CCB), second stage (Centaur), and any additional boosters (SRBs) propel it far out of our atmosphere and into low Earth orbit (LEO). For this launch, we'll simulate the Atlas V 500 Series rocket with 4 additional boosters, and carrying a 10,000 kilogram satellite to low Earth orbit.
The velocity necessary for orbit changes with altitude. So to insert a payload on orbit, the Atlas V must be going a certain velocity at a certain altitude. Launching from Cape Canaveral, the payload must reach 500 kilometers and 7.2 kilometers per second.
The pressure, temperature, and density of air fluctuate with altitude. After finding data sets, these can be fitted to find he best approximation for any given altitude. This will help to calculate drag later.
As the rocket ascends, it must turn to face the horizon. This causes it to accelerate horizontally, which if going fast enough, allows the rocket to enter orbit. Here, the direction of the rocket is represented mathematically as coefficients of thrust and drag. Note that these produce a non-optimal flight path.
Forces are summed to analyze the motion of the Atlas V. Thrust is opposed by drag and gravity, and provides acceleration based on the rockets changing mass, velocity, position, and tilt.
Force (F) is dependent on thrust (T), drag (D), gravity (G), and the directional coefficients (C).
Euler's Method for position and velocity
At this point, the rockets flight cannot be solved by hand. Using Euler's Method for iterative solutions, the flight of the Atlas V can now be simulated. Here are the results:
Standing very far away, this is the actual path one would see traced in the sky by the rockets flight:
This simulation uses a non-optimized flight path, and thus the Atlas V didn't go exactly where I wanted it to. It did however, reach the qualifications for orbit. For that reason, I am declaring this simulation a successful launch.
In the future, I'd like to learn how to optimize the flight path of the vehicle such that the entirety of the useable energy goes toward putting the rocket on the orbit. Next time, I'll experiment with a round Earth model for the launch and work on determining launch conditions from variables for the orbital path.