From the starting point of the geosynchronous orbit code, turn the script into a function that takes four inputs $(1)$ a starting velocity, $(2)$ a starting position, $(3)$ the size of the time step, and $(4)$ a total run time and returns the ratio of the distance from the satellite to the Earth at the end of the simulation over the distance from the satellite to the center of the Earth at the start of the simulation. Also make a second version of this same function that does not use the Verlet algorithm, but instead uses Euler's method by $(1)$ updating the position using the velocity and then $(2)$ updating the velocity using the force.

The deliverables are $(1)$ a plot with two lines where the $x-$axis shows the varying time steps and the $y-$axis shows the ratios from the two functions $($Verlet and Euler$)$ plotted on a log scale for the geosynchronous condition and $72h$ of run time and $(2)$ a short description in words of your conclusions based on the results. The goal is to get some idea of at what time step the two methods deliver comparable results $-$ say within $1\%$ or another threshold you specify. Your plot should be at least $100$ points and cover a non$-$trivial parameter space of time steps $($meaning enough range that something changes$).$ You should make a small MATLAB script that creates this plot, but no code will be graded.