For this homework I implemented both Euler's method and the midpoint method. I found that most methods were remarkably stable (I was not expecting such stability). I hypothesize that the reason for this is that the systems we were implementing (the bullet and mass-spring) were both very simple with only a few forces acting on the particles. I suspect, functions would be much more likely to explode at larger timesteps in more complex systems. I was able to use h = .1 for this excersize without having any numerical difficulty (though the system did take more steps to converge).
As for accuracy, the fourth order Runge-Kutta is clearly the most accurate of the three because it's order is O(h^4) and midpoint and explicit Euler are both O(h^3). I suspect that the midpoint is slightly more accurate than the Explicit Euler because it attempts to predict that extra half step every iteration but there was no noticable difference in my results between the two methods.
The Euler method is more computationally efficient and easier to code since it requires less steps. Since both ran in real-time in my simulation, there was no significant difference in program latency between the two.
The artillary code is zipped here and the spring-mass code is zipped here.