[rom16384]

RK4 is indeed a popular and standard method. However for this particular equation or other equations with a conserved quantity a Symplectic integrator [1] should be used, as they are much better at keeping the numerical solution close to the real solution since they conserve the quantity of interest (energy, etc.).

[1] https://en.wikipedia.org/wiki/Symplectic_integrator

[Gravityloss]

Interesting.

Especially the Leapfrog integrator seems cool:

http://en.wikipedia.org/wiki/Leapfrog_integration

[csense]

RK4 is hard to write. Symplectic integrator looks even harder. (I'm not familiar with symplectic integration, but I've just read the Wikipedia article and it's not immediately obvious how to go from the math in the article to working code. I have written an RK4 integrator in the past, and the RK4 Wikipedia article looks like it's much easier to apply if you're writing the integrator for a physics engine.)

Euler, Euler midpoint, and Verlet are all very easy to code.

Also, you have to keep in mind the operation count versus the numerical stability. For the case under discussion -- constant acceleration -- Euler midpoint is perfectly suitable, as it gives the same answer as an exact analytical solution for the case where x(t) and y(t) are quadratic polynomials. RK4 or the like would only result in longer and slower code with no actual benefit in this particular application.