As noted, Euler's method is little used in practice, as there are much better ways of solving initial value problems. By better, we mean, "able to achieve a result of the same precision using a larger step size". [Euler's method is also unstable for some problems where the step size can take you outside the physical domain of the function and the solution runs away to infinity.]

To improve on Euler's method, we will use the fourth-order Runge-Kutta method. This method requires four evaluations of the differential at each step, but often allows a much larger step size to achieve the same result. The method can be summarised as:

where we have written h in place of the step size Δx.

• Write a program to use the Runge-Kutta method of Eq. to solve the equation of motion of the projectile given in Eq.
• Repeat your analysis of the step size required to achieve a given accuracy in the result. Doyou see an improvement? By how much?