In my present course of numerical analysis we have been given an assignment to solve a problem involving applying Runge Kutta of the fourth order to a differential equation of the second order. The problem is as follows:

A spaceship is hanging immovable at the height H above earth's surface despite the engine being throttled all the way up. It is decided to angle the spaceship 90 degrees from the current trajectory (i.e. point the nose parallel to earth's surface) in order to reach outer space. Will the spacecraft make it without crashing?

Newtons motion equations expressed in polar coordinates states:


The angle beta is zero before the maneuver, but is changed to 90 degrees at T=0 and alpha = 0. R is the radius of the earth, g is the gravity acceleration at earths surface, G is the gravity acceleration at height H, G = g(R^2)/(R+H)^2. With the proper units, length in eart's radius, and time in hours, g = 20 and R =1. The small r is the varying height from earths core, and alpha is the varying angle between the vertical x axis that the spaceship was following.

Lets say H = 2 earths radius. Calculate and plot the trajectory. The starting values of the differential equation are given by the fact that the spacecraft was at standstill at t =0.

I get somewhat confused when we have to different motions, part the varying height, and part the varying angle. How to solve it?