Differential equation

xs = G * mE * -xs / |xs| + G * mm * Xm-xs/ |xm-xs|^3
non-linear

x(t)=sin(t)
x(t)=sin(x(t))
x(t)=cos(y(t))
t=(y(t))^2

differential equation that govern the motion
simple solution method

import math
from xxx import *

moon_distance = 384e6

def orbit():
	num_steps = 50
	x = numpy.zeros([num_steps + 1, 2])

	for i in range(num_steps + 1):
		angle = 2.*path.pi * i / num_steps
		x[i, 0] = moon_distance * math.cos(angle)
		x[i, 1] = moon_distance * math.sin(angle)

	return x

x = orbit()

@show_plot
def plot_me():
	matplotlib.pyplot.axis('equal')
	matplotlib.pyplot.plot(x[:, 0], x[:, 1])
	axes = matplotlib.pyplot.gca()
	axes.set_xlabel('Longitudinal position in m')
	axes.set_ylabel('Lateral position in m')
plot_me()

import math
from xxx import *

h = 0.1
g = 9.81
acceleration = numpy.array
initial_speed = 20. # m / s

@show_plot
def trajectory():
	angles = numpy.linspace(20., 70., 6)

	num_steps = 30
	x = numpy.zeros([num_steps + 1, 2])
	v = numpy.zeros([num_steps + 1, 2])

	for angle in angles:

		matplotlib.pyplot.plot(x[:, 0], x[:, 1])
	matplotlib.pyplot.axis('equal')
	axes = matplotlib.pyplot.gca()
	axes.set_xlabel('Horizontal position in m')
	axes.set_ylabel('Vertical position in m')
	return x, v

trajectory()