# A rocket is started from the surface of the earth. Calculating the height as function of its mass and fuel consumption. 
# Approximation of acceleration with Taylor series
# Equation of motion
# r'' = (alpha / m0)*v0 + (alpha/m0)²*v0*t - gamma * ME / r²
#   alpha: fuel consumption, e.g. 2000t in 2,5 min = 13,333*10^3 kg/s
#   m0: initial mass of rocket, e.g. Saturn V: 2900 t = 2,9*10^6 kg
#   v0: the velocity of the exhaust of the rocket: 3,180*10^3 m/s
#   gamma: gravitational constant: 6,6743E-11 m³/(kg*s²)
#   ME: mass of earth: 5,97E+24 kg

# include idivide
include CompoundFunctions.LACE 

coefficient.1(+1) -> alpha/m0*v0
coefficient.2(+1) -> (alpha/m0)^2*v0
coefficient.3     -> scale 
coefficient.4(-1) -> -RE
coefficient.5(+1) -> 10*gamma*ME

# generate t-ramp
iintegrate (-1) -> t

# calculating altitude
iintegrate (alpha/m0*v0, (alpha/m0)^2*v0*t, -gamma*ME/r^2) -> -v # input is a ### 3. Term fehlen noch ###
cmultiply (scale, -v) -> -v.scaled
iintegrate (-v) -> r
  IC: -RE

# calculating acceleration
multiply (r,r) -> r^2
idivide (10*gamma*ME, r^2) -> -10*gamma*ME/r^2

cmultiply ((alpha/m0)^2*v0, t) -> (alpha/m0)^2*v0*t

# inverting velocity for display
invert (-v.scaled) -> v.scaled

# subtracting the radius of earth in order to get altitude above ground for display
isum (-RE, r) -> -z
invert (-z) -> z

output (t) -> out.x
output (v) -> out.y
output (z) -> out.z