# Calculate rocket trajectory

Lets say we shoot a rocket from Earth and assume it starts with a base velocity. For something close to Earth one would use this to calculate its height depending on time etc. $$\frac{1}{2}at^2 + v_0t + h_0 = h(t)$$ If we try to calculate its height, we have to consider that the acceleration drops depending on its height. Thus the output of the function depends on itself (recursive). Normally I would solve this by programming, but I am interested if there is a "prettier" method. I know about using energies to calculate the height, but this method cannot be used if i have something like air friction or a continuous thrust.

Does somebody know about ways to calculate such trajectories or is it common to do it using programs with small timesteps.

• Do not forget that the mass of the rocket is variable, as per en.wikipedia.org/wiki/Variable-mass_system – ZeroTheHero Apr 14 '17 at 18:24
• @ZeroTheHero Absolutely, thats a factor which would change the acceleration by far. Thanks for pointing that out! – RIJIK Apr 16 '17 at 12:33

For a start substitute in $acceleration=\frac{GM}{height(time)}$ and rearrange for height(time). Can you solve this?