Derivation of Tsiolkovsky Rocket equation

[mardlamock]

Hello everyone, I would like some help trying to derive the rocke equation. I tried to do it my way, so sorry for the mistakes.

Ok, so I started off by saying that f=ma, then a=f/m, so the acceleration at any given time is = F/(m0-mr*t) , m0 being the initial mass, mr being the mass flow rate. Velocity is the integral of acceleration therefore if i integrate f/(m0-mr*t) I should get the velocity as a function of time. I did the integration by substitution part and i finally got that v(t)=(f/mr)*ln(m0-mr*t), which is wrong because the rocket equation says it is (f/mr)*ln(m0/mr*t). What did i do wrong?

[tavert]

Your integral has the wrong sign. The indefinite integral of f/(m0-mr*t) with respect to t is (-f/mr)*ln(m0-mr*t). You want the definite integral from t1 to t2, which equals (-f/mr)*ln(m0-mr*t2) + (f/mr)*ln(m0-mr*t1) = (f/mr)*ln((m0-mr*t1)/(m0-mr*t2)) = (f/mr)*ln(m1/m2).

[mardlamock]

Oh woaw, thanks man! I hadnt noticed the wrong sign nor thought about where i wanted the integral from. Thank you very much! Do you know what i should read to derive the equations for eliptical orbits and such? I learnt some calculus on my own (two years until i learn it at school), but im not smart enough to find out the laws of motion for an object in a eliptical or hyperbolic orbit. Thanks!

[tavert]

No problem. You probably read the Wikipedia article on the rocket equation and saw that the common derivation method is to note that force is the time derivative of momentum (= m v), but since both mass and velocity change with a rocket, you get both mdot and vdot terms. Your way works fine here though.

For derivation of Kepler orbits, are you able to follow either of the derivations here: http://en.wikipedia.org/wiki/Kepler_orbit#Mathematical_solution_of_the_differential_equation_.281.29_above ? You need to do a few not-immediately-obvious variable substitutions and apply conservation of angular momentum to get the central body inverse square force problem into an easy to solve form.

Edited January 16, 2014 by tavert