Solving the Tsiolokovsky Eqn for Velocity w/Time

[James Brady]

Hello! I'm doing a little derivation of the Tsiolokovsky equation where I'm trying to find velocity with respect to time, here's what I got so far:

F=ma, a = F/m

Here I replace the force term and the mass term, taking into account that the rocket is losing mass:

$a = \frac{v_e\cdot \dot{x}}{m_0 - \dot{x}\cdot t}$

where:
v = exhaust velocity
m dot = mass flow rate
m naught = initial mass

After this I take the integral of acceleration to get velocity, it's a pretty easy one since the the top two terms, mass flow rate and exhaust velocity, are both constants:

$\int \frac{v_e\cdot\dot{x}}{m_0 - \dot{x}\cdot t} = -v_e\cdot ln(m_0- \dot{x}\cdot t) + c$

Which is the velocity. My problem with this though is that when t = 0, velocity is $-v_e \cdot ln(m_0)$ which doesn't make any sense, right from the start there is an instantaneous velocity? Maybe the constant of integration is suppose to fix that? Any help would be appreciated.

 

[D.Biswas]

if i am correct then , yes , the constant of integration is there to fix that. i haven't checked your calculations as they look fine. c is supposed to be (v_e * ln(m_o ). if you evaluate the definite integral of the function between t = 0 and t = t_i (for any t_i) then you'll see that the velocity function
v = -v_e * ln(m_o - m't_i) + v_e*ln(m_o)

 

[James Brady]

Ah OK, I was just beginning to suspect that I as I was writing that post. Thanks for the help.