# Tsiolokovsky Equation

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.