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acceleration, velocity and position of a rocket

Hi everyone,

Quick question I may just not be thinking right here but I was trying to find the acceleration, velocity, and position of a rocket as a function of time. I started with acceleration:

$a =\frac{T}{m-\dot{m}t} - g$ where T is the Thrust, m is the initial mass, mdot is the mass flow rate, and g is gravity. This equation seems to work out with dimensional analysis and logically it seems to make sense, but maybe I'm wrong there. So from there I integrated wrt to time to get the velocity:

$v = -\frac{T}{\dot{m}} ln(m-\dot{m}t) -gt + v_0$ Here is where the problem comes in, while I'm pretty sure I did my integration right, the units don't work out properly and velocity doesn't start out at 0 either, unless you set the $v_0$ term to some value. Finally I tried to get position by integrating v:

$s = \frac{T}{\dot{m}^2} ((m-\dot{m}t)ln(m-\dot{m}t) - (m-\dot{m}t)) - 0.5gt^2 + v_0t + s_0$ Again the units don't work out properly.

I'm just considering where the rocket goes vertical for now, no horizontal components.

What am I not seeing here? This should be fairly straight forward. Thanks in advance for any help.
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