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## Homework Statement

Consider the case of a rocket taking off vertically from rest in a gravitational field g. The differential equation is given by

[tex]m\dot{v} = -\dot{m}v_{ex}-mg[/tex]

Assume the rocket ejects mass at a constant rate, [itex]\dot{m}=-k[/itex] (where k is a positive constant), so that [itex]m=m_{0}-kt[/itex]. Solve equation for v as a function of t, using separation of variables.

## Homework Equations

[tex]m\dot{v} = -\dot{m}v_{ex}-mg[/tex]

## The Attempt at a Solution

[tex]m\dot{v} = -\dot{m}v_{ex}-mg[/tex]

[tex](m_{0}-kt)\frac{dv}{dt}=-kv_{ex}-(m_{0}-kt)g[/tex]

[tex]\frac{dv}{dt}=\frac{-kv_{ex}}{m_{0}-kt}-g[/tex]

[tex]dv=(\frac{-kv_{ex}}{m_{0}-kt}-g)dt[/tex]

[tex]v(t)=v_{ex}ln(m_{0}-kt)-gt-c[/tex]

Setting t = 0, it appears that [itex]c=v_{ex}ln(m_{0})[/itex].

However, the next step of the problem is to plug in values, which revealed that this clearly isn't the right solution. I attempted to solve it again and ended up with the same solution, so I'm obviously having some difficulty recognizing my mistake.