1. The problem statement, all variables and given/known data Find the speed v of the rocket when the mass of the rocket = m. The rocket starts from rest at with mass M. Fuel is ejected at speed u relative to the rocket. 2. Relevant equations m1v1 = m2v2 3. The attempt at a solution In the textbook, it starts off with a moving rocket with mass m and speed v. The fuel is given the mass (-dm) which is positive. So after a short time dt, the mass of the rocket changes to m+dm and speed v+dv. The mass of the ejected fuel is (-dm) and since it ejects with speed u relative to the rocket travelling at speed v, the ejected fuel travels at speed v-u, which can be positive or negative depending on which of v or u is larger. Writing down the equation of conservation of momentum: mv = (m+dm)(v+dv) + (-dm)(v-u) which then leads to m dv = -u dm After a few steps, we get: v2-v1 = u ln(m1/m2) That all looks fine and understandable. However, it mentions in the book that I am free to define dm to be positive, and then subtract it from the rocket's mass, and have dm get shot out the back. So I have decided to try it. Writing down the equation of conservation of momentum: mv = (m-dm)(v+dv) + dm(v-u) which then leads to m dv = u dm (note that at this point, the equation is already different from before) Moving the variables around and integrating v from v1 to v2, m from m1 to m2 as before, I obtained: v2-v1 = u ln(m2/m1) which is clearly wrong because m2<m1, so ln(m2/m1) < 0 but v2 > v1 So what went wrong there?