I'm having some issues with the rocket equation. I'm deriving the velocity as a function of time for a descending rocket (so the rocket is accelerating upwards in order to slow the descent). The result I should obtain is v=v0+gt+uln(mf/mi) where mf is the final mass, mi is the initial mass, u is the relative speed between rocket and the fuel it releases, v0 is the the speed at t=0, and v is the speed after a time t. Now I can obtain this if: I take positive downwards and consider the scene from an inertial frame on, say, the ground. At time t consider a rocket with mass M and speed v downwards. At time t+Δt consider the rocket with mass M-Δm and speed v+Δv downwards. A small piece of fuel has mass Δm and speed u+v downwards. Then let ΔP be the change in momentum, so ΔP=MΔv+uΔm. Divide through by Δt and let t→0 so dP/dt=Mdv/dt+udm/dt. dm/dt=-dM/dt so dP/dt=Mdv/dt-udM/dt. But dP/dt=Mg so this becomes Mg=Mdv/dt-udM/dt dv/dt=g+(u/M)dM/dt Solving by integrating between t=0 when the speed is v0 and the mass mi to a time t when the velocity is v and the mass mf, and we get the desired result. However, if I instead say that when the rocket emits the fuel at t+Δt, its velocity becomes v-Δv, then the whole thing messes up... I can't see why this shouldn't be a valid method of solving it, as in reality the rockets velocity really does decrease if I have positive downwards, rather than increase as in my above method.