Simple Rocket Equation in Gravity

Homework Statement

I'm having trouble deriving the equation for the velocity of a rocket in constant G given that it has constant exhaust velocity

Homework Equations

I know that a=dv/dt=U/M(dM/dt)-g

The Attempt at a Solution

so from here dv=U/M dM-gdt

v=U ∫1/M dM - g ∫dt v from 0 to v, M from Minitial to Mfinal, and t from 0 to t

which leads to v=-ln[Minitial/Mfinal]-gt

HOWEVER, everywhere I look, the answer is positive ln[Minitial/Mfinal] -gt and I can't seem to get why this is so

ehild
Homework Helper

Homework Statement

I'm having trouble deriving the equation for the velocity of a rocket in constant G given that it has constant exhaust velocity

Homework Equations

I know that a=dv/dt=U/M(dM/dt)-g

The Attempt at a Solution

so from here dv=U/M dM-gdt

v=U ∫1/M dM - g ∫dt v from 0 to v, M from Minitial to Mfinal, and t from 0 to t

which leads to v=-ln[Minitial/Mfinal]-gt

HOWEVER, everywhere I look, the answer is positive ln[Minitial/Mfinal] -gt and I can't seem to get why this is so

From conservation of momentum you get the equation MΔv-UΔm=-Mg, where Δm is the exhausted mass during Δt time. But it is negative of the change of mass of the rocket. So the differential equation for the rocket mass is M dv/dt+UdM/dt=-Mg ---->## v= -\int _{Minitial}^{Mfinal}(\frac{dM}{M})##

Ah, i see what I was doing wrong, my answer is correct if U is taken to to be negative