# 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

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

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## 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

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 ---->$v= -\int _{Minitial}^{Mfinal}(\frac{dM}{M})$