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Simple Rocket Equation in Gravity

  • Thread starter Dextrine
  • Start date
  • #1
102
7

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
 

Answers and Replies

  • #2
ehild
Homework Helper
15,427
1,823

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})##
 
  • #3
102
7
Ah, i see what I was doing wrong, my answer is correct if U is taken to to be negative
 

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