Simple Rocket Equation in Gravity

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SUMMARY

The forum discussion focuses on deriving the velocity equation for a rocket under constant gravitational acceleration (g) with a constant exhaust velocity (U). The user initially arrives at the equation v = -ln[Minitial/Mfinal] - gt but finds discrepancies with the expected positive form, ln[Minitial/Mfinal] - gt. The resolution involves recognizing that the exhaust velocity (U) must be treated as negative in the context of mass loss, leading to the correct formulation of the rocket's velocity equation.

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  • Understanding of basic calculus, specifically integration techniques.
  • Familiarity with the concepts of conservation of momentum in physics.
  • Knowledge of rocket propulsion principles, including exhaust velocity and mass flow rate.
  • Basic understanding of differential equations and their applications in physics.
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  • Study the derivation of the Tsiolkovsky rocket equation for a deeper understanding of rocket dynamics.
  • Learn about the implications of varying exhaust velocities in rocket propulsion systems.
  • Explore the effects of gravitational forces on rocket trajectories and velocity calculations.
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Students in aerospace engineering, physics enthusiasts, and anyone involved in rocket design or propulsion analysis will benefit from this discussion.

Dextrine
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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

which leads to v=-ln[Minitial/Mfinal]-gtHOWEVER, 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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Dextrine said:

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
 

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