I know the formula for the velocity of a rocket after the on board fuel has been used up:

$$ \Delta v = v_{ex} \ln\left( \frac{m_i}{m_e} \right) - gt $$

where:

  • $v_{ex}$ is the exhaust velocity

  • $m_i$ is the initial mass of the rocket (with fuel)

  • $m_e$ is the dry mass of the rocket (without fuel)

  • $gt$ is gravitational acceleration multiplied by the amount of time the rocket has been accelerating

What is the velocity of the rocket before all of the fuel is used up? In other words, what is the velocity of the rocket at a specific time during the rocket's flight?

closed as off-topic by DanielSank, Kyle Kanos, ACuriousMind, Neuneck, Martin Aug 7 '15 at 15:28

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    On this site we do not give answers to problems like this. Instead, the person asking the question must describe how they have approached the problem, or ask a specific conceptual question. Take a look at the guidelines for asking questions like these in the help center. – DanielSank Aug 6 '15 at 16:44
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    Note that this is the expression for a rocket going straight up. Real rockets don't do that. – David Hammen Aug 6 '15 at 17:11
  • Why would this only apply when all of the fuel is used up? – HDE 226868 Aug 6 '15 at 18:01

Well, since unburned fuel is not distinguishable from dry mass, we could replace the above equation with $$ \Delta v = v_{ex}\ln(\frac{m_i}{m_i - m_b}) - gt $$

where $m_b$ is the mass of the burned fuel. Assuming constant burn rate $r_b$, it would just be

$$ \Delta v = v_{ex}\ln(\frac{m_i}{m_i - r_bt}) - gt $$

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