# Burnout Velocity of Upward Rocket under Constant Gravity

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## Theorem

Let $R$ be a rocket whose structural mass is $m_1$.

Let $R$ contain fuel of initial mass $m_2$.

Let $R$ be fired straight up from the surface of a planet whose gravitational field exerts an Acceleration Due to Gravity of $g$, assumed constant.

Let $R$ burn fuel at a constant rate $a$, producing a constant exhaust velocity $b$ relative to $R$.

Let all forces on $R$ except that due to the gravitational field be neglected.

Then the burnout velocity of $R$ is given by:

- $v_b = b \ln \left({1 + \dfrac {m_1} {m_2} }\right) - \dfrac {g m_2} a$

## Proof

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## Sources

- 1972: George F. Simmons:
*Differential Equations*... (previous) ... (next): Miscellaneous Problems for Chapter $2$: Problem $29$