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A rocket is launched with an initial mass $m_0$. As the engines use up the fuel and eject the exhaust gasses, the rocket gradually loses mass as $m(t) = m_0(1 - kt)$, where $k$ is a constant.

Assuming the rocket rises with a constant velocity, $v$, from the surface of the Earth (radius $R$, mass $M$) to a distance $r$ from the Earth's center, how much work must the rocket engines do against gravity?


$W = Gm_0 M\left (\cfrac{1}{R} - \cfrac{1}{r}\right )$


$W = Gm_0 M\left (\cfrac{1}{R} - \cfrac{1}{r} + \cfrac{k}{v}\ln{\left (\cfrac{R}{r}\right )}\right )$


$W = Gm_0M\left (\cfrac{1}{R} - \cfrac{1}{r}\right )\left (1 - k\cfrac{r}{v}\right )$


$W = m_0\left (1 - k\cfrac{r}{v}\right )gr$

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