Past exam of the mathematics course of the University of Cambridge / 2019 / ia / Paper 4 / 3A / Solution

During a short interval $dt$, let the rocket's mass change by $dm<0$ and its speed by $dv$. The expelled mass $-dm$ has inertial-frame speed $v-u$ to first order. Conservation of linear momentum, including the impulse $Fdt$, givesDiscarding the second-order product $dmdv$ leavesand hence the Tsiolkovsky rocket equation

In deep space $F=0$, so $dv=-udm/m$. Burning from initial mass $m_0$ to $m_0/2$ increases the speed by