Momentum analysis for a rocket. A rocket has a mass of \(m_{\mathrm{R}}\) and carries fuel with mass \(m_{\mathrm{f}}\) at a given instant of time. Thus the total mass of the system at the current time is \(m_{\mathrm{R}}+m_{\mathrm{f}}\).

The fuel is consumed at a rate of \(\dot{m}\). The initial velocity of the rocket is zero and the initial amount of fuel is \(m_{\mathrm{f} 0}\).
The motion of the rocket is directly upwards, and the air resistance is represented as a force component \(F_{y}\) (in the negative vertical direction). The exit velocity from the nozzle is \(v_{\mathrm{r}}\) relative to the rocket.
Derive an equation for the acceleration of the rocket.