Optimal Firing Rate to Maximize Vehicle Velocity
Suppose that you are in a vehicle in space with a power source generating a constant power of $P_0$ that you can use to fire iron pellets. The iron pellets compose 99% of the ship's mass, $m$. The rate at which you can fire the pellets is controllable, but is limited by the vehicle's power plant.
I have tried solving this, but I seem to get an answer for the optimal rate $\dot{m}$ that is non physical. I think I've done something conceptually wrong, here is my attempt:
We know that for our mass we have, over the time interval $t_0 < t < t_f$: \begin{equation} m(t) = m_0 - \frac{dm}{dt}(t-t_0) \end{equation}
And that the power is: \begin{equation} P = \frac{1}{2\Delta t} m(t)v^2(t)=P_0 \end{equation}
This implies that: \begin{equation} v(t) = \sqrt\frac{2P_0 \Delta t}{m(t)} = \sqrt\frac{2P_0 \Delta t}{m_0 - \frac{dm}{dt}(t-t_0)} \end{equation}
Where $\Delta t = t_f - t_o$. 512
So now I assume that maximizing the distance traveled over the time period $\Delta t$ is equivalent to maximizing the final velocity $v(t_f)$. And we have: \begin{equation} x(t_f) - x_0 = \int_{t_0}^{t_f} v(t)dt = \sqrt{2P_0 \Delta t} \int_{t_0}^{t_f}\frac{dt}{\sqrt{m_0 - \frac{dm}{dt}(t - t_0)}} \end{equation}
The Euler-Lagrange equation then becomes: \begin{equation} \frac{\partial f}{\partial m} - \frac{d}{dt}\frac{\partial f}{\partial \dot{m}} = 0 \implies \end{equation} \begin{equation} -\frac{d}{dt}[\frac{1}{2(m_0 - \dot{m}(t-t_0))^{\frac{3}{2}}}] = 0 \end{equation} Or that: \begin{equation} \frac{1}{2(m_0 - \dot{m}(t-t_0))^{\frac{3}{2}}} =C \end{equation} Then we have that (with some algebra): \begin{equation} \dot{m}(t) = \frac{1}{t-t_0}(m_0 - (\frac{1}{2C})^\frac{2}{3}) \end{equation}
This solution doesn't make sense in that near $t = t_0$ it implies that all of the iron pellets are expelled. Over all I'm just having a difficult time understanding how to setup the fundamental physics equations to get to the correct functional. Please advise. 563