# 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 $\Delta t = t_f - t_o$. 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.

• There is no dependence of the final velocity on the available power. Take a look at the rocket equation. That's the correct solution to the problem. – CuriousOne Sep 15 '15 at 20:10
• I'm not sure if I understand what you mean, the problem states the rate at which pellets can be fired is dependent upon the power source. I know of the thrust equation for a rocket but I'm still not really sure how to express the correct functional that needs to be maximized. Also, since the problem statement says the firing rate, and therefore the change in mass is dependent on the fixed power supply I thought it would need to be included in the solution. – Captainj2001 Sep 15 '15 at 20:16