## Rocket problem

One of my weaknesses has been interpreting a physics word problem into math. Wanting to practice this, I thought I’d tackle a problem I’d seen before, the rocket problem, from scratch, knowing I could always look up a version of it in Halliday and Resnik.

Assuming a rocket with no exterior forces acting on it, propelling itself by chemical combustion with a constant force $F_r$.  Fully fueled, the rocket weighs $m_o$, and it is burning fuel at a constant rate of $m_r$.  This gives us an equation for the mass as a function of time, and the acceleration

$m(t) = m_o - m_rt$
$\ddot x = F_r / m$

Integrating both sides by $dt$ once yields

$\dot x = v_o - {F_r \over m_r } \ln | m_o - m_rt |$

Integrating a second time yields

$x = x_o + v_ot + \left( {F_r \over m_r^2} \right) \left[ m_rt - (m_o+m_rt) \ln (m_o -m_rt) \right]$

I learned several things from working on this.
1) that I understood that I could set up a system of first-order ODEs
2) I was surprised that the velocity was not proportional to t

I think it will be useful to watch some of the MIT open course math lectures, especially this one on solving first order linear ODEs.