According to Kleppner and Kolenkow in "An Introduction to Mechanics" it's better to burn the fuel of a rocket quickly since this will net a greater final velocity. I'm a bit confused with this.
Using the rocket equation we have:
$\vec{F}_{ext}=\frac{d\vec{P}}{dt} \leftrightarrow M\vec{g}=M\frac{d\vec{v}}{dt}-\vec{u}\frac{dM}{dt}$
After integrating with respect to time and setting $\vec{v}_0=0$ and $t_0=0$ we obtain:
$v_f=ulog(\frac{M_o}{M_f})-gt_f$
It appears that when reducing $t_f$ we indeed end up with a greater velocity. My confusion is that I'm integrating the expression as something like this (with the same initial conditions):
$v(t)=ulog(\frac{M_o}{M(t)})-gt$
Which means we are only selecting a time at which we measure the velocity? I think I'm not understanding the integration step properly perhaps. I would very much appreciate some help with this.