Answer
$v=580ft/s$
Work Step by Step
We can determine the required speed as follows:
$\int_0^v dv=\int_0^5(\frac{T+qv_{D/e}}{m_{\circ}-qt})$
$\implies v=(\frac{T+qv_{D/e}}{q})\ln(\frac{m_{\circ}}{m_{\circ}-qt})+v_{\circ}$
$\implies v=(\frac{T+2\frac{dW}{dt}\frac{1}{g}v_{D/e}}{2\frac{dW}{dt}})\ln(\frac{W_{\circ}}{W_{\circ}-2qt})+v_{\circ}$
We plug in the known values to obtain:
$v=(\frac{15000+2(150)(\frac{1}{32.2})(3000)}{2(150)})ln(\frac{40000}{40000-2(150)})+440$
This simplifies to:
$v=580ft/s$
