Rockets$-$Variable Mass

Please give the equation for the following:

$1.$ Please write the equation for the Force or $($dp$/$dt$)$ for a rocket that is expelling mass where M is the mass of the rocket and m is the mass of the gas getting expelled and v is the speed of the rocket and u is the speed of the gas getting expelled. Please put it in terms of dM $($no dm$\text{'}$s left$)($I believe dm$/$dt$=-$dM$/$dt$)($only terms in answer should be M$,$ dv$/$dt$,$ u$,$ dM$/$dt$)$

$2.$ With no external forces, Does Mdv$/$dt$=$udM$/$dt or does Mdv$/$dt$=-$udM$/$dt$?$

$3.$ Similar to above, please give the equation for the final velocity of the rocket with no external forces.

$4.$ please give the equation for the final velocity of the rocket in the presence of gravity.

$5.$ please give the equation for y$($t$)$ for the rocket