A rocket, fred from rest at time $t=0,$ has an initial mass of $m0($including its fuel$).$ Assuming that the fuel is consumed at a constant rate $k,$ the mass $m$ of the rocket, while fuel is being burned, will be given by $m0-kt.$ It can be shown that if air resistance is neglected and the fuel gases are expelled at a constant speed $c$ relative to the rocket, then the velocity $v$ of the rocket will satisfy the equation

$m\frac{dv}{dt}=ck-mg$

where $g$ is the acceleration due to gravity.

$($a$)$ Find $v(t)$ keeping in mind that the mass $m$ is a function of $t.$

$v(t)=\frac{c}{k}ln(\frac{x_0}{x_0-kt})-gt\frac{m}{sec}$

$($b$)$ Suppose that the fuel accounts for $65\%$ of the initial mass of the rocket and that all of the fuel is consumed at $130$ s $.$ Find the velocity of the rocket in meters per second at the instant the fuel is exhausted. $[$Note: Take $g=9\mathrm{.}8\frac{m}{s^2}$ and $c=2500\frac{m}{s}.]$

$v(130)=\frac{m}{sec}[$Round to nearest whole number$]$