(Solved): A differential equation describing the velocity v of a falling mass subject to air resistance propo ...

A differential equation describing the velocity $v$ of a falling mass subject to air resistance proportional to the square of the instantaneous velocity is $m\frac{dv}{dt}=mg?k{v}^2,k>0$ vhere $k>0$ is a constant of proportionality. The positive direction is downward. (a) ( 8 points) Solve the equation subject to the initial condition $v(0)=v_0$. b) (2 points) Use the solution in part (a) to determine the limiting, or terminal, velocity of the mass. c) (10 points) If the distance $s$, measured from the point where the mass was released above ground, is related to velocity $v$ by $ds/dt=v(t)$, find an explicit expression for $s(t)$ if $s(0)=0$. $\text{ te: }\sinh x=\frac{e^x?e^{?x}}{2},\cosh x=\frac{e^x+e^{?x}}{2}$