Separable Differential Equations: Videos & Practice Problems

A small object is dropped into a viscous fluid. The forces acting on the object are gravity pulling it downward and a resistance force from the fluid opposing the motion. According to Newton's Second Law, the velocity $v\left(t\right)$ of the object satisfies the differential equation $m\frac{dv}{dt}=mg+F\left(v\right)$, where $m$ is the mass of the object, $g$ is the gravitational acceleration, and $F\left(v\right)$ is the drag force exerted by the fluid, with positive velocity defined downward. Assume the drag force is proportional to the velocity and acts opposite to the direction of motion, modeled by $F\left(v\right)=-Rv$, where $R>0$ is the drag coefficient. Find the velocity function $v\left(t\right)$ given the initial condition $v\left(0\right)=0$, and assume the velocity satisfies .