A body of mass $5$ kg is projected vertically upward with an initial velocity $23$ meters per second.

We assume that the forces acting on the body are the force of gravity and a retarding force of air resistance with direction opposite to the direction of motion and with magnitude $k|v(t)|$ where $k$ is a positive constant and $v(t)$ is the velocity of the ball at time $t$. The gravitational constant is $g = 9.8 m/s^2$.

a) Find a differential equation for the velocity $v$ (in terms of $k$):
$v'(t) =$

b) Solve the differential equation in part a) and find a formula for the velocity at any time ( in terms of $k$ ):
$v(t) =$

Find the limit of this velocity for a fixed time $t_0$ as the air resistance coefficient $k$ goes to $0$. (Enter $t_0$ as t0 .)
$v(t_0) =$

How does this compare with the solution to the equation for velocity when there is no air resistance?

This illustrates an important fact, related to the fundamental theorem of ODE and called 'continuous dependence' on parameters and initial conditions. What this means is that, for a fixed time, changing the initial conditions slightly, or changing the parameters slightly, only slightly changes the value at time $t$.

The fact that the terminal time $t$ under consideration is a fixed, finite number is important. If you consider 'infinite' $t$, or the 'final' result you may get very different answers. Consider for example a solution to $y'=y$, whose initial condition is essentially zero, but which might vary a bit positive or negative. If the initial condition is positive the "final" result is plus infinity, but if the initial condition is negative the final condition is negative infinity.