Problem -22(UP1 (10 points) Let the drag force on a ball of mass $m$ moving through air in the gravity-free
Bubble consist of viscous drag $F_v = -kv$, where $k$ is a constant with appropriate units. At time $t = 0$, the ball is
thrown from initial position $x = 0$ with initial velocity $v_0$. (a) Write the $F = ma$ equation for $t > 0$. (b) Use
dimensional analysis to find a rough time scale $\tau$ over which you expect the ball's velocity to change by a
significant fraction of its value. The time scale could depend on any or all of $m$, $k$, and $v_0$. (c) Solve the equation
exactly to find the ball's velocity as a function of time. (d) Did dimensional analysis yield an appropriate time
scale $\tau$? Explain your answer. (e) Integrate numerically using Euler's method, for five time steps of $0.2\tau$ each,
to find the velocity of the ball after a total elapsed time of $\tau$. Use $m = 100g$, $k = 1 kg/s$, and $v_0 = 1 m/s$. You should
do this part with the help of a calculator, a spreadsheet, or even a short piece of code. (f) Compare the numerical results
with the exact results for the ball's velocity at time $t = \tau$.