Particles of plant pollen are often nearly spherical, with diameters ranging from about $10$ to $100m$ and densities of about $1000k\frac{g}{m^3}.$ Suppose that a sudden gust of wind dislodges such particles and lifts them to a height of $5m,$ followed by a steady breeze at $4\frac{m}{s}.$

$($a$)$ Calculate the terminal velocities for two sizes of pollen, $D=10$ and $100m.$ Hint: you can solve this problem iteratively, but we recommend that you define a function using the necessary equation$($s$)$ and use an existing Python solver to solve. Please show your iterative work or attach your code as part of your solution.

$($b$)$ Assuming that the terminal velocity is reached immediately, how far will each be carried before reaching the ground?

$($c$)$ Confirm that for both particle sizes the time needed to attain the terminal velocity is negligible.