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Raindrops - Modeling Large Raindrops

David A. Smith and Lawrence C. Moore

For large raindrops, say with diameter 0.05 inches (or 0.004 feet), a size typical of drops in a thunderstorm), the force of air resistance is better modeled as a multiple of the square of the velocity. The differential equation now has the form

where   a  is another constant. In this case, the experimental evidence yields a value for   a  of   0.115. With the same initial condition,   v(0) = 0,  we have a new initial value problem. We will use Euler's Method to approximate the solution of this new problem, this time over the time interval from 0 to 2 seconds.

  1. What are the units for the constant   a?

  2. Plot a slope field for the new differential equation, and confirm the reasonableness of the selected time interval. Does it look as though the solution will reach terminal velocity in 2 seconds?

  3. This time calculate the terminal velocity from the differential equation first, before finding a solution. Express your answer in both feet/sec and miles/hour.

  4. Enter in your worksheeet a general formula for   v in terms of   vk - 1. Create and plot all the points   (tkvk)  for   ranging from   to   n = 100.

  5. Check your results by overlaying the solution plot on the slope field from Step 2.

  6. Estimate the terminal velocity from your computed solution, and compare the result with your calculation in Step 3.

  7. Compare your terminal velocity with what you obtained in Part 1 as the velocity when a raindrop hits the ground after falling 3000 feet. Which model seems more reasonable?

  8. As you have seen, a thunderstorm drop approaches its terminal velocity quite rapidly -- but not as rapidly as a drizzle drop. Assuming that the velocity is constant during the whole duration of the fall, estimate the time it takes the drop to fall to the ground from 3000 feet. How does this time compare to your time-of-fall answer on Page 2, where no air resistance was assumed?

David A. Smith and Lawrence C. Moore, "Raindrops - Modeling Large Raindrops," Convergence (December 2004)