10/2001

Oh, no! More rain!

Objective:

In a previous lab, we investigated the position, velocity, and acceleration of a falling raindrop, first ignoring air resistance, and then with air resistance proportional to the first power of the velocity -- a condition which is compatible with a tiny drizzle drop. In today's lab we will compare the numerical result from that lab with the exact solution of that problem, and will look at how the problem changes when we encounter larger raindrops.

Recall from the Raindrop lab that, when a raindrop was very small, the air resistance was proportional to the first power of velocity, and we had the differential equation

We solved the problem this way:

Here is a graph of the values of v_i vs. time:

You have learned in a classroom project that an exact solution of this problem is

Move a copy of this definition of the function velocity(t) above the graph, and then modify the graph so that it plots velocity(t_i) along with the computed values v_i. Gradually change n from 20 to 100, and compare the exact and numerical results. Comment.

Look carefully at the formula for the exact velocity function. What happens to e^-at as t gets bigger and bigger? So what happens to velocity(t) as t gets bigger and bigger? How does your answer compare to the value you found for terminal velocity in the Raindrop lab?

Bigger raindrops...

For large raindrops, say with diameter of about 0.05 inches, a size typical of drops in a thunderstorm, the force of air resistance is proportional to the square of the velocity. If we use an uppercase V for velocity (to keep it distinct from the velocity of the drizzle drop), the differential equation now has the form

where a is another constant. In this case, experimental evidence yields a value for a of 0.115.

We want to approximate the solution to this problem by using a recursive formula. Using a method similar to the one you used in the Raindrop lab, determine a recursive formula for

V_i+1 in terms of V_i. This problem takes a little longer to stabilize, about 2 seconds rather than 0.2 seconds, so we will let t take on bigger values.

(Enter your formula for V_i+1 above.)

Graph the computed values of V_i below.

Increase m by stages from 20 to 100. Describe the resulting plots and your impression of this velocity function. How does it compare to the drizzle situation? In particular, is there a terminal velocity? Estimate it. How long would it take a big raindrop to fall 3000 feet, if it fell the entire time at this terminal velocity? Compare your answer to both of your previous answers: the problem without air resistance, and the drizzle drop problem.

This lab, together with the Raindrops lab, will become the subject of a group report. Your professor will provide instructions. Meanwhile, you may want to print out a copy of this lab for future reference.