Author(s):

David A. Smith and Lawrence C. Moore

Here again is our initial value problem: Find ** ** *v* = *v*(*t*)** ** so that

We have specific values for ** ** *g * and ** ** *c*, both obtained experimentally: and .

We will use Euler's Method to calculate approximate values for the velocity ** ** *v* at ** ** *n* equally spaced points in a fixed time interval. The Euler procedure gives a better approximation to the exact solution if ** ** *n* is large rather than small. Thus, for convenience, we set ** ** *n* = 100. Our time interval will be ** ** 0 < *t* < 0.2 seconds -- the reason for this choice is suggested by the slope field on Page 3. Thus the distance between consecutive *t* values will be

- Enter the constants and starting values that you find in your worksheet. Calculate
** ** *v*_{1}, ** ** *v*_{2}, and ** ** *v*_{3 } to make sure you understand how the steps start out. (Note: The notation in your worksheet for the subscripted variables * t* and * v * may be different from the mathematical notation here.)
- Write down the numbers
*t*_{0}, *t*_{1}, *t*_{2}, and *t*_{3}. Then write a general formula for *t*_{k}. Enter this formula in your worksheet.
- Enter in your worksheeet a general formula for
*v*_{k} in terms of *v*_{k - 1}. Check to make sure your formula produces the same starting values as in Step 1.
- Create and plot all the points
** ** (*t*_{k}, *v*_{k}) for ** ** *k * ranging from ** ** 1 to ** ** *n* = 100.
- Check your results by overlaying the solution plot on the slope field from Page 3.
- There is something different in this graph -- something that did not occur in the model without air resistance on Page 2. Describe the difference.
- Estimate the limiting value of the velocity as time increases. This is called the
**terminal velocity**. Express your answer in both feet/sec and miles/hour.
- Compare your terminal velocity with what you obtained on Page 2 as the velocity when a raindrop hits the ground after falling 3000 feet. Which model seems more reasonable?
- As
* t* increases and velocity * v*** ** approaches terminal velocity, what happens to the *slope* of the velocity versus time curve? What happens to the derivative ** ** *dv/dt*?
- Using your answer to the preceding question, calculate the terminal velocity directly from the original differential equation,
** ** *dv*/*dt * = *g * - *cv*.
- As you have seen, a drizzle drop approaches its terminal velocity quite rapidly. Estimate the time it takes the drop to fall to the ground from 3000 feet by assuming that the velocity is the constant terminal velocity during the whole duration of the fall. 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 Small Raindrops," *Convergence* (December 2004)