This is a module for beginning calculus students to study differential equations that model falling objects subject to air resistance -- for example, raindrops -- and to develop Euler's Method, a tool for approximating solutions of initial value problems.

Notes for the Instructor

About this Module and its Authors

Choice of Computer Algebra System

Click on the button corresponding to your preferred computer algebra system (CAS). This will download a file which you may open with your CAS.
 

Ver. 7 or higher

Ver. 5.0 or higher

Ver. 5.1 or higher

 

Copyright 1998-2004, CCP and the authors

Published June, 2004

Our models for velocity of a falling object will all be based on Newton's Second Law of Motion, which states that force equals mass times acceleration:

Here  F  is the force exerted on an object of mass  m,  causing the object to have an acceleration  a.

Acceleration is defined to be the derivative of velocity, i.e.,

where   v = v(t)  is the velocity at time   t. Thus Newton's Law can be rewritten as

The primary force on a falling body is gravity, the pull of the Earth's mass on the object. Our first model for a falling body will consider gravity to be the only force on the object.

It is known through experimental observation that (near the surface of the Earth) the force of gravity on an object is proportional to the mass of the object, i.e., there is a constant   g  such that

The value of the constant  g  is known by experimentation to be approximately  32.2 sec².

Equating our two formulas for the force   F  and dividing by   m,  we find a differential equation:

1. Make a mental image of what this differential equation says -- what its slope field looks like, and what its solutions look like. Then execute the relevant commands in your worksheet to confirm your image.

If we assume that our object was initially at rest at time   t = 0,  then our initial condition is  
v(0) = 0. Together with the differential equation, we have an initial value problem for the velocity function   v = v(t):

2. Enter the solution of this initial value problem in your worksheet.

This solution for a velocity function leads to a second differential equation: The velocity   v  is itself the derivative of the distance function  s = s(t),  i.e.,

3. Substitute your formula for   v  into this equation, and construct a slope field for this differential equation. Does it have the appearance you expect?

4. Write down an appropriate initial condition for  s.  What is the solution of this initial value problem?

5. Using this model for distance as a function of time, compute how long (in seconds) it would take a raindrop to fall from a height of 3000 feet. How fast would it be traveling when it hit the ground? Give your answer first in feet per second, and then convert it to miles per hour.

6. What would happen if you got hit with a raindrop traveling at this speed? Is this consistent with your experience with rain?

We now investigate a more sophisticated model for a falling body, one that takes into account the resisting force of the air through which the object falls. The usual physical assumption is that the force of air resistance is proportional to some power of the velocity, but the particular power (first, second, or other) depends on the particular object.

We consider raindrops falling from a cloud 3000 feet above the ground. If the raindrop is small, say a drop of diameter 0.003 inches (or 0.00025 feet), a size found in a drizzle, the force of air resistance is modeled well by a multiple of the first power of the velocity. In other words, the resisting force can be described by

for some positive constant   k.  (The minus sign indicates that the force is in the direction opposite to the velocity, i.e., upward.) When combined with the force of gravity,

this yields the total force on the raindrop:

We recall Newton's Second Law of Motion:

Equating our two formulas for the force  F  and dividing by  m,  we find a new differential equation for velocity:

We'll let   c  represent the quotient   k/m.  When we attach our initial condition,   v(0) = 0,  we obtain our new initial value problem:

Experimental evidence gives an approximate value of  52.6 sec^-1  for  c,  when distances are in feet and the drops are drizzle size.

1. Why must the unit for   c  be  sec^-1  (reciprocal seconds)?

2. Enter the new formula for the derivative  dv/dt  in your worksheet, and construct a slope field for this formula. (Note that the time and velocity scales provided in the worksheet are very different from those on Page 2.)

3. Using the slope field as a guide, make a guess as to what type of function might be a solution to this differential equation, and enter your guess in the worksheet. (If you know how to solve the equation exactly, enter the actual solution.) Does your proposed solution satisfy the initial condition?

4. Plot your proposed solution -- either in your worksheet or on your calculator -- to see if you are on the right track. Does your proposed solution have the right shape to fit the direction field? Why or why not? (Note: There is no plot command for this in the worksheet, but you can enter a command on your own. Don't worry at this point if you can't find or don't know a symbolic solution -- the rest of the module is about an alternative to such solutions.)

Guessing a formula for   v  as a function of time may be more difficult for this problem than actual calculation was on Page 2. You may or may not know a systematic way to solve this type of problem at this point. On the next page we will take a different approach and use a numerical technique called Euler's Method. This technique will only approximate the desired solution, but it has the distinct advantage of applicability to any initial value problem.

We ended Page 3 with an initial value problem to be solved: Find   v = v(t)   so that

More generally, our problem is to solve any initial value problem of the form

We will calculate approximate values for the velocity   v  at   n  equally spaced points in some fixed time interval. Our procedure is simple: We repeatedly calculate a rise in   v  as slope x run. Then we add the rise to the current value of   v  to get the next value of   v.

Our goal is to estimate the velocity  v(t)  at times

Our estimated velocity values at these times will be denoted by

Our method for estimating the velocity values will be recursive, i.e.,   v_k  will be calculated from the preceding   v_k--1  for each   k = 1, 2, 3, ...  .

How do we obtain   v₁  from   v₀,  the initial velocity? We will answer this in a geometric fashion. We will look at the graph of velocity versus time on the   (t,v)-plane. The following figure shows a graph of the starting situation: the initial velocity   v₀  is shown as a vertical line segment of length   v₀  at the starting time   t₀  = 0.

We now include the graph of   v  versus   t. Our next velocity value,   v₁,  is shown as the length of a vertical line segment at time   t₁.

However, the value of   v₁  is not known to us, and hence we will estimate its value. We do this by drawing the tangent line to the graph at   t  = t₀. Follow this tangent line to the point   P, the top of a vertical line segment that approximates   v₁.

We can compute the length of this new line segment: We separate the line segment into two pieces -- the bottom piece having length   v₀,  and the top piece being the  rise  of a right triangle with Using

we see that  rise  equals  slope    times  . Hence,

(We use the symbol to mean "almost equal to".)

This is the key to Euler's Method for approximating the solution of an initial value problem. It's valuable because the slope (of the tangent line) equals the derivative  dv/dt,  which is given by our original differential equation when   t  = t₀  and   v  = v₀:

Substituting this value of the slope into the preceding equation, we find

Great! This gives us a method for going from   v₀  to   v₁.  But how do we go from   v₁  to   v₂?  Easy -- we use the same equation, only with   v₀  and   v₁  replaced by   v₁  and   v₂:

In general, to go from   v_k - 1  to   v_k  we have

This equation, along with the initial value   v₀  = 0  and the assignment of a value to the step size  ,  plays the central role in our computations.

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

1. Enter the constants and starting values that you find in your worksheet. Calculate   v₁,   v₂,  and   v₃  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.)

2. Write down the numbers  t₀,  t₁,  t₂, and  t₃. Then write a general formula for  t_k. Enter this formula in your worksheet.

3. 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.

4. Create and plot all the points   (t_k, v_k)  for   k  ranging from   1  to   n = 100.

5. Check your results by overlaying the solution plot on the slope field from Page 3.

6. There is something different in this graph -- something that did not occur in the model without air resistance on Page 2. Describe the difference.

7. 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.

8. 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?

9. 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?

10. Using your answer to the preceding question, calculate the terminal velocity directly from the original differential equation,   dv/dt  = g  - cv.

11. 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?

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_k  in terms of   v_k - 1. Create and plot all the points   (t_k, v_k)  for   k  ranging from   1  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?

1. Why is it important to consider air resistance when modeling raindrops as falling objects?

2. What important feature did you find in both resistance models that was lacking in the no-resistance model? How did the slope fields reveal this feature? How does it appear symbolically in the differential equations?

3. Explain in your own words how Euler's Method generates a solution of an initial value problem. In particular, explain how Euler's Method uses the same information that is used to generate a slope field.

4. Explain why

 

is an exact solution of the drizzle drop problem. How does this formula reveal the terminal velocity you know already? If you guessed the form of the solution on Page 3, compare this symbolic form with your guess. Are the two proposed solutions the same? If not, describe how they differ.

Final comments

• You may wonder why we didn't study a model for raindrops between very large and very small drops. For most of the size range, no one knows an accurate model as simple as those studied here. However, the evidence at both ends of the range suggests that one might as well assume that all raindrops fall at terminal velocity most of the time -- a velocity very much dependent on the size of the drops.

• In Step 4 above, you confirmed an exact formula for the solution of the falling body problem with linear resistance. You may have already encountered this formula in a calculus course. There is also an exact formula for the solution of the quadratic resistance problem (thunderstorm drops), but it is much more complicated and not likely to appear early in a calculus course. Observe that the information we get from Euler's Method is the same in both cases -- and that it doesn't depend on whether there is a formula for the solution, or on whether we know the formula if there is one.