Raindrops

 

(Adapted from the Raindrops project developed by Duke University’s Connected Curriculum Project, see http://math.duke.edu/education/ccp)

 

This project may be done in a group of up to 4 people.  Each member of the group will receive the same grade.  The project is due Tuesday, January 29.

 

Part 1: Background: Falling Bodies

 

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:

F = m a.

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.,

a = dv/dt,

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

F = m dv/dt.

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

F = m g.

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

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

dv/dt = g.

 

1.        Make a mental image of what this differential equation says -- what its slope field looks like, and what its solutions look like.  Then use the SLOPES program on your calculator (attached to the end of this project) to confirm your image.  Sketch the slope field.  (Use a window of [0, 3] by [0, 60].)

 

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):

dv/dt = g, v(0) = 0.

 

2.        Find the solution to this initial value problem.

 

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) (the distance fallen by the raindrop), i.e.,

v = ds/dt.

 

3.        Substitute your formula for v into this equation, and construct a slope field for this differential equation. Sketch the slope field.  Does it have the appearance you expect?  (Use the same window as in question 1.)

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?

 

Part 2: Falling Bodies with Air Resistance

 

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, 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 0.00025 feet (or 0.003 inches, 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

Fr = -k v

for some 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,

Fg = m g,

this yields the total force on the raindrop:

F = Fg + Fr.

We recall Newton's Second Law of Motion:

F = m dv/dt.

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

dv/dt = g - (k/m) v.

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:

dv/dt = g - cv, v(0)=0.

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.        Construct a slope field for the formula for dv/dt. (You will need to use a different window from part 1.  Use [0, 0.1] by [0, 1].)

3.        Make a guess as to what type of function might be a solution to this differential equation. Does your proposed solution satisfy the initial condition? (If you know how to solve the equation exactly, give the actual solution.)

 

Guessing a formula for v as a function of time may be more difficult for this problem than actual calculation was in Part 1. You may or may not know a systematic way to solve this type of problem at this point. In the next part 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, regardless of whether an exact solution can be found.

 

Part 3: Euler's Method

 

We ended Part 2 with an initial value problem to be solved: Find v = v(t) so that

dv/dt = g - cv and v(0) = 0.

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

dv/dt = f(t,v) with v(0) = v0.

We will calculate approximate values for the velocity v at n equally spaced points in some fixed time interval.  To do this we will repeatedly calculate a rise in v as slope x run.  Our goal is to estimate the velocity v(t) at times

t0 = 0, t1 = Dt, t2 = 2 Dt, ... , tn = n Dt.

Our estimated velocity values at these times will be denoted by

v0, v1, v2, v3, ... , vn.

Our method for estimating the velocity values will be recursive, i.e., vk will be calculated from the preceding vk-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 first diagram ...

...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 add ...

... 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. Moreover, 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 run = Dt. Using

slope = rise / run,

we see that rise equals slope times Dt. Hence, since v1 is approximated by v0 + slope x Dt,

v1 ~ v0 + slope x Dt.

(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:

slope = dv/dt = g - cv0.

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

v1 ~ v0 + (g - cv0) x Dt.

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:

v2 ~ v1 + (g - cv1) x Dt.

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

vk ~ vk-1 + (g - cvk-1) x Dt.

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

 

Part 4: Modeling Small Raindrops

 

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

dv/dt = g - cv and v0=0.

We have specific values for g and c, both obtained experimentally: g = 32.2 ft/sec2 and c = 52.6 sec-1.

 

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 (why?). 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 in Part 2. Thus the distance between consecutive t values will be Dt = 0.2/n = 0.002 sec.

 

1.        Calculate v1, v2, and v3 by hand to make sure you understand how the steps start out.

2.        Write down the numbers t0, t1, t2, and t3. Then write a general formula for tk.

3.        Find a general formula for vk in terms of vk-1. Check to make sure your formula produces the same starting values as in Step 1.

4.        Create and plot all the points (tk, vk) for k ranging from 1 to n = 100 using the program EULER1 (attached at the end of the project).  Give the coordinates of the last 3 points (you don't need to give me all 100!), and sketch the plot.

5.        Check your results by overlaying the solution plot on the slope field from Part 2.  You can do this just by running the SLOPES program again after plotting the points.  Sketch the results.

6.        There is something different in this graph -- something that did not occur in the model without air resistance in Part 1. 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 in Part 1 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 - c v.

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 in Part 1, where no air resistance was assumed?

 

Part 5: Modeling Large Raindrops

 

For large raindrops, say with diameter 0.004 feet (or 0.05 inches, 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

dv/dt = g - a v2,

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 (use a window of [0, 2] by [0, 25]). 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.        Find a general formula for vk in terms of vk-1. Create and plot all the points (tk, vk) for k ranging from 1 to n = 100, as you did in Part 4 (note that the time interval is now [0, 2], so Dt will be different). Give the coordinates of the last 3 points (you don't need to give me all 100!), and sketch the plot.

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

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 in Part 1, where no air resistance was assumed?

 

Part 6: Summary

 

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 algebraically 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 v(t) = (g/c) (1 - e-ct) is an exact solution of the drizzle drop problem. How does this formula reveal the terminal velocity you know already?

 

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.

 

Calculator Programs

 

PROGRAM:SLOPES

:ClrDraw                                 (2nd PRGM (DRAW) / DRAW / ClrDraw)

:FnOff                                                     (VARS / YVARS / On/Off / FnOff)

:7(Xmax-Xmin)/83®H                             (VARS / WINDOW / Xmax or Xmin; ® is STO®)

:7(Ymax-Ymin)/55®K                              (VARS / WINDOW / Ymax or Ymin)

:1/(0.4H)2®A

:1/(0.4K)2®B

:Xmin+0.5H®X

:Ymin+0.5K®Z

:For(I,1,12,1)                                          (PRGM / CTL / For)

:Z®Y

:For(J,1,8,1)

:Y1®T                                                    (VARS / Y-VARS / Function / Y1)

:1/Ö(A+B*T2)®C

:T*C®S

:X®U

:Y®V

:Line(U-C,V-S,U+C,V+S)                     (2nd PRGM (DRAW) / DRAW / Line)

:V+K®Y

:End                                                        (PRGM / CTL / End)

:U+H®X

:End

 

SLOPES will draw a slope field.  Before running the program, the differential equation should be entered as Y1 on the Y= screen, using both X and Y.  You will need to set the window coordinates yourself before running the program.  You can then overlay other graphs over the slope field - to clear the screen, select 2nd PRGM (DRAW) / DRAW / ClrDraw or change the window coordinates.

 

PROGRAM:EULER1

:FnOff

:ClrList L1, L2                                         (STAT / EDIT / ClrList)

:A®L1(1)

:B®L2(1)

:For(I,1,N,1)

:A+I*D®L1(I+1)

:L1(I) ®X

:L2(I) ®Y

:Y+Y1*D®L2(I+1)

:End

 

EULER1 will use Euler's method to approximate points on a solution curve to a differential equation dy/dx = f(x,y).   It will compute a list of n points (x0, y0), (x1, y1), ... , (xn, yn), storing the xi's in list L1 and the yi's in list L2 (since the list numbering starts at 1, the list entries will be numbered from 1 to n+1).  Before running the program, you need to enter x0 into A, y0 into B, Dx (the step size) into D, and n into N.  You also need to enter the differential equation f(x,y) as Y1, using X and Y.  After running the program, you can plot the points as usual.

 

PROGRAM:EULER2

:FnOff

:For(X,X,Xmax,H)

:Y+H*Y1®V

:Line(X,Y,X+H,V)

:V®Y

:End

 

EULER2 (courtesy of Dr. Belogay) will draw an approximate solution curve for a differential equation equation dy/dx = f(x,y) using Euler's method (rather than storing the points).  Before running the program, you need to place the cursor in the graphing window at the point (x0, y0), you need to store Dx (the step size) into H, and you need to enter the differential equation f(x,y) as Y1, using X and Y.  The curve will be drawn until it hits the right side of the screen.