Project Help Notebook by Keith Stroyan

Help NoteBook Copyright 1998 by Academic Press, Inc. - All rights reserved.
Modify this Notebook to prepare your project report.

Project 24: Air Resistance

by A. Student and B. Pupil

with technical help from Da LabMonitor and Help Notebook

Initialization

Needs["Graphics`Colors`"];

Introduction: The Wind in the Willows

Write this last.  Begin with the basic question and state your main finding clearly, but with few technical details.  You might even modify the title of this section to make it more interesting.

Background Information: Galileo & Newton

In the Gravity.ma Notebook for Chapter 10 of the main text, you saw that data for the fall of a lead cannon ball from a high cliff satisfies a simple second order differential equation

(((d^2) s)/(dt^2)) = g (a universal constant)

where  s  is the distance travled and  t   is time.  This equation says that the acceleration due to gravity is constant - independent of the mass of the object.

Acceleration is the name for the second derivative of distance traveled, a[t] = (((d^2) s)/(dt^2)).  We can express Galileo's law in terms of the velocity,   v[t]  = (ds/dt)  (or speed with sign).  Since acceleration is the derivative of speed, Galileo's Law says, "The speed increases at a constant rate."   Galileo's Law may also be written

(dv/dt) =  g

Read about Newton's "F = m a" law and his formula for the force of gravity in the project.  This will let you interpret terms added to Galileo's Law to account for air resistance in terms of forces.

An areodynamic lead ball dropped a relatively short distance allows us to neglect the effect of air resistance.  In this notebook we will simulate the fall of a wooden ball, subject to air resistance.  The sanitized data we will use is defined in the next cell.  This lighter object dropped a long distance is affected by air resistance, so the acceleration is not constant.

Gravity Data (with air resistance)

Gravity data with air as a `matrix' of (time,distance) pairs in (second,meter) units:

gravityDataAR = {{0., 0.}, {0.5, 1.199}, {1., 4.702}, {1.5, 10.37},
	{2., 18.06}, {2.5, 27.67}, {3., 39.07}, {3.5, 52.15},
	{4., 66.82}, {4.5, 82.97}, {5., 100.5}, {5.5, 119.4},
	{6., 139.5}, {6.5, 160.7}, {7., 183.1}, {7.5, 206.4},
	{8., 230.7}, {8.5, 256.}, {9., 282.}, {9.5, 308.9},
	{10., 336.5}} ;

Gravity Data as a Continuous Plot

As we did in the Gravity Notebook, we first draw a graph of the data.

distDataPlot=ListPlot[ gravityDataAR, PlotJoined -> True, AspectRatio -> 1,
			PlotRange -> {0, 500}, AxesLabel -> {"time", "distance"},PlotStyle->Red ];

This graph is drawn with the same PlotRange as the one in the Gravity Notebook, which you should also open for comparison.  Note that the wooden ball doesn't fall nearly as far or fast as the lead cannon ball in the 10 seconds of flight we recorded.  As the velocity increases, the air slows the INCREASE in the speed.  The change in the speed is the acceleration, so a negative term needs to be added to the acceleration due to gravity.

Hint 24.2: Falling with Air Resistance

The Conjecture

For an areodynamic body falling under the effects of air resistance, the accelerations should satisfy the following equation (where a[t] denotes acceleration at time t and v[t] denotes velocity)

a[t] = g - b v[t]

for some constant k.  Read the print project and explain why adding a term "- b v[t]" to the formula for acceleration is plausible.  What does it say about the force caused by moving through air?

We can test this hypothesis by using the differences routine we developed in the Gravity Notebook.

Refresher on List Computations

datalist = Prepend[ gravityDataAR, {"t", "s"} ] ;
Print[ TableForm[datalist] ]

differences function

differences[ datalist_List ] :=
	Block[ {diffs},
		diffs = {} ;
		Do[
			AppendTo[ diffs,
				(datalist[[i, 2]] - datalist[[i-1, 2]])/
				(datalist[[i, 1]] - datalist[[i-1, 1]]) ],
			{i, 2, Length[datalist]}
		] ;
		diffs
	]

Apply the differences function to the data above, to obtain a list of average speeds.

avespeedsAR = differences[ gravityDataAR ]

These best approximate the speed of the ball at times 0.25, 0.75, 1.25, etc.

times = Table[ t, {t, 0.25, 9.75, 0.5} ] ;
velocityDataAR = Transpose[ {times, avespeedsAR} ]

If we now plot a graph of this data we will see a significant feature of the fall of an object under the effects of air resistance.

speedDataPlot=ListPlot[ velocityDataAR, PlotJoined -> True, AspectRatio -> 1,
				AxesLabel -> {"time", "speed"},PlotStyle->Blue ] ;

Look at the Velocities!

What feature does the graph of the velocities reveal that is different from free-fall in vacuum?

Compute the Accelerations

aveaccelsAR =

Hint 24.3: Compute b's

The value of b in the conjectured formula above can now be estimated.  Remember that we can divide lists by other lists as in the gravity notebook so the computation

b = (g - a[t])/v[t]

is easy to perform on the lists of average speeds and acelerations above.  We must be careful to note, however, that we have one less acceleration that speed.  Hence you need to drop the last velocity when we perform the above calculation.

g = 9.8 ;
bEstimates = ?

Does the Air Resistance Conjecture Work?

Have your computations verified the conjecture of how the wooden ball falls with gravity and air resistance?  Put your write-up here.  Include an explanation of what the conjecture is mathematically speaking, where it comes from in terms of Newton's law of acceleration and whether or not you have verified that the conjecture holds.

The Symbolic Solution

The conjecture above

a[t] = g - b v[t]

for some constants b and  g  can be expressed as a differential equation in velocity,

(dv/dt) =  g  -  b v

Exploration 24.4: Terminal Velocity

Use the differential equation above to investigate behavior of a function satisfying it.  If  v  is very small, show that  v[t]  is an increasing function. (What calculus condition makes a function like v[t] increase?)  If  v  is very large, show that  v[t]  is decreasing.  What value of  v  makes it stay constant? (What calculus condition makes a function stay constant?)  Read the printed hint, too.

Prove that  v[t]  tends to  g/b  as  t  tends toward infinity.

The rest of the differential equation

Velocity is the derivative of position, (ds/dt) = v, and we release our object from rest, v[0] = 0, when it hasn't yet fallen, s[0] = 0.  This means that the conjecture about air resistance means the wodden ball falls according to the "law"

s[0]  =  0\(TextForm\`v[0]\  = \ 0
	\)\(TextForm\`ds\/dt = v
\)(dv/dt) =  g  -  b v

Hint 24.5

Work Hint 24.5.

Mathematica's Solution of the Differential Equations & Initial Conditions:

\(Clear[s, v, t, g, b]; 

    soln = DSolve[{s[0] == 0, v[0] == 0, \(s'\)[t] == v[t], 
          \(v'\)[t] == g\  - \ b\ v[t]}, {s[t], v[t]}, t]\)s[t_] = Simplify[Expand[s[t] /. soln[[1]]]]v[t_] = Expand[v[t] /. soln[[1]]]

Once you know  b, you can see the terminal velocity.  Following is a sample with a wild guess at  b.  Of course, we can use termianl velocity to measure  b.

\(g = 9.8; 
b = 0.1; 

    symbolicSpeedPlot = 
      Plot[v[t], {t, 0, 50}, PlotStyle -> Purple, PlotRange -> All]; 

    symbolicDistPlot = Plot[s[t], {t, 0, 50}, PlotStyle -> Brown]; \)

Show[symbolicSpeedPlot, speedDataPlot];

Show[symbolicDistPlot, distDataPlot];