Hands-On Math
Last Revision March, 2011
Prologue
Introduction
Background
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1D Calculator
2D Calculator
Postscript
References
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Chapter 1
Chapter 2
Chapter 3
Chapter 4
Chapter 5
Chapter 6
Chapter 7
Chapter 8
Chapter 9
Chapter 4
An Interactive Spreadsheet Emulator for the Solution of a Differential Equation for Bodies Falling in Air
About the Spreadsheet Emulator
There are nine default parameters eight of which may be changed by the user. Five parameters, (g, m,
p
, r,
Cd
), are used to calculate the terminal velocity,
vt
.
Two parameters, step size and number of steps, are employed in calculating the table and its graphs.
Another two parameters, a velocity v, and
μ/
p
along with
r
are used to calculate
R
, the Reynolds number. (On the spreadsheet,
μ
is represented as mu.) The default value of
μ/
p
has been frozen at a value thought suitable for the atmosphere at sea level.
Were
R
to have a value less than ~1000, the implication is that the Rayleigh approximation is not representing the fall very well for the given v and lower velocities.
Input boxes are given solid outlines, except for that of
μ/
p,
which is dotted. The reason is that its default value applies to air near the earth's surface, the situation addressed by this calculator.
R
does not contribute to the solution. It is calculated to indicate whether or not the Rayleigh approximation might apply at a selected velocity given the other parameters that have been chosen.
The table and graphs are calculated using g, m, vt and the step size Dt.
The number of steps, together with Dt, determine the time period that is covered. There is a limit, 1000, to the number of steps that can be input by the user.
Rather than input the area
A
, a radius r is input from which the spreadsheet calculates
A.
A reason for this is that the value for r is again used for the calculation of R.
A
is displayed and is used in the calculation of vt.
Shape
The shape of the falling object may not be a sphere to which a radius can be directly attributed, and for which a value of
Cd
may be selected. In the absence of a vertical wind tunnel, the user might estimate the area that a body projects on to the horizontal plane and calculate r as the radius of a disk of that area. Then, if the body were quite thin in the vertical direction as compared with the disk diameter, search for a
Cd
that was appropriate to a flat disk.
The default value of
Cd is
0.1. For an upright man,
Cd
may have a value of ~1.2. A reference for some drag coefficients is provided in
Approximations
of Part A of this topic.
Step Size
Although the user may input step size, calculation accuracy can be quite poor if it is chosen as less than vt/100. The user might not make a suitable guess in the absence of knowing vt, in which case the calculation will be rejected so that the user may enter a smaller step size. The trick is to choose it quite small in the first place and adjust it later if desired.
There is another aspect to the choice of step size. The user may often prefer to read the table at exact decimal intervals of seconds such as 10 milliseconds or 10 seconds.
The Plots
The plots, being small, are a bit primitive to avoid clutter.
The end value shown for the ordinates or abscissas can be mentally divided by 10 and then taken as the amount between tick marks. Scroll the table to obtain more precise values.
Calculate - Reset - Input Errors
The button, C
alculate
, to the right of the table of parameters, causes the spreadsheet and its graphs to be calculated or re-calculated. The
Reset
button restores the parameter default values and then re-calculates.
Browsers are not standardized. With many web browsers, calculation will occur after the user's
Return
that follows the entry of a parameter value. We have no
fix
at this time for that behaviour.
The calculation process catches many errors. These result in an error message that appears just below the table of parameters. With these errors the spreadsheet table is blanked. To see an example, type 1001 as the
Number of steps
.
A user input that results in a "divide by zero" is caught by the System and a less than friendly page pops up. To return from the error page, should this happen, click its "back" button and then correct the error or use
Reset
to restore the default parameters
.
Individuals preparing their own spreadsheet would pay little attention to introducing warnings against their inappropriate entries. There is a saying in the computer world "Garbage in then Garbage out".
The Differential Equation
The equation used by the calculator, developed in Part B of this topic, is:
a = g * [1 - v
2
/ vt
2
]
The cell expressions for the table are provided at the top of the table on the 1D calculator page.
Click
this
to access this one-dimensional, 1D Calculator. Or, reach it from the top row of navigation tabs. Enjoy!
Next
The use of the calculator will be illustrated by applying it to solving the differential equations that characterize falling rain and hail.
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