Hands-On Math

Last Revision March, 2011

Last Revision March, 2011

Modeling applications that come to mind are:

1.The behaviour of weather balloons;

2. Aspects of piloting netted gas balloons such as that used for the Kittinger ascent;

3. Estimating the influence of the atmosphere on Escape Velocity;

4. HARP, High Altitude Research Program, projectiles;

5. The ballistics of a Grenade Launcher.

The foregoing applications break down into 5 cases each with slightly different reporting needs. We have called them Cases 0 through 4 as follows:

0 - Expandable Balloons, e.g. Weather Balloons;

1 - Non-Expandable Balloons, e.g. The Kittinger Ascent;

2 - The Kittinger Descent, a special case;

3 - High Altitude Projectiles, e.g. HARP, Escape Velocity;

4 - Ballistics, e.g., Grenade Launcher.

The Parameter Table caters to the variety of reporting needs of the cases by providing some fixed labels, some case dependent labels, and some result dependent labels.

Our model employs
**Lord Rayleigh's** approximation for the drag force on a moving
object. In Chapter 2 we arrived at the expression
for the acceleration of such a body due to gravity as:

**
a
= g - 1/2*1/m* ρ*A*Cd*v2 =
g - k*v2.
**

We see that there are only three factors in the expression
that are related to the physical characteristics
of the object, i.e., **m**,
* A, *and

The user of the model has the task of ** distilling
the characteristics of the balloon, falling object, or projectile into values for
these three factors**.

In Chapter 3 we took buoyancy into account as a modification to g in the differential equation. That was because buoyancy creates a force that is opposite to gravity:

One may think of an effective or buoyant mass mb for an object with mass m where:

mb = m * (1 - *p*fluid/*p*object)

Under gravity the object will fall if its density is greater than that of the fluid and rise if less.

The force on the body will be:

Fb = mb* g

The user does not directly input a value for the mass. He does provide an overall density for the object
or system of connected objects and leaves it to the macro to calculate the corresponding
mass from the frontal area *A*.

The frontal area *A* is the cross section of the body in its direction of travel.
We imagine a cylindrical object for which the user provides a radius. It may be thought
of as a sphere that is elongated at its diameter to become a cylinder with ends consisting
of hemispherical caps.

When there are a group of connected objects such as, in line, a balloon, parachute and instrument package, one object such as an inflated balloon, will provide the greatest part of the drag and its area may be chosen. The user specifies the frontal area by inputting the radius of the hemispherical cap.

The drag coefficient is shape dependent and can take on a huge range of values, encompassing at least three orders of magnitude. See Guns for an interesting article.

Drag coefficients are obtained by experimental methods and are frequently kept as industrial and military secrets. We are often left to guess the drag coefficient that could apply to any particular case being studied. Roughness of the frontal area tends to increase drag. Objects can be streamlined to reduce drag.

In some cases we can infer a drag coefficient through consistency checks with various reported observations. In many cases we are quite relaxed about the precision of the results and have greater interest in gross behaviour.

Besides, our model neglects all sorts of potential variables such as temperature, water vapour, rain, snow, wind, up drafts, down drafts, hail and so on.

A combination of balloon and payload will have an effective mass that is the sum of the respective masses. The combination will have a volume that is the sum of the two volumes and an effective density equal to the ratio of the effective mass to the total volume. If that effective density is less than that of the surrounding atmosphere, the balloon and its payload will rise.

For ballooning, the important factor is the effective density that is achieved and its relationship to the density of the surrounding atmosphere.

For projectiles, a long thin object will generally be assigned a higher density in accordance with its length.

Slender projectiles moving in air tend to tumble and have an erratic trajectory. For this reason they are either spun at launch by a rifled barrel or employ tail fins.

A weather balloon is made from a highly flexible material, often Latex. It is inflated with a light gas such as hydrogen or helium and carries a radiosonde and, sometimes, a parachute.

Similar devices are employed by amateur groups in hobby ballooning.

Weather balloons may attain altitudes of ~ 40,000 metres before they burst from over expansion.

The Expandable Balloon case requires a special user input, the Balloon Expansion Factor. This factor is used by the macro to adjust the density of balloon and payload as atmospheric density changes during the flight. The radius of an expandable balloon is increased by the macro as less dense atmosphere is encountered and vice versa. The limit provided by the factor is a restriction on radius. A factor greater than 0.0 restricts the radius change to that multiple. The factor 0.0 indicates that an object may neither expand nor contract.

In Case 0, the Weather Balloon, the balloon disintegrates when it reaches the expansion limit. In Case 1, the balloon simply expands no further when the limit is reached.

See Weather Balloon for other details.

The Web-based 2D Calculator version of the spreadsheet is used for the examples in this topic. That calculator is available at the end of this chapter and from the upper row of navigation tabs.

The parameter table follows:

The user is interested in a launch from sea level and puts a 0 in cell B1. The macro then provides the values for B2 through B6. The value in B6 is a reminder that we are presuming that drag force is proportional to the square of velocity.

The user puts the value of 0 in B7 and B8 as the balloon is not being given an initial push at launch. It is simply being released.

The user puts the limit by which the balloon's radius may expand before bursting in B9. This is a value that is generally provided by the balloon's manufacturer.

The user puts the desired step size for the calculation in cell B10. It may be borne in mind that the first step will always consist of 10 graded smaller steps that have the purpose of, improving computation when initial values are subject to rapid change as can occur when a projectile decelerates in atmosphere.

The user provides the value 0.4 in D1. This value would be consistent with a rough-surfaced spherical shape for the balloon. Perhaps 0.4 will also take into account the drag of the attached parachute and instrument case.

The balloon as inflated at sea level has a radius of one metre. The value 1 is placed in D2. (Weather balloons can be obtained in a variety of sizes as needed for different payload masses.)

The macro provides values for D3 and D4.

The user provides a density value for the system that is smaller than that of the atmosphere at sea level. The smaller the difference the slower will be the ascent.

The macro then provides values for D6 and D7.

D7 gives the sea level value of the k value that appears in **Lord Rayleigh's**
approximation for the drag force on a moving object:

**
a = g - k*v2. **

Our k is not a constant but is a variable that is dependent on altitude.

A value for D8 can be more useful in other cases but a value placed here for Target Altitude will cause the macro to report the number of calculation steps taken to just exceed that altitude at D11.

The user puts a 0 in cell D9 to indicate to the macro that the data applies to the Weather Balloon case.

The value the user places in D10 constrains the total number of steps that will be calculated by the macro. The macro will stop computation at that number of steps even though an objective may not have been reached.

There may be cases for which the velocity reached at a selected altitude is of interest. A user altitude value placed in B17 will cause the macro to report the associated velocity in cell D17.

Macro operation has two phases. In the first phase the macro provides cell values in the parameter table in accord with user inputs and any changes made thereto, thus acting as a handy calculator. In the second phase the macro calculates all the required steps and produces a step-by-step table in which intermediate values can be seen and from which graphs are constructed.

Examples of the possible application of the handy calculator can be as simple as calculating sphere properties such as cross section area volume or mass, to providing a value for g at a given altitude or the value for atmosphere density at a given altitude.

In this case the balloon bursts after 23,091 steps at an altitude of 20,783.5 metres at ~ 38.5 minutes after launch.

Its upward velocity at the 20,700 metre level was ~11.64 metres per second.

It reached a Target Altitude of 5,000 in 6486 steps. At 0.1 second per step, that's at nearly 11 minutes after launch. From these values we deduce an average ascent rate to the 5,000 metre level as 7.71 metres per second, less than that seen at the 20,700 metre level.

A portion of the table that is output showing the 5,000 metre level follows:

*
*

Had the balloon a greater expansion factor it would have reached a higher altitude before bursting.

Had the overall density been greater, e.g. more payload, the balloon would have risen at a slower rate.

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