Hands-On Math

Last Revision March, 2011

Last Revision March, 2011

Briefly, his balloon reached an altitude of about 31,330 metres at the point when he exited his gondola. After about 12 seconds of fall, he opened a 1.8 metre stabilization parachute and continued falling until his main parachute, 8.5 metres, deployed at a height of about 5500 metres. He landed about eight minutes later.

A picture of his gondola, below, taken at the Air Force Museum can be found here.

Two excerpts from the foregoing link follow:

1)

2)

From Excerpt 2) it seems that the peak of his acceleration was reached prior to reaching 90,000 feet of altitude. Measured by his altimeter? Has he corrected, in this statement, for its offset from the ground radar altimeters? Is his altimeter referenced to sea level or earth surface level?

Is the ground radar altimeter reading referenced to sea level or earth surface level?

What is his method of measuring velocity? Could the method be the rate of change of his altimeter observations?

We presume that he is referring to his altimeter with the statement "

His stabilization chute had a radius of ~ 0.9 metres. This chute would be expected to be only partly full until some time after maximum decent speed was reached. The effective radius of the combination of chute and Kittinger must be found, as must a value for the drag.

A density is also required for the combination of chute and man.

We need to solve for values of the sphere radius, density

This can be done by adding a steepest decent routine, introduced in Chapter 2, to the spreadsheet. For those readers interested in adding the routine to the downloadable spreadsheet it is seen next.

Although iteration to find the solutions is simple enough to be done "by hand" the process is tedious. Moreover, a more capable steepest decent routine will be employed in a later Topic and this application serves as an introduction to its use.

The three solutions, one for each suggested mass value, 121.8, 140, and 158.2 kg are seen next.

For all three solutions the low radius values seem more representative of Kittinger and his equipment than of the chute. This suggests that the stabilization chute is streaming out above Kittinger but not open to any significant extent. The most significant drag may stem from Kittinger and his pack of instrumentation.

All three solutions show the time taken to reach the maximum fall rate of ~ 274.5 m/s as ~ 41 seconds at an altitude of about 25,778 metres.

A by-product of obtaining the three solutions was a range of possible drag coefficients, 1.49, 1.56, and 1.62 for the assemblage of a streaming chute, and Kittinger with his equipment.

How well did the actual atmosphere over the New Mexico desert on August 16, 1960, correspond to the 1976 U.S. Standard Atmosphere? Although it is readily conceivable that seasonal adjustments could be made to our atmospheric model, this has not been attempted.

For interest, the web-based calculator linked in Chapter 6 and also found on the upper row of tabs is used to provide graphs of altitude and velocity during the fall, to ~25,000 metres for the 140 kg case, seen following:

Notwithstanding the uncertain parameters used in modeling the Kittinger decent, the spreadsheet, with its inclusion of a model of the standard atmosphere, should be quite useful for estimating the behaviour of projectiles and falling bodies.

As has been mentioned, the macro varies some of the labels on the Par Sheet in accord with the value, 0 through 4, of the Parameter Exper, which in this case has the value 2 as is seen in the next table for modeling Kittinger's descent.

The ">", greater-than, character indicates that the value to its right is to be provided by the user. These values show in bold type.

Three values, B4:B6, are permanent reminders of underlying assumptions.

The values B2 and B3 are calculated immediately when the platform altitude is entered as possibly of interest to the user. Similarly, on entry of r in D2, D3:D4 and D6:D7 are calculated and on entry of d in D5, D6:D7 are immediately recalculated.

Some cell labels are applicable to other Experiments. Their associated values will remain blank. Some labels and their values are common to all or most experiments.

The value for Target Altitude in D8 causes calculation to terminate when that altitude is reached. Similarly calculation will terminate when the number of calculation steps given in D10 is reached.

Notice the value 2.0 in B7 and 45 in B8. These are there to represent Kittinger's little jump, up and out, to exit his gondola.

Row 0 shows the initial conditions.

Row 1 is special in that is the culmination of calculating eleven sub rows consisting of: two rows calculated using 1/1024 of the step size, followed by 9 rows each employing twice the step size used in the previous step. This is equivalent to a whole step at the specified step size.

The reason for doing this is that projectiles may have a quite large initial velocity that drops quite quickly when an atmosphere is encountered. The precision of the calculation benefits from smaller step sizes in that initial period of encountering the atmosphere when deceleration is most rapid.

Data rows around the maximum fall rate are shown, in two sections, next:

One might note that the acceleration, Dv(y)/Dt, changes sign after the maximum fall rate is reached,

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