Trombone Slide Positions
Theoretical distances calculated from the speed of sound 

I've calculated the length of tubing corresponding to wavelengths of different pitches to determine the theoretical distance a trombonist must move the slide for the various positions.

The calculation for this page is available at this spreadsheet

A calculator that provides the slide position(s) for any pitch over a range of 5 octaves is available on this page.

First the speed of sound is calculated using the "more accurate" equation from this Wikipedia page. At 72 F (22.2 C) the speed of sound is 344.5 m/s. The speed of sound is essentially independent of pressure, so this calculation should work up in Denver as well as in the low lands.

Then wavelength is determined for a particular frequency by dividing the speed of sound by the frequency. For A at 440 Hz the wavelength at 72 F is 78.3 cm. 

To go from A to Bb (a half-step) on a well-tempered scale, the wavelength is divided by two raised to the 1/12th power.  This wavelength is multiplied by 4 since a pedal Bb on a trombone is three octaves below the Bb just above A at 440 Hz, and a pedal Bb is produced by a half-wavelength.  

At 72 F, this calculation says the length of a tubing of a Bb trombone is 295.6 cm, or 116.4 inches, or 9.7 ft. I have seen websites that say a trombone is 9 ft long, but this length corresponds to a temperature of 0 F. Slide positions are usually irrelevant much below 30 F because the slide tends to freeze.

Here is a graph showing the slide distances in cm at 72 F for a Bb trombone, as well as for trombones in F, E, Eb and D. These are all calculated using a well-tempered scale, as above. The distances are given for the 7 positions going from left to right.  Most slides are about 62 cm long, so anything greater than that is off the horn. Click on the graph for a larger view.

Thanks to Benny Leonard for a correction on the explanation for the wavelength of a pedal Bb.