When taking off, why are we required to climb at least 35 feet over the departure end of runway? Why not other numbers? This comes from the FAA regulations. I believe there must a clear explanation for this but can't find one. Is there any historical or mathematical reason?
The best explanation I've seen for the logic behind the 35 foot crossing height requirement can be found in the FAA Instrument Procedures Handbook (IPH). It's available from the FAA website at this link: https://www.faa.gov/regulations_policies/handbooks_manuals/aviation/instrument_procedures_handbook/
In Chapter 1, Departure Procedures, under Design Criteria on page 1-16: "The aircraft climb path assumption provides a minimum of 35 feet of additional obstacle clearance above the required obstacle clearance (ROC), from the DER outward, to absorb variations ranging from the distance of the static source to the landing gear, to differences in establishing the minimum 200 FPNM climb gradient, etc."
So, the 35 feet is in place to account for a multitude of variables. The phrase "differences in establishing the minimum 200 FPMN climb gradient" covers a whole host of potential errors such as variations in individual aircraft performance, early rotation, late rotation, under rotation, over rotation, poor airspeed control, etc. This error proofing is important since the 35 foot requirement is used when building all performance data for go and no-go decisions. That performance data is all based on set assumptions. Reasonable deviations from the assumptions are mitigated (somewhat) by the 35 foot margin.
As with most safety margins, there isn't necessarily a precise reason for the size of the buffer. However, even increasing the crossing height a slight amount, say to 50 feet can have significant effect on performance calculations. At a minimum climb gradient of 200 FPNM, the difference between a screen height of 35 feet and 50 feet is just over 450 feet of required horizontal distance. In a tight performance situation such as high pressure elevation or short runways, a 50 foot crossing height could be too much for some large commercial aircraft.
Of note, the regulation that created the 35 foot crossing height, SR422, came about in the late 50s during the beginning of jet-powered commercial aviation. The aircraft being designed and fielded back then, the Boeing 707 for example, would have had a hard time making a higher restriction. That likely had influence on the required crossing height.
As I understand it, from when I was attending flight engineer school with the Air Force, the screen height has to do with a 50' obstacle. The 35' screen height for FAA TERPSd fields translates to being able to clear a 50' obstacle at 200ft/min providing you are 35' AGL at brick last of the runway.
For Army TERPSd fields the screen height is typically 16' and for Air Force and Navy TERPSd fields the screen height is 0', meaning that if you maintain 200ft/min climb, you can rotate at brick last and then you will clear the 50' obstacle by the required minimum clearance. The 35', 16' and 0' screen heights are assumed if no other screen height is given, and based upon who has evaluated the airport.