The World record for both speed and height by an air-breathing aircraft (not a rocket) was 85,135 feet. It was set in an SR-71 Blackbird in 1976. The speed record is also held by an SR-71, at 2,193 mph. This is not as high or as fast as the airplane can fly, however, it's absolute speed and altitude limits are classified.
Most US military aircraft can exceed 50,000 feet, if they really really try. Again, the limits are classified. Most commercial jetliners cruise somewhere between 30,000 and 45,000 feet above mean sea level. At higher speeds and altitudes, there isn't enough oxygen in the air to continuously burn the jet fuel required to stay up there.
Engines designed to work very well that high, have serious limitations when they are operated closer to the surface. There are aircraft that have flown higher and faster (the X-15) but they really aren't aircraft, they are rockets, because they carry their own source of oxygen, instead of using the air. However, the fastest and highest airplanes are the American SR-71 Blackbirds.
As aircraft climb to higher altitudes, the air outside gets thinner. The actual equation for the change in pressure with altitude gives an exponential rather than linear decay. This is essentially because air is squashable so is compressed together at lower altitudes.
Therefore pressure is decreased more rapidly near the ground than at higher altitudes. It is approximately halved by 18,000ft. Airplanes are pressurised to maintain a comfortable living environment for human beings.
Ideally an airplane would be pressurised to ground level pressure, but this is not practical as the fuselage of a plane would have to be incredibly strong (and hence very heavy and expensive to fly) to withstand the outward force caused by the high pressure inside.
Therefore modern commercial jets compromise and are pressurised to an altitude of 5000-8000ft. This explains the phenomenon of 'ears popping' on take-off and landing as the air trapped in the ear is effected by the change in pressure from ground level to this effective altitude.
A typical commercial jet (most standard flights) cruises at around 28-35,000ft (up to 6.6 miles of altitude). The main exception is Concorde which was designed to fly at a higher altitude (and hence lower wind resistance) at around 45,000ft. Although many jets could fly at higher altitudes, they are usually certified to an altitude giving a wide safety margin. For example the new generation Boeing 737 is certified to 41,000ft (7.8 miles).
Many military jet aircraft are able to fly considerably higher. Often, the plane itself is not pressurised and instead the pilot wears a pressure suit that provides him with a pressurised environment.
Originally, very high altitude military planes were used for surveillance ½ the current versions of the famous U2 spy plane, originally designed in the 1950s can cruise at up to 90,000ft (17 miles). The Stealth Bomber cruises at up to 50,000ft (8.3 miles) and many other combat planes can now also attain significant altitudes.
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