You're a marine biologist concerned with the effect of sonic booms on plankton, and you need to estimate the altitude of a supersonic aircraft flying directly over you at 2.2 times the speed of sound. You hear its sonic boom 19 s later. Assuming a constant \(340 \mathrm{m} / \mathrm{s}\) sound speed, find the plane's altitude.

The speed of sound, an essential factor in the context of sonic booms, refers to the rate at which sound waves travel through a medium. It varies with the medium's properties such as temperature, density, and state (solid, liquid, or gas). In the case of our marine biologist, the medium is air, and the speed of sound at sea level and at a temperature around 20°C (68°F) is approximately 340 meters per second (m/s).

This value is crucial when calculating how far away something is based on the length of time it takes for the sound to reach us. For the observer on the ground, the time delay between seeing a plane pass overhead and hearing its sonic boom can be used to calculate the aircraft's altitude, assuming the sound speed is known.

Supersonic flight occurs when an aircraft travels faster than the speed of sound. When aircraft exceeds this threshold, it begins to produce a sonic boom, a loud and distinctive sound that can be heard by observers below. This boom is the result of shock waves created by the aircraft as it pushes through the air at speeds greater than sound can travel.

The term 'Mach number' is used to describe the ratio of the object's speed to the speed of sound. In our exercise, the aircraft was traveling at 2.2 times the speed of sound, also known as Mach 2.2. This high-speed flight has various implications, from aircraft design to environmental impact, as evidenced by the biologist's concern over the effect on marine life.

The distance-speed-time relationship is a fundamental principle in physics, underpinning many calculations involving motion. The formula relating these three quantities is: \[ \text{Distance} = \text{Speed} \times \text{Time} \]

For the biologist's problem, we can apply this formula to determine the altitude of the plane. We know the speed of sound (340 m/s) and the time the sonic boom took to reach the observer (19 seconds). Multiplying these together gives us the distance the sound traveled, which is the same as the plane's altitude at the time of the boom, because the boom travels downwards from the plane to the observer:
\[ \text{Distance} = 340 \, \text{m/s} \times 19 \, \text{s} = 6460 \, \text{m} \]

It is important to note that it is assumed the plane was flying at a constant altitude with a consistent speed of sound. Variations in either could alter the calculated altitude.