Thursday, April 26, 2012

Airplanes That Seem to Hover

Last year I posted an article about the fundamental principles behind how a subsonic aircraft works.  In summary, an airplane takes off once a critical speed is reached: the greater the speed of the wings relative to the air being cut by them, the greater the pressure difference between the air on the top and bottom of the wings, and the greater the lift force.

Now, I would like to address a particular concern about airplanes that some of my students have had.  Have you ever looked up at an airplane, and had the impression that it was barely moving?

Helicopters, hot air balloons, and some supersonic aircrafts possess the ability to hover, but subsonic aircrafts do not.  As already mentioned, the only reason that a commercial plane can maintain a given altitude is because it is moving horizontally with respect to the air around it.

So, what is going on here?  I too have looked up at aircrafts as they descend and have remarked, on occasion, that the plane seems to be moving at a snail's pace.  Sometimes an airplane flies above me as I cruise along the highway, and my impression is that I am in fact moving faster than it relative to the ground.  Is it an optical illusion?  Let us investigate.

Whenever we observe airplanes from the ground, they are always either taking off or landing.  Airplanes typically cruise in the area of ten kilometers above the Earth, and at that distance, we do not try to assess their speed.  It turns out that airplanes do travel much faster while they cruise than they do during ascent or descent.

Jet aircrafts are designed to cruise at a high speed for an obvious reason: to travel great distances in reasonably small times.  Typically, a jet aircraft will cruise in the range of 700 - 900 km/h.  The particular wing cross-section design for a given aircraft determines its cruising speed at a given altitude, which can then fluctuate a bit (for example, more cargo requires a higher cruising speed).  Note that lift is also proportional to air density, which is substantially lower at 30,000 ft than it is at sea level.

It is not desirable to move at a cruising velocity during takeoff or landing for a number of reasons.  A slow approach allows any deviations from the desired course (due to high winds, say) to be handled safely.  Also, the tires would not appreciate rolling along at 800 km/h supporting a five hundred ton vessel.

During both takeoff and landing, airplanes usually travel around 200 km/h.  It is important to note that just as in cruising, there is no vertical acceleration going on.  During ascent, there is a roughly constant upward velocity, and during descent, a roughly constant downward one.  So, as was the case for cruising, the lift force is simply equal to the weight of the plane while taking off and landing (and, of course, the weight of the plane is not changing).  How then is the lift force equivalent during takeoff and landing at about 200 km/h as it is when cruising at 800 km/h?

One reason is the aforementioned higher air density near the ground.  The other is due to wing flaps, which are activated during takeoff and landing.  When the flap is raised, it increases the coefficient of lift associated with the airplane significantly.  So, on account of both higher air density and a greater lift coefficient, you get the same bang for fewer bucks (equal pressure gradient causing equal lift for less relative velocity).

Clearly we are not entirely wrong when we discern that an airplane is travelling rather slowly.  Whenever we observe it, it is moving at about 25% of the value that it cruises at.  Still, it is moving faster than most any land vehicle typically does: about twice as fast as a car on the highway.  Yet, there are times when I look up at such planes and estimate them to be moving at 40 km/h.  There is no question that I am wrong, but what is the source of the optical illusion?  It has to do with distance.

People are good at estimating the size of things that are near them.  The angle swept by our eyes from end to end of an object yields a very accurate measurement for their length.  The accuracy of this measurement decreases significantly as objects move far away.  When objects are very far away from our eyes, the only way in which we can estimate their size with reasonable accuracy is if other objects of known size are near them.

When we observe an airplane, we improperly estimate its size as well as its distance from us.  Furthermore, if the plane is moving away from or towards us, the only way we can try to estimate its speed is by observing the rate at which its cross-section appears to be changing.  Such an exercise turns out to be futile.

If you wish to estimate the speed of an airplane with decent accuracy, independent of its direction of travel with respect to you, you may like to try the following technique...

Pick any fixed object between you and the flight path of the airplane (for example, electric power lines).  Then, with a timer, measure the amount of time that the plane takes to cross that fixed point - that is, start your timer when the nose meets the fixed point, and stop it when the tail leaves it.  Then, simply take the plane's full length in meters (nose to tail) and divide it by your recorded time in seconds.  You'll have measured its speed in m/s.  Multiply by 3.6 for km/h.

If the commercial plane is a 'big one', like a Boeing 747, then its length is about 70 m.  A smaller airplane, with a 100 passenger capacity, is about half that length (35 m).  It is here that our eyes cause us to underestimate airplane speed.  The length of a typical airplane spans half that of a football field, and yet our eyes see them as winged buses moving across the sky.

I used my technique with a mid-sized plane that my senses judged to be moving at 40 km/h.  It crossed a power line in 0.86 seconds (with some error of course).  If it was indeed about 50 m in length, then its speed was (50 m / 0.86 s) about 58.1 m/s or 209 km/h, which is typical for an aircraft on final approach.  If the airplane were only 10 m in length (like a bus), then its speed would have been calculated to be 42 km/h.  This is the calculation our senses make when we misjudge airplanes to be moving at a crawl.

The next time you perceive a plane to be moving oddly slowly, remember that it is as long as a plane, and not a bus.  When it covers its own length in under a second, imagine a football player trying to run half the length of the field in under a second.

In life, we make quick physical measurements on a regular basis.  And, as our assessments are often reasonably accurate, we learn to trust our senses.  The case of the 'hovering airplane' reminds us that our senses are limited.  Favoring reason over gut feeling empowers us to deduce beyond our limited senses and avoid being fooled.

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