Aviation Magic

Consider an aircraft flying a clockwise circle at 100 knots airspeed at a constant altitude within an air mass that has a wind speed relative to the ground of 25 knots from 270 degrees. The pilot observes, relative to the air, that nothing changes except the aircraft’s direction of travel.

An object’s kinetic energy is the product of ½ its mass and its squared speed. Kinetic energy is proportional to squared speed.

From the pilot’s perspective, the aircraft’s speed is constant at 100 knots, hence its kinetic energy does not change and is proportional to 100*100 = 10,000.

From a ground observer’s perspective, the aircraft’s speed is 125 knots when its heading is 90 degrees and 75 knots when its heading is 270 degrees. He figures the aircraft’s kinetic energy is proportional to 125*125 = 15,625 on a heading of 90 degrees and 75*75 = 5,625 on a heading of 270 degrees. According to him, the aircraft’s kinetic energy drops by (5625-15625)/15625 = 64% during the turn from 90 degrees to 270 degrees. 64% of the aircraft’s kinetic energy has vanished – where did it go? The ground observer also sees the aircraft’s kinetic energy increase by (15625-5625)/5625 = 178% during the turn from 270 degrees to 90 degrees. Where did this kinetic energy come from?

For every full circle of the aircraft from 90 degrees to 270 degrees and then from 270 degrees to 90 degrees, the ground observer sees the aircraft’s percentage kinetic energy change by -64% followed by +178%. This is an average change in the aircraft’s kinetic energy of (-64+178)/2 = +57%. This corresponds to a net gain of 57% in the aircraft’s squared speed, e.g., from 100*100 to 100*100+0.57*100*100 = 1.57*100*100 = (1.25*100)*(1.25*100), or a gain of 25% in its speed.[1] For example, from 100 knots to 125 knots on the first circle. Evidently, all it takes to achieve high ground speeds when there is a wind is a few circles before setting out on your desired heading.

[1] A gain of 57% is an increase by a factor of (1+0.57) = 1.57. The square root of 1.57 is 1.25.