Energy cost of flying

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Gliders and planes glide similarly. How similar?

Big Ideas: 
  • By observing how an aircraft glides, you can infer how much
    power is required to keep it flying horizontally. This can be done for a
    small model glider, and any full-sized aircraft.

http://www.youtube.com/watch?v=0UpPjAst5mw&feature=channel_video_title

Glider_final9.avi (right-click and choose "Save Link As..." to download to your computer)

Physics of gliding

All aircraft, from a small hand-thrown glider to a massive Boeing 747 act the same way when they are gliding. Planes use their potential energy to overcome the air drag and the speed creates lift to counteract its weight. As the plane descends at a constant speed, the potential energy is being lost at a constant rate, which can be measured by observing the glide.

Consider the instantaneous potential energy (PE) of a plane gliding at constant speed and angle, at a height h above the ground:

PE = mgh

where m = mass of the plane, and g = constant of gravity.

The glide path is shown by the inclined line, of length x. The angle θ is that of the glide path with respect to the horizontal. Therefore the total PE at the start of the glide is:

PE = mgh = mgxsin(θ)

Consider a flight segment of length Δx, the change in potential energy is given by:

Δ(PE) = mg Δx sin(θ)

If it takes a time Δt to fly Δx, then the power P required to maintain a constant speed v = Δx/Δt is the rate at which PE is lost:

P = Δ(PE)/Δt = mg (Δx/Δt) sin(θ) = mgvsin(θ)

Now you can make your own glider, and observe that when the glide slope is constant (i.e. θ is constant), the velocity is constant as well. Since the mass and gravity don't change, we can assume that the power,  mgvsin(θ), is also constant.

Make a glider at home

If you want to make a glider at home, you only need to get a 2-mm thick styrofoam sheet, and draw the following shapes:

Afterwards, you cut the shapes and glue the bottom part to the middle of the top part with wood glue. The glider should look similar to this:

Once glued together, let it dry for as long as necessary and attach one paperclip on each side of the nose (for a total of two) with tape. These paperclips are to give stability to the glider so it can fly without stalling.

Once you are finished with this, you can practice throwing the glider and making adjustments to the shape of the wings (as shown below) until you get a smooth and straight flight. See the video in the article on flying.

 

Power of a small hand thrown glider

If you want to calculate the approximate power required to keep this glider flying at a constant speed, you only need to measure its mass, angle of glide slope, and velocity when it glides. It is worth mentioning that this power is the change in the glider's potential energy per time (measured in Joules per second). To do this you can use an ordinary kitchen scale to get the mass, and record the glider flying with a camera that is positioned horizontally and looking from the side to the glider's path. From the video, you can measure the angle the gliding path makes with the horizontal, as well as calculate the velocity from the distance and time the glider travels in the video.

For reference, the glider we made has a mass of approximately 6.1 grams (6.1x10-3 kg), its velocity was around 1.6 m/s, and the glide angle was about 14°.

If we calculate the power with the equation P = mgvsin(θ) = (6.1x10-3 kg)(9.81 m/s2)(1.6 m/s)(sin(14°)) = 0.0232 watts = 23.2 mW. This explains that in order to maintain this glider flying horizontally, we would need a motor with a power of 23.2 mW.

With this we can show how easy is to determine the power a glider has when is thrown. These calculations also work for large jetliners.

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