Monthly Archives: November 2009

triangle-trigonometry

Simple Geometry and Height Estimation for a Model Plane

A Squirrel can fly very high but it’s not easy to measure it’s height directly.

We can take advantage of geometry in order to estimate the height of a Squirrel (or a tree or a building for that matter).

If we look at the above right angle triangle you can see a way of naming the sides. Angle marked in red is the angle Θ (Pronounced Theta, it is the Greet letter Capital Theta). The side of the triangle that is right next to the angle Θ is called “adjacent”. The one across from it is called “opposite”. And the one that is opposite the right angle (90 degrees) is called “hypotenuse”.

From this there are three basic trigonometry functions involving sine, cosine and tangent:

sinΘ = opposite/hypotenuse

cosΘ = adjacent/hypotenuse

tanΘ = opposite/adjacent

It’s possible to get two observers to form a right angle triangle with the Squirrel as follows.

If the observer on the left is running underneath the Squirrel then the triangle will approximate a right angle triangle.

So it is possible to have the observer on the right (or some other observer) estimate the angle Θ. It is also possible to measure the “adjacent” side of the triangle since it is just the distance between the two observers. We are wanting to know the length of the “opposite” side of the triangle.

If we look at the trig functions above, there is one that includes Θ, “opposite” and “adjacent”. It’s interesting for our problem because it references the two entities we know (“adjacent” and Θ) and the one we don’t know (“opposite”).

tanΘ = opposite/adjacent

Using algebra we can rearrange this equation to issolate the unknown variable:

tanΘ adjacent = opposite (multiply both sides by adjacent)

opposite = tanΘ adjacent (reversing order)

We now have a formula for calculating the height of the Squirrel. We take the tangent of the angle Θ and then multiply it by “adjacent” which is the distance between the two observers.

For example, if the angle observed was 28 degress, and the two observers are 45 meters apart, then we get a height of 0.5317 times 45 meters which is 24 meters.

History of Aviation and Rubber-Power

Orville Wright
Wilbur Wright

Few people are aware that rubber-power aviation was a key motivator or the Wright Brothers.

The Wright Brothers received a simple free flight rubber-powered aircraft when they were 8 and 12 years old. This fuelled their fascination for flight and they eventually embarked on a journey resulting in the invention of man made flight.

This may seem surprising but its true. In 1871, French scientist Alphonse Penaud astounded members of the French Academy of Sciences by flying a rubber-powered aircraft he called a planaphore for 131 feet. It was the first recorded flight of an inherently stable, heavier than air aircraft.

For the next 50 years, rubber-powered (then called torsion-powered) airplanes were a key research tool for aerodynamic engineers. It allowed them to test numerous configurations of flying surfaces for airworthiness without having to build full-size airplanes.

Wilbur Wright 1902.

Torsion-powered aircraft also became a popular toy in the late 1800s. After Penaud’s demonstration, toy makers immediately started to create rubber-powered flying toys. One of the most popular of these toys was a torsion-powered helicopter. In 1878, Bishop Milton Wright brought this toy home to his sons Wilbur (age 12) and Orville (age 8) and started them dreaming of flight. It soon wore out, but they made copy after copy. They were still making copies to delight their nieces and nephews in 1903 just before they made their first powered flights in a real airplane.

Here are a few failed attempts at flight. You can see that Wilbur and Orville were superior problem solvers.

Elastic Rubber Band Facts. Amazing But True!

A rubber band is an extremely capable engine for an aircraft, and the science of torsion motors for model airplanes has progressed at the same amazing pace as gasoline and jet engines in real airplanes.

In 1909, as Wilbur and Orville Wright were coming home from a triumphant tour of Europe, the American record for distance flown with a rubber band-powered airplane was just over 200 feet.

In 1916, as World War 1 was at its height, Thomas Hall if the Illinois Model Aero Club flew a model 5337 feet — over a mile!

In 1924, just a few years before Lindbergh flew the Atlantic, Robert V. Jaros (also from the Illinois Model Aero Club) flew a model 7920 feet in 10 minutes and 14 seconds.

Today, model airplanes in the competitive F1D class can fly for more than 40 minutes