Rate of Descent

The triangle of forces is geometrically similar to the glide triangle. You can see this by drawing the glide triangle, as above, and plotting the lift, drag and weight forces acting on a point along the hypotenuse. Pick any point about midway along the hypotenuse of the glide triangle. This point represents the glider’s position at some time. In our example we know that the drag force is one unit long for every five units of the lift force. Draw an arrow 1/4″ long pointing uphill along the glide path from the point. This represents the drag force. Draw an arrow perpendicular to the hypotenuse, pointing upwards, 5 times as long as the drag arrow, or 11/4″ long. This arrow represents the lift force. These two arrows form two sides of a rectangle. Carefully draw in the other two sides of that rectangle. Now draw in the diagonal of the rectangle that passes through the original point. This diagonal will be vertical.

This diagonal represents the total aerodynamic force that acts on the glider and it is equal and opposite to the weight force in a steady glide. The triangles formed by the diagonal within the rectangle are geometrically similar to the glide triangle. Both triangles have the same angles. Corresponding sides are in the same ratio to each other. In particular, the lift (long side of force triangle) is to the weight (hypotenuse of force triangle) as the horizontal distance covered (long side of flight path triangle) is to the flight path length (hypotenuse of flight path triangle). Let’s assume the glider weighs 10.2 grams.

If the 10.2 grams of the glider’s weight corresponds to the 5.1 inches of the scale drawing of the flight path, then the lift must be 10 grams to correspond to the 5 inches of the scale drawing of the horizontal distance covered. The ratio is two to one.

This can be expressed algebraically, and it can be expressed in terms of the original measured distances. We also know that the drag is one fifth of the lift, so it must be 2 grams. This establishes a scale for our force diagram of 2 grams per quarter inch or 8 grams per inch. The scale force diagram is known as a vector diagram. The arrows representing the forces are known as vectors and weight is the vector sum of lift and drag. A vector is a mathematical entity that has both magnitude and direction, just like forces.

 Gravitational Energy of Glider

A glider uses gravitational energy to fly. Gravity doesn’t just pull the glider down, it also pulls it against air resistance. Gravity powers the glider. From the information we have, we can calculate the energy and power provided by gravity.

 Definition of Work

When you lift a weight, you do work. When you lift twice as much weight, you do twice as much work. When you lift the weight twice as far, you do twice as much work. Work is energy. Energy has units of a force (weight) times a distance (height). The energy that the glider had at the top of its path, relative to the ground, is the weight of the glider times its height above the floor.

That’s how much energy it would take to lift it that high and that’s how much energy it takes for it to fly against air resistance on the way down.

Energy Units and Conversion

Energy may be expressed in different units, depending on what units were used to measure the weight and the distance.

In the example above we had a 10.2 gram glider launched 4 feet above the floor. This would result in an energy value of 10.2 x 4 = 40.8 gram feet. It is more usual to express energy in foot pounds. There are 453.59 grams in a pound, so we must divide 40.8 gram feet by 453.59 grams per pound to get 0.0899 foot pounds of energy. Gravity expended that much energy to bring the glider down to the floor.

Calculating Horsepower

Power is the rate at which energy is expended. We know how long it took for that 0.0899 foot pounds of energy to be expended because we timed the flight. Let’s assume the flight lasted 3 seconds. Then we know that energy was expended at a rate of 0.0889 foot pounds in 3 seconds, or 0.0899 / 3 = 0.0300 foot pounds per second. It is more usual to hear power expressed in horse power. One horse power is 550 foot pounds per second, so we must divide 0.0300 foot pounds per second by 550 foot pounds per second per horsepower to get 0.0000545 horsepower.

Gravity pulled the glider against air resistance. We measured the air resistance as 2 grams of drag. We also measured the distance the glider moved through the air against that drag, 20.4 feet. The amount of energy expended by gravity pulling the glider against the drag is the force times the distance, or 2 grams x 20.4 feet = 40.8 gram feet, the same as we got before.

This took the same 3 seconds, so the horsepower is also the same when calculated this way. The interesting thing about this way of calculating the power is that we multiplied the drag force by the length of the flight path and divided by the time of flight.

But the length of the flight path divided by the time of flight is the glider’s airspeed. This tells us that another way to calculate the power of a glider is to multiply its’ drag by its’ airspeed.

Science and Math Lessons from Simple Gliders

This illustrates how simple gliders can motivate a discussion of measurement, length, time, speed, unit conversion, ratio, angle, relative speed, piloting, scale, scale drawing, graphical solution, right triangles, angle tangent, Pythagorean theorem, weight, force, force diagram, vectors, vector addition, geometrical similarity, energy, rate and power. It also illustrates how scientific concepts begin with simple operational definitions in measurement processes and are elaborated through geometrical and algebraic manipulation. You can draw a flow diagram showing how the measured quantities lead to the derived quantities.

With four simple measuring tools, a tape, a protractor, a stopwatch and a balance, we have measured four quantities and derived ten. We have demonstrated numerical and graphical solution methods.

To Build Simple Gliders From Foam Plates & Straws check article on this website.