###### By Don Lincoln Ph.D., University of Notre Dame

## Bernoulli’s equation is based on energy conservation and fluid movement. It is widely used to explain how planes fly: “The air pressure under the wing is higher than the pressure above the wing since the speed there is higher. The difference of pressure creates the lift needed to push the plane off the ground, and the plane flies.” Right? No, very wrong.

Normally, when teachers and even many scientists are asked how planes fly, they use Bernoulli’s equation. Bernoulli’s equation was developed in 1738 by Swiss physicist Daniel Bernoulli. It was rewritten, in 1752, by a fellow Swiss physicist Leonhard Euler working on energy conservation, in the form used today. The equation focuses on fluid movement, using pressure and speed. According to Bernoulli’s equation, as the speed of fluid (gas or liquid) increases, its pressure decreases. Is this what happens around the wings of a plane?

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### Types of Energy Involved in Flying

Both kinetic (moving) and potential (non-moving) energy are involved when a plane starts to move. The potential energy here is of two types: the lifting kind and the pressure kind. If symbols were used to explain the movement and numbers, the results would be as follows:

Kinetic energy is half the mass of an object multiplied by its velocity squared, i.e., one-half mv-squared. Lifting potential energy is mass times gravity times the height objects are lifted, i.e., mgh. The pressure potential energy is pressure times a change in volume; i.e., P times V. These are the three components of energy.

The energy conservation rules apply as well. Hence, the energy before the plane starts to fly, is the same as the energy when it is flying. The types of energy change, but the sum does not. The next item in the equation is mass.

If electrons want to move in a wire, they have billions of barriers ahead: other atoms and electrons. For example, a piece of copper wire is a string of copper atoms sitting in place and not moving that much. Some electrons around each atom can move around freely and jump from one atom to the next. These electrons make the electrical current happen.

When a battery is inserted in a circuit or a lamp is switched on, the electron moves forward but immediately gets hit by the next atom and is deflected. This keeps happening, so the electron cannot move forward easily. Thus, the speed of an electron in a typical household wire is less than a tenth of a millimeter per second. In other words, it takes ten seconds for an electron to move one millimeter.

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### Mass Involved in How Planes Fly

A plane flies through the air, and the mass of air is hard to know. Therefore, it is easier to consider a specific volume of air, called V. With the specific volume, comes a specific mass, hence, a density by dividing the two. Density is denoted by the Greek letter rho. If the energy equations are divided by volume, similar equations will result, with P instead of PV.

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### Applying Bernoulli’s Equation to Flying

In the case of how planes fly, the point where energy should be calculated before and after movement is the wings: flat, huge surfaces, sometimes thicker on the front and thinner towards the back. The top of the wing is slightly curved, and the bottom is flat, and these are the two points Bernoulli’s equation has to calculate energy between.

As the wing is thin and the air density is low, the rho-gh term can be ignored. The air that flows from the top of the wing has to meet the wind from the bottom when it hits the end of the wing. The air should flow faster on top, and the pressure should be lower, as the distance is more compared to the bottom, due to the curve. Pressure is force divided by area, or force equals area (here of the wings) times pressure. This difference of pressure causes the force to lift the plane. Or does it?

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### Where Bernoulli’s Equation Cannot Fit

If the difference of pressure causes the force to fly, no plane should be able to fly upside down, but stunt planes do. If a Cessna plane is considered, the equation will have the following numbers:

The speed is about 220 kilometers per hour, the weight 1100 kilograms, the wings about 30 feet long and 6 feet wide, with a total area of about 18 square meters. The length difference of the wing’s top and the bottom is about 3%. The air’s density can be considered as rho: 1.2 kilograms per meter cubed. The velocity is about 60 meters per second on the bottom and 62 meters per second on the top. The force 146 Pascals, or Newtons per meter squared, which is the unit of pressure. Thus, the upward force to the wing is 2628 Newtons, and the weight (downward force) is 10,780 Newtons.

There is a huge lack of force to lift the plane when the equation is used. Consequently, it cannot explain how planes fly.

### Common Questions about How Planes Fly

**Q: Can an airplane stay in the air without moving?**

To explain how planes fly, one must first know the forces that cause the movement. The weight of the plane pushes it down, and lift pushes it up. If these two are exactly equal, and the drag and thrust are also equal, the forces in both directions cancel each other, and a plane can stay motionless in the air. However, this is not a realistic situation.

**Q: Why do airplanes fly so high?**

Air resistance has a leading role in how planes fly. The optimum air resistance is at 35,000 feet, which is the altitude commercial airplanes fly. The reason is that planes have minimal fuel expenditure at this height.

**Q: How do planes gain altitude?**

When a plane gains enough velocity on the ground, the wing shape and angle will create enough lift with the air resistance, and the plane gets lifted. Similar to the force one feels when an arm is stretched outside a speeding car’s window, the air pushes the plane, and the wings keep pushing back; hence, enough lift is created, and that is how planes fly and move up.

**Q: What happens if an airplane goes too high?**

The air density is an important factor in how planes fly and how much fuel they need: the thinner the air, the less fuel the plane needs. However, if the density gets lower than a specific amount, the plane will fall due to a lack of lift force.