Yves Rousseau attempted his first human-muscle-powered flight with flapping wings in 1995. On April 20th, 2006, on his 212th attempt, he succeeded in flying a distance of 64 metres. On his 213th flight attempt, a gust of wind led to a broken wing, causing him to be gravely injured and rendered paraplegic.

After physics class finished, I approached my teacher and asked him how much power is required for an “ideal” helicopter to hover.

He grabbed a piece of paper drew a helicopter, with an arrow pointing down and another point up and said, “Well, clearly the thrust has to be equal to weight of the helicopter.” He then added, “But the helicopter isn’t moving. That means there is no work being done and no power is required”. As he was saying these last words, his face transitioned from confusion to excitement, it was this simple but challenging problem that has awaken his the passion for the field.

Let me show you the equation that changed Yves life, it show that the larger the propeller, less energy is required to generate thrust. (read this great article by Rhett Allain for a detailed proof)

After certain time, a cylinder of air has being pushed down.

Imagine our propeller covers a circle of area A, after a certain time, the propeller has pushed down a cylinder of air of size A * h.

This will generate a reaction F pushing the propeller upwards, and making are helicopter hover.

Newton’s equation:we multiply the mass of air in the cylinder by the acceleration induced on these particles.

Auxiliary equations

We want to substitute the variables in the force with our auxiliary equations to arrive to something that depends on the radius of the propeller and the speed.

My teacher’s answers after 45 minutes.

If you are like me, equations mean nothing until I plot them. Let’s try to answer, “How large does my propeller have to be, and what is the speed of wind required to lift a 1kg helicopter?”

import numpy as np

import matplotlib.pyplot as plt F = 9.8

r = np.arange(0.01, 1.0, 0.01)

v = np.sqrt(F / (0.5 * 1.225 * np.pi )) / r

plt.plot(r, v)

plt.xlabel('radius [m]')

plt.ylabel('air velocity [m/s]')

plt.grid(True)

plt.show()

All the points in the curves are configuration that produce a 1 kilogram force.

The answer is that a propeller of 0.05 cm has to spin fast enough to move the air at 50 m/s and make a kilogram helicopter hover. We can instead choose to use a very large propeller with a radius of 0.5 meters, which will require air moving at 2.7 m/s.

Both options produce the exact same amount of lift. And you read up to here, congratulations! The exciting part is about to start. The amount of power needed is wildly different.