What are the equations to describe the physics of a falling leaf?

Pennsy Dutch: What are the equations to describe the physics of a falling leaf?
When I was in high school in the 1970s, the physics teacher said that the physics of a falling leaf cannot yet be fully described mathematically. Is that still true?
Feel free to be technical.

Answers and Views:

Answer by renzokukenreddragon
Depends what he or she meant by fully. The first equation would be working out how long it takes to reach the ground, and how fast it’s going when it arrives. This is actually going to be a differential equation since the air resistance on the leaf will cause it to reach terminal velocity very quickly, and will have the form:

F = mg – k(dx^2/dt^2).

Terminal velocity will be reached when the net force is zero, i.e. the leaf is not accelerating but falling at a steady speed.

Secondly, you may want to consider how the leaf rotates. This would call for an angular momentum equation. I can’t think why it would be useful to do this, but you could come up with an equation describing the leaf’s rotation by considering the position of its centre of mass. If the leaf was a ball, this would be trivial. The leaf isn’t the same shape all over and the equation gets a bit harder, involving integral calculus.

However, this may have been what your teacher was getting at. It’s unlikely the leaf continues spinning the same way the whole way down in a predictable pattern, unless it’s evolved to do so for some reason (“helicopter” seeds things come to mind). Such random motion would be described as “chaotic” motion, which is the same principle as people saying a butterfly flapping its wings can cause a hurricane on the other side of the planet. The chaotic motion is caused because you can’t predict exactly where an air molcule will strike the leaf and slow it down, nor how often.

Chaotic motion of the leaf can’t really be described with conventional equations where you’d get a perfectly defined result. You can, however, get statistically expected behaviour, which would be more useful.

So no, you can’t *perfectly* describe the falling leaf, but you can come close enough to calculate something useful, like how long it will take an entire tree to go from full bloom to all its leaves on the ground.