Do Rocks and Feathers Really Fall at the Same Speed? The Answer May Astonish You.

Recently I’ve found myself revisiting some of the basics of physics during my longer term goal of putting a lid on my understanding of quantum mechanics. I went back to study Newton, Lagrange, and Hamilton. And something struck out to me as obvious but unaddressed in pop science. The original force equals mass times acceleration (f = m × a) equation is typically proof by definition that all objects, regardless of mass, fall at the same speed. Why wouldn’t they? In a gravitational field acceleration is constant and independent of mass. If it wasn’t then the equation would read something like, f = m × a(m). And whether its the Royal Institute or Bill Nye the Science Guy no one ever adds qualifications to the proclamation that feathers and rocks fall at the same speed in vacuum.

However, f = m × a is not always appropriate for objects in a gravitational potential. For more accuracy you want to use Newton’s Law of Universal Gravitation,

It’s nearly the same and indeed substituting the mass and radius of earth results in the original f = m × a. However, what we typically fail to account for is the removal of mass #2 from mass #1. Let me explain. Mass doesn’t just appear. If you are performing the rock and feather experiment on earth you’re removing those rocks and feathers from mass #1 and assigning them to mass #2. Conversely, if your rocks and feathers came from planet Vulcan then you’re creating mass #2 from scratch. Are you bringing the rock, performing the test, and then returning the rock? Or are you bringing the rock and the feather at the same time? The list of experiments gets quite long.

Lets say for simplicity that you’re not getting anything from planet Vulcan. All your materials for the rock and feather experiment are coming from earth. This leaves two different classes of experiment. Experiment class #1 where you lift the feather first, for example, run your drop test, and then lift the rock and run your drop test. Experiment class #2 involves lifting the rock and feather at the same time.

To the astute, you may notice that these are not actually the same type of experiment. The first one takes place in two parts. Part A is a feather drop test and mass #1 contains the earth and the rock. Part B is a rock drop test and mass #1 contains the earth and a feather. Experiment class #2 isn’t even a two body problem any more. Now you have three bodies free-falling in vacuum. We’ll come back to that later because it involves some trigonometry.

In the simplest case of experiment class #1 and the sequential drop tests we see that mass #1 differs between tests which results in a different acceleration for the rock and the feather meaning that they do not fall at the same speed. So, lets run the numbers and see just how different it actually is.

Using reasonable values for the earth, the rock, and the feather we calculate the time to fall 10 meters above the surface of earth (6,378.1 kilometers radius). Most calculators won’t handle this equation because there’s not enough significant digits. However, Matlab contains a feature called Variable Precision Arithmetic (VPA) which allows for arbitrary precision. With our new tool we find that the rock and feather fall at different speeds, in fact about 0.12 yoctoseconds different. But, what is absolutely mind blowing is that the feather actually falls faster!

It’s such a non intuitive result that it takes a second look to believe it. But indeed the feather drops faster than the rock because it is also being attracted by the rock.

So, you might say, well that’s not fair, the experiment always assumed that the rock and feather would fall at the same time. Goodie. Lets take a look.

If both the rock and the feather are dropped at the same time then they cannot be dropped in the same position. By necessity they’ll be dropped at different positions on planet earth. If somehow they could be dropped at the same position then yes, they would fall at the same speed. But, because they are not dropped at the same position this is now a three-body problem instead of a two. If the rock feather interaction was orthogonal to the gravity vector then the rock and feather would fall at the same speed. Indeed it may seem like if the rock and feather are dropped on opposite sides of a table that the rock feather interaction is perfectly orthogonal to the earths gravitational vector. But that is not the case. The gravitational vector points towards the center of earth and at different points on the table there is a measurable angle between the gravity vectors. So, the feather is being pulled to the ground not just by the mass of the earth but also by downward component of the rocks gravitational attraction. Once again, the feather beats the rock to the ground.