Why Do All Objects Fall At The Same Rate?

I can bet that when asked if “heavier” objects fall faster than “lighter” objects, the majority of people will say “yes, of course they do!”. However, a small, physics-interested minority will say “ha, good try, all objects fall at the same rate”. While the latter is correct, I wouldn’t blame anyone for saying the first, because that’s what seems to be the case in everyday life. Well, I’m now going to explain why that’s wrong!

Constant acceleration

When objects fall, they don’t fall at a constant speed. If you simply let go of a ball from a height and it maintained a constant speed, it wouldn’t go anywhere as its initial speed is zero. Thus, it accelerates (its speed increases with time). And so what we see when we drop two objects from the same height and one hits the ground before the other, is that the one which lands first has a higher acceleration.

Let’s imagine that in the diagram above that the grey ball on the left is a bowling ball and the brown/orange ball on the right is a basketball. Most would describe the bowling ball as feeling “heavier”. But what that means is that it has a greater weight, which is the force acting down due to the gravitational pull of the Earth. And the reason it has a greater weight is because it has a greater mass. Weight changes depending on where you are in the universe — it is just the force that is felt. The mass is a constant property of the object. As one might expect, mass and weight are proportional. To turn this into an equation, we add a constant of proportionality. On the surface of Earth, this is called the acceleration due to gravity, usually written as g. So, we can write W = mg. Simple enough.

So we know that gravitational force (weight) depends on mass. But there’s something else that depends on mass: acceleration. Now let’s imagine a mass being pushed by a single force horizontally on a flat surface. From Newton’s second law, we can say F = ma (force = mass x acceleration). Here we can see that for a given force acting on an object, mass and acceleration are inversely proportional to one another.

So, if we take F = ma, and now let the force be the force due to gravity, W, we can write W = ma. And we know W = mg, meaning that mg = ma. The masses cancel leaving g = a. In words, that means that in the case of gravity, acceleration does not depend on anything… it is constant. In other words, “heavier” things have a greater gravitational force AND heavier things have a lower acceleration. These two effects exactly cancel out meaning that regardless of the mass of an object (how heavy it feels), it will fall at the exact same rate. As mentioned, on the surface of the Earth this acceleration is g, which is approximately equal to 9.81 metres per second per second (ms^-2).

Air resistance

So why don’t all objects fall at the same rate in daily life? When I accidentally push a glass off the side of the kitchen table, it definitely hits the ground harder than the shopping list! Well, remember when I said earlier about a single force acting on an object? Turns out that was quite an important word. All that I described above assumes that the gravitational force acting on an object is the only force. In a vacuum, such as space, this is the case. But on Earth, when an object is dropped, there is also the force of the air particles it collides with on the way down. This is known as air resistance or “drag”. This resistance changes based on how fast an object is moving, meaning it increases as the object’s speed increasing until an equilibrium is found: a terminal velocity. That’s why you’ll eventually hit a constant speed during a skydive! Not only that but the resistive force is affected by the shape of the object. The more streamlined an object, the less air resistance. That’s why fighter jets are as narrow as possible! And that is exactly why different objects fall at different rates: it’s all because of their shapes!

If you image a feather and bowling ball as in the image above. Even though their downwards acceleration is the same, the ball is clearly more streamlined, and so feels far less air resistance, meaning it has a smaller force counteracting its acceleration and it accelerates faster. Remove the air (a vacuum), and neither have to deal with resistive forces, so they’ll fall at the same rate!

Conclusion

So, the only reason why objects fall at different rates is because of their shapes! You can go and test it for yourself: get two identical bottles (of shampoo, for example). Ensure one is full and one is empty (one feels heavier than another). Now lift them up and drop them from the same height. You’ll find they fall at the exact same rate. Or you could test the opposite effect. Take two pieces of identical paper. Now scrunch one up into a ball and drop them from the same height. The scrunched-up one will fall faster, even though they feel as “heavy” as each other.

Don’t believe me? Here’s proof:

Originally published at http://thephysicsfootprint.wordpress.com on December 30, 2021.