Fall Is Here! Time to Learn the Physics of... Falling

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Take a pencil,stretch out your arm and let go. We all know that the pencil will fall. OK, but what about dropping a bowling ball? Is that the same thing? No wait! How about a watermelon dropped off a tall building? Why would you do that? I would do it to see it splat. Or maybe even more extreme, a human jumping out of an airplane. These examples could all be considered "falling," but not every fall is the same.

So let's get to this. Here is all the physics you need to know about falling things. Hold on to your seats. This is probably going to be more than you asked for. Don't worry, the math will (mostly) be at a simple level.

Falling Without Air Resistance

I'm going to talk about air resistance down below. However, I want to start with the simplest case of an object falling near the surface of the Earth that has a negligible air resistance force. Really, this simplification isn't just approximately true in many cases, it's also one of the key components of the nature of science. If we want to build a scientific model (science is all about building models) the best bet is to start off with something without extra complications. If you want to model a mass on a spring, assume the spring is massless. If you want to model a cow, you have to assume it's a sphere (mandatory spherical cow joke). These simplifications are the first step to building more complicated models.

Is Gravity Constant?

This is one thing that comes up quite a bit. People say that if you drop two objects of different mass, they have the same gravity. OK, the first problem is the word "gravity"—what does that mean? It can mean many different things. The two most common meanings are: the gravitational force or the gravitational field.

Let's start with the gravitational field. This is a measure of the gravitational effect due to an object with mass. Since the gravitational interaction is force between two masses, you can think of this as "half" of that interaction (with just one mass). If you have an object near the surface of the Earth, then that object will have a gravitational interaction depending on the Earth's gravitational field. Near the surface of the Earth, the gravitational field is represented by the symbol "g" and has a value of about 9.8 Newtons per kilogram.

No. The value of "g" is not the acceleration due to gravity. Yes, it is true that 9.8 N/kg has the equivalent units of meters per second squared. It is also true that a free falling (no air resistance) object falls with an acceleration of 9.8 m/s2—but it's still just the gravitational field. It doesn't matter what object you put near the surface of the Earth, the gravitational field due to the Earth is constant and pointing towards the center of the Earth. Note: it's not actually constant. More on that below.

What about the gravitational force? Here is a picture of two objects with different mass.

If you hold these two objects up, it should be clear that the gravitational force pulling down is not the same. The big rock has a bigger mass and a bigger gravitational force. That small metal ball has a much, MUCH smaller mass and also a much smaller gravitational force.

Yes, the gravitational force is also called the weight—those are the same things. But the mass is not the same as weight. Mass is a measure of how much "stuff" is in an object and weight is the gravitational force. Now to connect it all together. Here is the relationship between mass, weight, and gravitational field.

Technically, this should be a vector equation—but I'm trying to keep it simple. However, you can see that since g is constant an increase in mass increases the weight.

Force and Acceleration

OK, so you drop an object with mass. Once you let go, there is only one force acting on it—the gravitational force. What happens to an object with a force acting on it? The answer is that it accelerates. Oh, I know what you are thinking. You want to say that "it just falls" and maybe it falls fairly fast. That isn't completely wrong—but if you were to measure it carefully, you would see that it actually accelerates. That means that the objects downward speed increases with time.

Let's forget about falling object for a moment. What about a small car on a horizontal frictionless track with a fan pushing it? Like this:

If I turn on the fan and release the car, it accelerates. There are two ways I can change the acceleration of this car. I could increase the force from the fan or I could decrease the mass. With just a single force on an object in one dimension, I can write the following relationship.

This is what a force (or a net force) does to an object—it makes it accelerate. Please don't say forces make objects move. "Move" is a four letter word (that means it's bad). Saying an object "moves" isn't wrong, but it doesn't really give enough of a description. Let's just stick with saying the object accelerates.

There are many, many more things that could be said about force and motion but this is enough for now.

Why do objects fall at the same time?

Now we can put together a bunch of stuff to explain falling objects. If you drop a bowling ball and a basketball from the same height, they will hit the ground at the same time. Oh, just in case you don't have ball experience—the bowling ball is MUCH more massive than the basketball.

Maybe they hit the ground at the same time because they have the same gravitational force on them? Nope. First, they can't have the same gravitational force because they have different masses (see above). Second, let's assume that these two balls have the same force. With the same force, the less massive one will have a greater acceleration based on the force-motion model above.

Here, you can see this with two fan carts. The closer one has a greater mass but the forces from the fans are the same. In the end, the less massive one "wins".

No, the two objects with different mass hit the ground at the same time because they have different forces. If we put together the definition of the gravitational force (on the surface of the Earth) and the force-motion model, we get this:

Since both the acceleration AND the gravitational depend on the mass, the mass cancels. Objects fall with the same acceleration—if and only if the gravitational force is the only force.

But does the gravitational force decrease with height?

Yes. The gravitational field is not constant. I lied. Your textbook lied. We lied to protect you. We aren't bad. But now I think you can handle the truth.

The gravitational force is an interaction between two objects with mass. For a falling ball, the two objects with mass are the Earth and the ball. The strength of this gravitational force is proportional to the product of the two masses, but inversely proportional to the square of the distance between the objects. As a scalar equation, it looks like this.

A couple of important things to point out (since you can handle the truth now). The "G" is the universal gravitational constant. It's value is super tiny such that we don't really notice the gravitational interaction between everyday objects—the force is too small to notice. The other thing to look at is the "r" in the denominator. This is the distance between the centers of the two objects. Since the Earth is mostly spherically uniform in density, the "r" for an object near the surface of the Earth will be equal to the radius of the Earth with a value of 6371 kilometers (huge).

So, what happens if you move 1 km above the surface of the Earth? The "r" goes from 6371 km to 6732 km—not a big change. Even if you go ALL the way up to the altitude of the International Space Station orbit (400 km), there isn't a crazy huge change. Here, I will show you with this plot of gravitational field vs. height above the surface. Oh, and here is the python code I used to make this—just in case you want it.

For just about all "dropping object" situations, we can just assume the gravitational force is constant.

But what about air resistance?

OK, now we are getting into the fun stuff. What if you drop an object and you can't ignore the air resistance? Then we have a more complicated problem because there are now TWO forces on the falling object. There is the gravitational force (see all the stuff above) and there is also an air resistance force. As an object moves through the air, there is a force pushing in the opposite direction of motion. This force depends on:

  • The object's speed.
  • The size of the object.
  • The shape of the object.
  • The density of the air.

The part that makes this complicated is the dependency of the air resistance on the speed of the object. Let's consider a falling object with significant air resistance. How about a ping pong ball? When I let go of this ball, it is not moving. This means there is zero air resistance force and only the downward gravitational force. This force causes the ball to increase in speed (in the downward direction)—but once the ball is moving, there is now air resistance force pushing up. This makes the net force a little bit smaller and thus a slighter increase in speed. Eventually the air drag and gravitational force have equal magnitudes. The ball then falls at a constant speed—this is called terminal velocity.

Since the net force on a falling object with air resistance isn't constant, this is a pretty tough problem. Really, the only practical (OK, not really the only way) to model this is with a numerical calculation that breaks the motion into tiny steps during which the force is approximately constant.

How about a model of a falling ping pong ball? Here you go. Click the "pencil" to see and edit the code and the "play" to run it.

You can see that the ping pong ball almost reaches a constant speed after dropping a distance of 10 meters. I put a "no air" object in there for reference. If you want to see what happens if you change the mass—go ahead and change the code and re run it. It's fun.

Do heavier objects fall faster?

Now we get to the interesting question. If I drop two objects from the same height, does the heavier one hit the ground first? The answer is "sometimes". Let's look at three examples.

Drop number 1: a basketball and bowling ball. Here is a slow motion view of this actual thing.

If you ignore air resistance then these two objects have the same acceleration because they have different masses (see above). But why can you ignore the air resistance in this case? Looking at the basketball, it has a significant mass and size. However, it is moving fairly slowly during the fall. Even at the fastest part of this drop the force from air on the ball is super tiny compared to the gravitational force. Now, if you dropped it from a much higher starting point the ball would be able to get up to a speed where the air drag makes it fall slower than the bowling ball.

Drop number 2: a small ball and a cardboard box top. Just to be clear, the mass of the cardboard is WAY higher than the ball. Here is the drop. Sorry, the ball is hard to see since it's small.

Does the more massive object fall faster? Nope. In fact it's the lower mass that hits the ground first. It's not just mass that matters, it's the size that matters too. Even though the cardboard has a greater mass, it's surface area is also GIANT. This produces a significant air resistance force to make it hit the ground it a longer time.

Drop number 3: two pieces of paper. Two sheets of paper are pretty much the same so that they should have the same mass. However, they can hit the ground at different times.

I tricked you. Both papers have the same mass, but by crumpling one up they have different surface areas. The crumpled up paper hits the ground first. It seems like this could be a good party trick. But again, it's about more than just the mass of the object.

What about different sized skydivers?

Two people jump out of an airplane (with parachutes because they aren't crazy). One person is large and one person is small. Which one falls with the greatest terminal velocity? Yes, you can assume they are both in standard free fall position (same shape).

I am going to invoke the "spherical cow" principle and look at two falling spherical humans instead. Human 1 is a sphere with a radius of 1 meter (yes, that would be huge) and human 2 has a radius twice the size with a radius of 2 meters.

How do the gravitational forces on these two spherical humans compare? Human 2 is obviously heavier. If the human density is constant, then the increase in gravitational force will be proportional to the increase in volume. If you double the radius of a sphere, you increase the volume by a factor of eight (volume is proportional to radius cubed). So human 2 has a weight 8 times that of human 1.

What about the air resistance on these two humans? Again, human 2 will have a bigger area and more air resistance. If you double the radius, the cross sectional area will be four times as much (since area is proportional to radius squared). Now you see that the bigger human will have a greater terminal velocity. Human 2 has a weight that is 8 times as much but air drag that is only 4 times as much as the smaller human.

Now let's take this to the extreme. An ant and an elephant jump out of a plane. The elephant is going to need a massive sized parachute, but the ant probably doesn't need anything. Since the weight to area ratio is super tiny for a super tiny object, the ant will have a very small terminal velocity. It can probably impact the ground with little injury. Note: to my ant-readers. Please be safe and don't try this in real life in the unlikely event that I am wrong.

But size matters—especially when falling with air resistance.

I think this might be my longest blog post. Congratulations if you made it all the way to the end.

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