Mass, Falling Speed, and Air Resistance
August 27, 2018 6:33 AM   Subscribe

I'm having trouble wrapping my head around the physics of falling speed. The Internet gives contradictory answers with half of all websites repeating Galileo's experiment as if feathers and bowling balls fall at the same speed even outside of a vacuum. The other half talk about drag and terminal velocity speeding up the fall of heavier objects but not in a way that makes sense to me. Intuitively my mind is even reaching for Newtons 3rd law to reason that heavier objects should fall slower than lighter objects. I can't find simple, intuitive, non-mathematical discussion of why a heavier object of matched size should fall faster with air resistance (or even if this is actually true -- since some physics websites seem to imply it isn't).
posted by fucker to Science & Nature (36 answers total) 5 users marked this as a favorite
 
For starters, it's not that heavier objects fall faster in an atmosphere. Its that objects that are more aerodynamic fall faster. Even if you had a gigantic feather that weighed the same amount as a bowling ball, the actual bowling ball would still fall faster than it, because bowling balls are more aerodynamic and face less air resistance.
posted by nebulawindphone at 6:38 AM on August 27, 2018 [3 favorites]


For starters, it's not that heavier objects fall faster in an atmosphere. Its that objects that are more aerodynamic fall faster.

OK, this is one critical source of confusion and seemingly contradictory (if not simply ambiguous) answers on physics websites/forums.

If matched for size and shape, will the heavier object fall faster?
posted by fucker at 6:48 AM on August 27, 2018


Perhaps you could link to some of the arguments that are *bad* examples of this being explained to you because, and I know I'm biased based on a background in mechanical engineering, it's not an awful concept for me to grasp compared to some.

The basic viewpoint is that of a statics equations (read stationary things like bridges, powerlines, floors, fences, etc) where the sum of forces must equal zero and the equation looks like this:

Sum of F = 0
where F is the forces in all directions

as it morphs into a dynamics equation where the sum of the force equation looks like this:

F = ma
where F being forces involved, or the sum thereof I should say, m is object mass, and a is acceleration.

So, getting to the point of your question, namely real objects falling in an atmosphere (as opposed to theoretical objects neglecting atmospheric considerations), we view it from a 2 dimensional (or even 1 dimensional really) physics point of view where the two forces (remember, Newton tells us that only forces do things to objects that have mass) acting on an object in opposite directions.

Gravity is pulling the object perfectly down.
Atmospheric Resistance is pushing in the opposite direction (much like friction would if the experiment were horizontal and involved blocks on a surface instead of falling objects in a fluid).

An aside, think of this mental experiment, if the force of friction is equal to the force pushing an object, it doesn't move and we're back to a statics equation, but I digress.

Back to the point, those forces are the only ones involved. If they were equal, the object wouldn't fall, it'd hover. But they're not so the object accelerates in a direction, down to be specific. This acceleration is finally maxed out (terminal you might say) when we factor in another aspect of real objects: turbulence or laminar vs turbulent flows.

That jump is the difference from freshman level engineering courses to junior/senior level coursework. Turbulent flows are hard to calculate and I'm not going to go into that math for obvious reasons but all you have to grok really is that things that are streamlined will take longer to develop a turbulent wake or drag force or whatever you want to think of it as that will become equal to the gravitational force acting on the mass in question and the a (as in acceleration) will go to zero and speed will become constant.

Sorry, I have to jet away for a conference call so I leave it as an exercise for the reader of my word salad above to do the math thinking work that will show how m (mass) plays into effect when summing those forces and how it would impact terminal velocity as an independent variable from drag (because mass has NOTHING to do with the shape of an object and that's what determines drag and the other stuff I mentioned above.

If matched for size and shape, will the heavier object fall faster?

I think I see where you're getting confused now. The key concept you need to examine deeper/differently is terminal velocity, maybe that'll help you grok why the same sky diver can fall at 200 mph or 120 mph.
posted by RolandOfEld at 6:52 AM on August 27, 2018 [1 favorite]


Think about two balloons. One filled with air and the other with water. They are now objects of the exact same size and shape, but with different weights. The water-filled balloon will fall faster than the air-filled one because it's greater weight means it can push the air out of it's way better than the air-filled balloon.
posted by zinon at 6:54 AM on August 27, 2018


All objects fall at the same rate in a vacuum because of a deep weirdness* in physics.

Objects have inertial mass, and the greater the inertial mass, the more an object resists changing its speed in response to a force.

But objects also experience a force of gravity that is proportional to their gravitational mass.

The deep weirdness is that inertial mass and gravitational mass are equal.

As a result, the effect of the greater inertial mass of the object is always precisely canceled out by the greater gravitational force on the object, so all objects fall at the same rate.

Now we introduce drag, which is not proportional to mass. In the presence of this force, inertial mass gives heavier objects an advantage; they are not slowed by drag as much as lighter objects are. If two objects are experiencing the same drag (because they are the same size and shape), the heavier object falls faster.

*This weirdness is explained very neatly by General Relativity.
posted by BrashTech at 7:05 AM on August 27, 2018 [9 favorites]


Imagine a bowling ball and a styrofoam ball of the same size, different masses (call them M and m).

In a vacuum, they fall at the same rate, because the gravitational force on them is proportional to their mass. That is, the force on the bowling ball is KM (where K = GMe/r^2), and the force on the styrofoam ball is Km. since F = Ma, a = F/M... for bowling ball, the acceleration is KM/M = K; for styrofoam ball, that's Km/m = K.

Now, put in air resistance. Air resistance is typically proportional to the velocity squared, and is some function of the cross-sectional area. It's going to be the same for the bowling ball as it is for the styrofoam ball. For the sake of simplicity, let's call it j(t).

Now the force equation for the bowling ball is F= KM - j(t) = Ma --> a = KM/M - j(t)/M = K-j(t)/M. For the styrofoam ball, a = Km/m - j(t)/m = K - j(t)/m.

So, you see the two equations are the same *EXCEPT* the air resistance term is divided by big M for the bowling ball, and small m for the styrofoam ball. The bigger the M, the less important the j(t)/M term is, meaning it behaves more like it's in a vacuum.

That is, a bowling ball with big M is going to fall almost unimpeded by the air. But a styrofoam ball with small m is really going to be feeling that air resistance term.

That is, the smaller the effect of air resistance relative to the pull of gravity, the less effect it will have on the falling object. A bowling ball will fall almost as if it's in a vacuum, while a styrofoam ball will be slowed down much more significantly by air resistance.
posted by cgs06 at 7:10 AM on August 27, 2018 [7 favorites]


Just realized that I oversimplified too much... j won't be the same as a function of time, but as a function of velocity. But the same logic holds overall.
posted by cgs06 at 7:12 AM on August 27, 2018


Ok, this is a lot easier with diagrams, but I’ll give it a shot.

Let’s consider a bowling ball and a soccer ball. We’ll assume that they have the exact same shape, but the bowling ball is heavier.

If there is no air, then there is only one force on each as they fall - gravity. The force of gravity is Fg=m*g, where m is the mass of the object, and g is the surface gravity, which is 9.81 m/s^2 on the surface of the Earth.

Here’s what the forces on the two objects look like:
(B)      (S)
|          |
|          |
|          |
|          V
|
V
Based on that diagram, you might think the bowling ball falls faster, right? It experiences a stronger force of gravity (as anyone who has picked up both types of balls can attest). However, to find the acceleration of the objects from the forces, we need to apply Newton’s second law. That tells us that the acceleration (a) of an object is Fnet = m*a, where Fnet is the “net”, or overall force.

Finding the net force is trivial here, because there’s only one force on each:

Fnet = Fg
m*a = m*g

Look at that - the “m”s cancel out leaving
a = g
So, the acceleration depends only on the surface gravity, which is the same for all objects on the surface of the Earth! It doesn’t depend on mass. Basically, the bowling ball feels more force, but it’s also harder to move, because it’s heavier. Those two “effects” balance out exactly. So, the bowling ball and soccer ball will fall at the same rate. You can find plenty of videos online with vacuum chambers demonstrating this.

However, that was without any air. When you add air to the situation, there’s another force involved - air resistance - Fair. Let’s redraw the diagrams:
^          ^
|          |
|          |
(B)      (S)
|          |
|          |
|          |
|          V
|
V
Notice, I’ve drawn the exact same force of air resistance on both, because air resistance only depends on the size and shape and surface properties of the objects, which we’re assuming are about the same for these two objects. It also depends on the velocity which starts out the same for both objects.

To find the acceleration of these objects, we need Newton’s second law again, but this time the net force Fnet is a combination of two forces - gravity and air resistance. So:

Fnet = m*g - Fair
m*a = m*g - Fair

If we divide by “m” on both sides, the “m”s don’t entirely cancel!

a = g - Fair/m

Look carefully at this result - it says that the acceleration of the object in this scenario that includes air is the surface gravity just like before but with an added “correction” - “Fair/m”! This correction depends on mass - for a light object, Fair/m is large - meaning it is slowed down a lot by air resistance. For a heavy object this correction is small, so it is not slowed down as much by air resistance.

Here is a great video that illustrates these concepts nicely.
posted by Salvor Hardin at 7:12 AM on August 27, 2018 [6 favorites]


There are 3 parts to the equation of force for an object:
Net force. For a falling object, this is the total of gravity (pulling down) and drag (pushing up). More on this in a minute.
Mass. How much stuff there is. More mass is harder to accelerate.
Acceleration. How much it increases speed. It accelerates in the direction of the net force. If the forces balance, it continues at the same speed and direction (and the speed might be zero).

So, for a given force, a more massive object will accelerate more slowly. BUT! The force of gravity is also proportional to the mass, so it gets pulled harder, too.

So, in a vacuum:
F = m * g = m * a
Gravity force is mass times a gravitational constant. Force is also mass times acceleration. The masses cancel out, so g = a.

What if you’re not in a vacuum? Now you have two forces, gravity (down) and drag (up). Drag is proportional to the area and shape, and also to the square of the velocity. It is not proportional to mass.

A heavy object:
m*g - [constants] * A * v^2 = m*a
The drag force won’t have much effect yet, compared to the size of the gravity force: the first term is bigger than the second, so for awhile the object mostly falls like it would in a vacuum. Eventually the speed gets high enough that the drag force catches up to the gravity force, and it doesn’t accelerate anymore. (This is terminal velocity.)

A light object:
m*g - [constants] * A * v^2 = m*a
Mass is small in this case, so the drag term starts to matter sooner. It still starts to accelerate, but pretty soon your velocity increases to the point that the drag stops you from accelerating.
posted by Huffy Puffy at 7:14 AM on August 27, 2018


Net force is the important idea here. A net force causes an acceleration (i.e. a change in speed). So, the reason something falls down is because the net force is pulling it down.

In a vacuum, the net force (and, in fact, the only force) is the force of gravity. As others have pointed out above, the force of gravity goes up with mass, but so does the force needed to accelerate an object. So, because both of those forces goes up in the same way, the mass doesn't matter and objects accelerate the same no matter what their mass is in a vacuum.

The drag force depends on things other than mass (e.g. size, velocity), so objects that are the same size (e.g. a balloon filled with water and a balloon filled with lead) will experience, roughly, the same drag force, pointing up. Since they're both experiencing the same drag force pointing up and the more massive object is experiencing a larger gravitational force pointing down, the larger object accelerates more and, eventually, falls faster.
posted by Betelgeuse at 7:35 AM on August 27, 2018 [1 favorite]


The velocity something falls through a fluid is dictated by a balance of forces the gravitational force pulling the thing downward and the forces counteracting that pull, namely bouyancy and drag.

Things that experience greater drag and/or have greater bouyancy will fall more slowly than those experiencing lesser drag and/or bouyancy. That's it.
posted by wierdo at 7:36 AM on August 27, 2018 [1 favorite]


Even in a vacuum, more massive objects will fall faster because the object they are falling toward will rise more rapidly to meet them -- which is a direct consequence of Newton's Third Law (action = reaction).

The Earth won't rise very much to meet a bowling ball, certainly, but it will rise more than it will to meet a feather, because the force exerted on the Earth by the bowling ball is many times the force exerted on the Earth by a feather.

And it's not just a matter of a shorter time to impact because the Earth is moving toward the bowling ball faster, either; the bowling ball will have greater velocity than the feather at every moment of its fall after the first because a closer Earth will produce greater acceleration of the bowling ball (due to the inverse square law).

Of course, if feather and ball are dropped at the same time, they will fall at the same velocity because the Earth will react to their combined mass.
posted by jamjam at 7:47 AM on August 27, 2018


You may want to look at how to calculate air resistance, no matter what direction the object is going. For example, if I'm riding my bicycle, the force I'm putting into the pedals is counteracted a bit by the amount of wind resistance I have. If I am wearing a cape and spread my arms out vs have it all pulled in, in one case I have more wind resistance than the other; I'm not lighter, but I'm experiencing more force counteracting my movement, more air resistance opposing the force making me go my intended direction.

pedalling force ------> Bicycle <------ air resistance

So, a bowling ball and a bowling ball sized styrofoam ball will try to fall at the same speed, but the bowling ball has more intertia, because it has more mass, so the pounds-per-whatever force of wind resistance will slow the bowling ball down less than the lighter, less inertia, styrofoam ball.

Maybe as an experiment, calculate how much wind is needed to completely counteract the acceleration of gravity in the styrofoam ball, vs the pull of gravity on the bowling ball. Then, calculate the wind resistance of a ball half the size of the bowling ball, but made of a material that's twice as heavy. Then figure the accelleration of gravity in each, minus the force counteracted by the wind, and see what you get.

I think you're thinking that how fast something falls in a vacuum, vs how fast it falls in air, is a single entangled "thing" when it's actually two different forces, from two different sources, operating in opposite directions and can be calculated separately.
posted by AzraelBrown at 8:05 AM on August 27, 2018


I don't feel like the answers above give you a non-mathematical intuitive answer. They are right I am sure, but do not make sense if you don't want to do equations.

Try this thought. Imagine a bowling ball and a light rubber ball of the same size. Sitting on a nice smooth slick flat surface.

Put an electric fan next to each one, a really powerful electric fan. Same fan for each. Turn them on, pointing at the balls.

I think the bowling ball will move slower, don't you?

So now think of air resistance as being conceptually like a fan. At a given speed of falling, the air resistance will do "less" to the bowling ball than to the rubber ball, just like the fan did. So the bowling ball keeps accelerating -falling FASTER - than the rubber ball does.

Now somebody will probably tell me that I am wrong, but that makes intuitive sense to me.
posted by sheldman at 8:20 AM on August 27, 2018 [1 favorite]


If matched for size and shape, will the heavier object fall faster?

Yes, given air resistance.

Let's take two bowling balls. One is made of steel. The other one is made of styrofoam. They're both coated with the same plastic so that we don't have to worry about the complicated effects of surface smoothness on air resistance.

Gravity pulls hard on the steel bowling ball. It is heavy.

Gravity pulls gently on the styrofoam bowling ball. It is light.

It takes a lot of force to accelerate the steel bowling ball. It has high mass.

It takes a little bit of force to accelerate the styrofoam bowling ball. It has low mass.

As it happens, those two things - how hard gravity pulls, and how much force it takes to accelerate a mass - are exactly proportional. As a result, all objects in the same gravitational field accelerate at the same rate, so long as there are no other forces acting on them. Gravity pulls a little on the styrofoam ball, and it only takes a little pull to accelerate it. Gravity pulls a lot on the steel bowling ball, and it takes a lot of pulling to accelerate it. In a vacuum, both bowling balls accelerate at the same rate.

It's important to remember that gravity is pulling hard on the steel bowling ball and pulling gently on the styrofoam bowling ball. You can feel the difference when you pick them both up. The steel bowling ball takes all your power to lift. You can lift the styrofoam bowling ball with your pinky finger. That's how hard gravity is pulling on each one.

Air resistance will push back on both bowling balls with the same force when they're travelling at the same speed, because they have the same size, shape, and surface smoothness. It doesn't push back any differently on the steel or styrofoam bowling balls.

So what makes the difference?

Gravity is pulling hard on the steel bowling ball. It easily overcomes the air resistance that's pushing back. It accelerates at almost (but not quite) the rate that it would've if the air resistance hadn't been there at all. Gravity is you, pulling with all your power. Air resistance is your hamster pushing back. The hamster doesn't stand a chance.

Gravity is pulling gently on the styrofoam bowling ball. It will just barely overcome the air resistance that's pushing back. Gravity is you, pulling with your pinky finger. Air resistance is the same hamster pushing back just as hard. This time, though, the hamster will slow you down. The hamster isn't pushing any harder, but your pinky finger is much weaker.

This is the same explanation that everybody else has given, but hopefully it adds a little bit to your understanding.
posted by clawsoon at 9:21 AM on August 27, 2018 [1 favorite]


It's not that heavier objects fall faster with air resistance. It's that lighter objects fall slower (they are more slowed down by the air resistance / drag than the heavier objects are). Everything falls slower in an atmosphere than it would in vacuum.
posted by heatherlogan at 9:24 AM on August 27, 2018 [2 favorites]


Wired has a really good article on this that has a teensy bit of math but not very much.

Scientific American also has a page with an experiment you can do. But the really useful part of this is the "Observations and results" section at the bottom of the page.
posted by donut_princess at 9:38 AM on August 27, 2018


Ok thanks for the initial responses, folks. But I'm sorry to say that I still don't understand the basic logic of the physics. As to some issues:

Contradictory Answers

RolandofEld asked for some examples of the Google's unhelpfulness to this (to him) totally boringly simple concept. A good example is the very first answer in this thread which denied that heavier objects fall faster! Two favorites.

Donut_princess's "really good" Wired link is another example of ambiguously (if not directly) contradictory explanations. While it does suggest heavier objects (sometimes) fall faster, it doesn't explain why, and the video and diagrams it provides suggest otherwise. It also recommends another video which again suggests otherwise (two similarly sized balls of different weights are shown to fall at the same speed.)

So first I'd like to thank all the people here who at least confirmed the truth. I don't exactly understand your explanations, but even finding a clear or consistent answer among physics people online was allusive.

Vacuum logic

So in a vacuum two different forces are acting on falling objects: "how hard gravity pulls, and how much force it takes to accelerate a mass".

And these forces are (coincidentally?) the same and cancel each other out? I don't think I get this. Do dropped objects in a vacuum even accelerate? Do they change speed as they fall...drop faster and faster? Until how fast? Or do they drop toward the ground at a steady speed as soon as you let go?

The Wikipedia article on terminal velocity notes the sky-diving record is falling at 840 mph because it was "at high altitude, where extremely thin air presents less drag force." But there is no drag force in a vacuum so all objects would presumably drop even faster than this... at some incredible speed.

Outside of a vacuum both Salvor Hardin and the Wired article provide a diagram showing that added air resistance is equal on both objects, which does not (visually) suggest they should fall at unequal speeds, so I do not understand the helpfulness of the diagrams as such.

That air resistance slows down lighter objects more than heavier objects seems pretty intuitive and straightforward, but this is not being offered unanimously as the simple answer and I don't know why.
posted by fucker at 11:59 AM on August 27, 2018


If given a uniform linear acceleration, an object at rest would continue to increase speed indefinitely, until its speed begins to approach the speed of light. Now, it’s more complicated than this in real life, because the force of gravity isn’t constant over long distances, but is dependent on distance (squared).

If the very weird and unlikely equality of “inertial” mass and “gravitational” mass confuses you: you are not alone, at all, and it literally took Einstein’s theory of (general) relativity to explain why.
posted by Huffy Puffy at 12:39 PM on August 27, 2018


Yes... dropped objects in a vacuum accelerate, and change velocity as they fall. That's the definition of acceleration -- and there's no speed limit (in Newtonian mechanics... the real speed limit, the speed of light, is not something you have to worry about for the sorts of problems we're discussing here.) When you drop an object, it starts out moving at the speed you're moving. In a vacuum, it moves about 10 meters/sec faster each second after you drop it: after 1 sec, it's moving 10 m/s; after 2 sec, 20 m/s; after 3 sec, 30 m/s, and so forth. There's no limit (until you hit the ground.)

With air resistance, it's more complicated. The acceleration changes over time, asymptotically going to zero as the velocity reaches terminal velocity. (This point is reached when the force of air resistance balances the force of gravity.)
posted by cgs06 at 12:42 PM on August 27, 2018


Everybody is giving you complicated answers and it sounds like you are looking for more Newtonian than Einsteinian, so let's stick with that.

In a vacuum - and assuming you are talking about ball near a planet - there are not two forces. There is one, gravity. The ball accelerates.

With air resistance, the amount of resistance depends on how fast the body is falling. It's like turning the fan on higher - more wind = more force. That's why there is such a thing as terminal velocity, in an atmosphere - because at some point the body reaches a speed where the force of wind resistance is equal and opposite to the force of gravity, so the body stops accelarating. It doesn't stop falling, but falls at constant speed.

What is the terminal velocity? Will depend on the atmosphere, and the mass of the planet, and maybe I guess the shape of the falling thing.
posted by sheldman at 12:46 PM on August 27, 2018


In case you haven’t seen it before, here is a video from Apollo 15 where they dropped a hammer and a feather on the moon.

Galileo figured out that falling objects accelerate by rolling them down an inclined plane. He’d string strings across his inclined plane and roll balls down, so they’d make a click as they hit each string. He’d sing a song to keep time, and adjust the string positions to click on the beat. The distance between the strings continued to increase the farther down the ball went.
posted by Huffy Puffy at 12:47 PM on August 27, 2018


Two objects with the same cross-section will fall at the same rate. Their weight/mass doesn't factor in at all. Here's a video featuring two basketball-shaped objects of different weights.
posted by panic at 1:39 PM on August 27, 2018


It sounds like you would be better served by starting with some more basic physics concepts like forces and acceleration - once you have those under your belt, this question will be easier to answer.
posted by Salvor Hardin at 1:43 PM on August 27, 2018 [1 favorite]


There are three things to understand here:
  1. Forces & acceleration
  2. Gravitation
  3. Air resistance
Forces & Acceleration
  1. A force is what changes the way a thing is moving
  2. It's like an arrow that pushes the thing in some direction
  3. The same amount of force affects the movement of some things more than others
  4. The mass of an object is the word we use to describe this property
  5. Things with a lot of mass are less affected by a given force
The equation for this is Newton's second law. It says

F = m * a

This means that the acceleration (change in speed of movement) is related to the Force applied through the mass. If you increase something's mass, it will be harder to accelerate. This is a familiar thing - imagine trying to push a railway train to get it to go at 1m/s, versus a football. You would have to push the train very hard to start it going, and once it was going very hard to stop, compared to the football, because it has a lot more mass.

Gravitation
    Gravity is one of the forces relevant for a falling object.
  1. Every object pulls on every other object through gravity.
  2. The amount of pull is related to the objects' masses, and how far away they are - for most objects the pull is very small, so we do not care. Most really big objects are a long way away (like the sun). The Earth however is both very big (massive) and quite close, so its gravity is big enough to notice.
The equation for the force of gravity from Earth onto a thing comes out as

F = m * g

The Force from the earth on some thing is given by its mass times some number g (about 9.81)

This is a simplification but it works because the Earth is very big and because we are all right next to it.

If gravity is all that is in the picture (because we are thinking about a thing a vacuum) then this is the only force going on, so we can say

F = m * a = m * g

Notice the m on both sides of the equation - this means that if you double the mass of a thing there are two effects:
  1. The thing gets pulled twice as hard by gravity
  2. The thing gets twice as hard to accelerate
This balance is why heavy things fall as fast as light things in a vacuum.
Whatever extra gravity they feel because of being more massive is soaked up by how much harder they are to push around.

Air resistance
Air resistance is another force.

It comes from having to push the air out the way when going along.
When an object is not moving, it feels no air resistance.
When it is moving, it feels air resistance in the opposite direction to whatever way it's going.
The size of the air resistance is related to two things:
  1. The speed it is going at - more speed means more air resistance
  2. The shape of the thing in the direction of motion. A bigger thing has to move more air out the way, which is harder
The formulae for this are more complicated and I will leave them out, but the important thing is what happens to the force equation

Now F = m * a = m * g - X

where X is the drag force from the air.
Imagine we have two balls, one twice as massive as the other, but otherwise in identical conditions.

For one:

m * a = m * g - X

for the second

2 * m * a = 2 * m * g - X

In the second, heavier ball, the effect of the drag force on the acceleration is - X / 2 * m. For the first, it is - X / m. So, the air resistance will have twice as much effect on how fast the lighter ball speeds up as it does on how fast the heavier ball speeds up.

Imagine m is very very small, nearing zero. In this case, the effect of the drag on the acceleration will be very large, because X/m will be very large.

Now imagine instead m is exceedingly large. In this case, X / m is almost zero, so the drag has very little effect.

One way to imagine this is to think about two things sitting in a vacuum in deep space.
The first is a table tennis ball, and the second a lead ball of the same shape and size.

Now imagine you blow a puff of air at each ball. This is like presenting the ball with the force it would be getting from air resistance when falling, but just for a second. The table tennis ball will shoot off, but the lead ball will not get going very fast.


In a way the conclusion here is that gravity is the special force that's making a difficulty - because gravity happens to be caused by mass, making something more massive doesn't make it any harder or any easier for gravity to shift it. However it does make it harder for all the other forces.
posted by larkery at 2:19 PM on August 27, 2018


Let’s do an example. Say I have a 4 pound ball and a 9 pound ball. And let’s say the aerodynamics are such that at 10 miles/hour, the balls have 1 pound of drag; at 20 miles/hour, the balls have 4 pounds of drag; and at 30 miles/hour, the balls have 9 pounds of drag. (Drag force increases with the square of the air speed.)

Drop both balls. Their initial speed is zero. There is no drag force yet.
The 4 pound ball is pulled downward by 4 pounds of gravity. It accelerates downward.
The 9 pound ball is pulled downward by 9 pounds of gravity. It accelerates downward.

Eventually, the balls accelerate to 10 miles/hour. The drag force at this speed is 1 pound.
The 4 pound ball is pulled down by 4 pounds of gravity, and is pushed up by 1 pound of drag. Net force 4-1=3 pounds. It continues to accelerate. It’s still getting faster, but its rate of speeding up has slowed down a little.
The 9 pound ball is pulled downward by 9 pounds of gravity and pushed up by 1 pound of drag. Net force 9-1=8 pounds. It continues to accelerate, almost as much as before.

Eventually, the balls accelerate to 20 miles/hour (not at the same time, but don’t worry about it). The drag force at this speed is 4 pounds.
The 4 pound ball is pulled downward by 4 pounds of gravity, and pushed up by 4 pounds of drag. Net force 4-4=0 pounds, so it won’t accelerate any more. It will continue to fall at the same speed of 20 miles/hour.
The 9 pound ball is pulled downward by 9 pounds of gravity, and is pushed up by 4 pounds of drag. Net force 9-4=5 pounds, so it continues to accelerate downward, though its speed is increasing only about half as much now as when it started (but its speed is still increasing).

Eventually, the 9 pound ball accelerates to 30 miles/hour. (The 4 pound ball never does, because the forces on it balance out at a lower speed.) The drag force at this speed is 9 pounds. Net force on the 9 pound ball is 9-9=0 pounds, so the 9 pound ball stops accelerating. It still falls, but it stays at the same speed of 30 miles/hour.

So the 4 pound ball falls at a maximum of 20 miles/hour, and the 9 pound ball falls at a maximum of 30 miles/hour, and even before they reach terminal velocity, the 9 pound ball accelerates faster along most of the path.

That’s why heavier objects fall faster than lighter objects when there is air (or other fluid) resistance.
posted by Huffy Puffy at 2:31 PM on August 27, 2018 [2 favorites]


I agree with posters above that you would probably really enjoy a basic physics class right now. You're asking all of the right questions and you'd probably have many epiphanies of the "Oh, THAT'S how it works" sort.

I enjoyed my high school physics course, and the first two semesters of college-level physics about more than any others I have ever taken from that perspective--figuring out some of the fundamentals about how the physical world works, from gravity, movement, acceleration, friction, and the like to lenses and rainbows.

If you can't take an actual class right now, maybe something like working your way through the Kahn Academy physics section?

You're going to find a systematic approach far more satisfying and helpful than watching a random video or reading a random article about some topic or other.

And just to set expectations, it's going to take maybe a few weeks (or months?) of working your way systematically through various interrelated concepts, making sure you are really thoroughly comfortable with each of them, before you have a solid enough background to be able to completely understand the answers to the fairly complex question you have asked.

There is a good reason that it took thousands of years of human history before we had a decent physical understanding of how gravity works--and you're asking a question that involves that and a few other things on top of it.
posted by flug at 5:12 PM on August 27, 2018


A good example is the very first answer in this thread which denied that heavier objects fall faster! Two favorites.

Sorry. That was me. The rest of this thread has convinced me that I was misremembering how this works. If I could take that answer down, I would. It is incorrect.
posted by nebulawindphone at 6:35 PM on August 27, 2018 [1 favorite]


Yeah this (the Internet at large) is sadly not a great venue for clarifying things for you. As you are discovering, here and elsewhere you get a potpourri of explanations that are, variously,

a) correct, clear and compelling
b) incorrect, clear and compelling
c) off-topic
d) focused on a corner case that doesn't clarify your issues
e) focused on a fascinating but ultimately non illuminating fact
f) or any of the above, combined with being unclear or misleading

You, meanwhile, are the last person who should have to tell the difference, or you wouldn't have asked the question.

Your question and its responses get right at the heart of a conundrum in physics education that was first brought to steaming, horrifying light by a study called the Force Concept Inventory. That paper is unfortunately 100 pages long but basically shows that misconceptions about Newtonian physics are (a) extremely deeply rooted and (b) basically unaffected by receiving actual physics education. (I am overstating the hopelessness slightly but as a person who teaches this stuff those results are terrifying.)

You might find the paper to be a useful prompt about how some of these concepts work but fundamentally I think the issue is that the underlying principles of Newtonian physics are rooted in acceleration, not velocity. We think of an object sitting at rest and an object traveling a steady 100mph as being in very different states but as far as Newton is concerned they are exactly the same, because in both cases the acceleration is zero and therefore the total force on the object must be zero as well. This (acceleration) is at first an extremely non-intuitive lens to use to look at the world, and the only reason we use it despite its discomfort is that it is correct.
posted by range at 6:49 PM on August 27, 2018 [1 favorite]


Huh, after reading some more answers, I take mine back. I felt so sure this was one of those things they drilled into us in high school physics, but in retrospect we were probably ignoring air resistance.
posted by panic at 11:05 PM on August 27, 2018 [1 favorite]


So in a vacuum two different forces are acting on falling objects: "how hard gravity pulls, and how much force it takes to accelerate a mass".

Well.... I mean.... I don't think of it that way at all. I think some folks (and maybe this is the difference in physics folks and engineering folks because the two fields really do seem to tackle this sort of thing differently, at least at the undergrad level where I experienced them) seem to be introducing inertia into this discussion when, to me anyway, it really isn't necessary and borders on not helpful.

That is to say, per the reference to an answer above that maybe seems to suggest that there's "two different forces" acting on a falling body in a true vacuum, I don't agree. The only force that's worth considering at this level of discussion (again, Newtonian vs. Einstenian and useful/relevant points only) is the force of gravity. If you introduce an atmosphere then you add in another force acting in the opposite direction to the object's movement, namely air resistance.

If you want to get really fancy is when you introduce the planet accelerating towards the object as well as relativistic concerns but those are so far from relevant as to be absolutely not worth mentioning that, blah.

And these forces are (coincidentally?) the same and cancel each other out? I don't think I get this. Do dropped objects in a vacuum even accelerate? Do they change speed as they fall...drop faster and faster? Until how fast? Or do they drop toward the ground at a steady speed as soon as you let go?

A) Again, not two forces (in vacuum), at least not in my book so no canceling.
B) Dropped objects in a vacuum that had a gravitational force acting upon them will accelerate infinetly until they hit the object/ground they are falling towards.

F_gravitational=Mass_Of_Object * Acceleration_Of_Object.

If we set F_grav as constant and Mass_Of_Object as constant (which are both reasonable constraints) then acceleration is the variable you solve for. It's easy math to do so. Newtons of force_grav divided by Mass of object in Kilograms gives m/s^2 of acceleration for that object in a vacuum under that gravity force. Truly easy. That acceleration value will be fixed until collision.

C) That' what it means to accelerate, change speed (literally that many meters per second of increase for every second the object falls). So an acceleration value of 1 m/s^2 means that a ball dropped from rest would be going 0 m/s at t=0, 1 m/s at t=1 second, 2 m/s at t=2 seconds. This is how one calculates velocity based on the kinematic formula v = at (number one a bit down here at Kahn). Note that that formula doesn't mention forces but I already covered that above, you get acceleration_of_object from the F_gravitational, not vice versa.

D) Until how fast? Until relativistic concerns begin to be noticeable, but that's outside the scope of this question. So, forever. In a vacuum I mean, cuz that's what we're talking about here just now. Throw in a atmostphere and you get terminal velocity at some point.

E) Or do they drop toward the ground at a steady speed as soon as you let go?

No, no they do not. They accelerate at a constant acceleration, that's important as that determines speed, not speed determining acceleration. Though I see how that could be confusing as speed does, in a way because of air resistance, determine acceleration when an atmosphere is introduced because that is the other force involved. In a vacuum there's just one force. Gravity. Down.

Edit to clarify: all the things i just said were speaking to your confusion regarding behavior in a vacuum, please don't get twisted there, it could be bad.
posted by RolandOfEld at 12:43 PM on August 28, 2018 [1 favorite]


Outside of a vacuum both Salvor Hardin and the Wired article provide a diagram showing that added air resistance is equal on both objects, which does not (visually) suggest they should fall at unequal speeds, so I do not understand the helpfulness of the diagrams as such.

Air resistance is equal for two objects of the same shape but different mass, which both the wired article and Salvor Hardin were clear to specify. This can easily be verified, in a sense, by sticking your arm out of a car window going 70 mph and noticing the force of air pushing it backwards (ignore the weight of your arm downwards) then doing the same while holding a roll of quarters or lead or whatnot. Your arm will be pushed backwards (ignore downwards again) exactly the same if it's the same shape as before. Air resistance (this time horizontally) is the same for both objects, though you changed mass.

Air resistance is the same for same shapes. Running with that you get back to the formula that Salvor Hardin put into terms of acceleration at the end of his comment/answer that shows how mass plays into things when air is introduced as a countering force. It's pretty clear to me but maybe you don't want to see it in those terms.

Well, weight, which is a function of mass * gravity goes up against the resistance of the air.

So you have the two relevant forces involved Weight and F_air_resistance. Weight is a force like any other and we already have that formula F=ma where a is (on earth, 9.81 m/s^2) and m is whatever your object's mass is. Those things multiplied together give Weight pushing downwards. We've already determined that Air Resistance is another force (in this case the same force for two objects of different mass but same cross sectional area (read: shape)).

So, the sum of the forces equation, at terminal velocity would sum to zero when Weight = Force_air_resistance. That's when acceleration of the object in question would be zero, aka terminal velocity. This point in time from t=0 would be later for a heavier object. Later means a higher terminal velocity based upon our kinematic formula of v = at which is velocity equals acceleration times time.

The helpfulness of the diagrams those two provided is maintained, though I wish the Wired article hadn't jumped from comparing baseballs to ping pong balls to then comparing basketballs to bowling balls, while saying assume the latter two are same size but different masses. That part didn't flow well for me either.
posted by RolandOfEld at 1:43 PM on August 28, 2018 [1 favorite]


"how hard gravity pulls, and how much force it takes to accelerate a mass". And these forces are (coincidentally?) the same and cancel each other out? I don't think I get this.

The concepts you're thinking of are inertial mass and gravitational mass. Wikipedia has this to say:
There are several distinct phenomena which can be used to measure mass. Although some theorists have speculated that some of these phenomena could be independent of each other, current experiments have found no difference in results regardless of how it is measured:
  • Inertial mass measures an object's resistance to being accelerated by a force (represented by the relationship F = ma).
  • Active gravitational mass measures the gravitational force exerted by an object.
  • Passive gravitational mass measures the gravitational force exerted on an object in a known gravitational field.
Do some searches on why are inertial mass and gravitational mass the same if you want to get into those very interesting weeds. There's no theoretical reason that they should be the same, but they are (at least to the limits of the most accurate possible measurements we've been able to make).
posted by clawsoon at 4:40 PM on August 28, 2018


Thanks for self-correcting, panic. :-) The drag forces in the video you posted will be very small at the speeds which the balls reach before hitting the ground, and they'll cause an effect which is much smaller than the margin of error introduced by having random people try to drop them from above their heads at the same time. (If you want an inexact physics experiment, I can't think of a better way to do it. :-) )

Do the same experiment with an air-filled balloon. (Maybe versus a water-filled balloon if you want drag to be precisely equal.) In the case of the air-filled balloon, the drag forces will be significant w.r.t. the force of gravity on the balloon, and it will fall much more slowly.

An easy experiment that you could probably try right now, OP: Put a penny in the centre of one piece of paper, and something heavy in the centre of another piece of paper. Fold up an inch or so of each edge into a tray shape so that the weights don't bend the papers and change their shape as they're falling. Drop them both at the same time.

I just did the experiment with a nickel and a roll of tape. As all of the correct answers have predicted, the heavy one fell much faster, despite the fact that the piece of paper gave them more-or-less the same shape/drag. The drag force was the same on both - because of the piece of paper - but the gravitational force on the roll of tape was much higher and easily overcame the drag force.
posted by clawsoon at 5:03 PM on August 28, 2018 [1 favorite]


The deep weirdness is that inertial mass and gravitational mass are equal.

Do some searches on why are inertial mass and gravitational mass the same if you want to get into those very interesting weeds. There's no theoretical reason that they should be the same, but they are.

Inertial mass == gravitational mass isn't particularly strange at all, but to understand it, you need to get your head around what the property of mass actually IS, which is a bit OT for this discussion.

That is to say, per the reference to an answer above that maybe seems to suggest that there's "two different forces" acting on a falling body in a true vacuum, I don't agree. The only force that's worth considering at this level of discussion ... is the force of gravity.

I disagree - if you disregard inertia, you're left with the different gravitational force on objects with differing mass resulting in different acceleration, which is objectively incorrect. The action of inertia is critical to the understanding of why the hammer and the feather accelerate at the same rate.
posted by HiroProtagonist at 8:09 PM on August 28, 2018 [1 favorite]


I'd agree that inertia doesn't count as "one of two different forces", and I'd also agree that it's critical to understanding what's going on. Inertial mass is what determines how much force (gravitational or otherwise) is needed to accelerate something.

Take gravity out of the the equation for a moment. Let's say you're standing at the end of a perfectly slippery shuffleboard table, and you're giving two different objects - let's say a small 1 lb puck and a 500 lb steel anvil - as hard and fast a push as you can. You're using as much force as you can to accelerate the objects as much as you can.

How much each object actually accelerates while you're pushing it will depend on its inertial mass. The anvil has a lot of inertial mass, and will accelerate slowly. By the time you let go of it, it won't be going very fast. The puck, on the other hand, will accelerate quickly, since it has a small inertial mass. It'll leave your hand at a very high speed.

The difference between you and gravity is that you're applying the same maximum force to both the anvil and the puck, a force that's limited by the power output of your muscles, while gravity scales its force to precisely match the inertial mass. You won't be able to accelerate the anvil and the puck at the same rate. If you drop them both, though - in a vacuum - gravity will accelerate them at the same rate.
posted by clawsoon at 5:49 AM on August 29, 2018


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