Why do objects fall at the same acceleration?

Question

I read these two posts and now I am more confused.

Do heavier objects fall faster?

Don't heavier objects actually fall faster because they exert their own gravity?

I was going to ask: if mass is an objects tendency to resist acceleration then why do two objects of different masses fall to the Earth at the same acceleration?

Then I read those posts and it seems that even though it is very small, the more massive object falls faster. Okay I understand, both objects attract each other.

If two cars of different masses collide doesn't the car with less mass accelerate more even though both cars received the same Force. Then that implies you need more force to accelerate a large mass than to accelerate a small mass. Because that is how I see it, the Moon attracts the Earth with the same Force as the Earth attracts the Moon but the Earth accelerates less due to its larger mass.

So then how is mass an object's tendency to resist acceleration? I am aware of $F_1 = F_2 = GMm/r^2$. So should we not really be able to see the difference in acceleration when dropping a massive object?

Answer 1

I hope this doesn't confuse you, but in one sense, yes, heavier bodies do fall faster than light ones, even in a vacuum. Previous answers are correct in pointing out that if you double the mass of the falling object, the attraction between it and the earth doubles, but since it is twice as massive its acceleration is unchanged. This, however, is true in the frame of reference of the center of mass of the combined bodies. It is also true that the earth is attracted to the falling body, and with twice the mass (of the falling body), the earth's acceleration is twice as large. Therefore, in the earth's frame of reference, a heavy body will fall faster than a light one.

Granted, for any practical experiment I don't see how you'd measure a difference that small, but in principle it is there.

Answer 2

Mass is an object's tendency to resist acceleration.

This applies when both masses you're testing are subjected to identical forces.

From Newton's Law of Gravity,

$$F = G \frac{M \cdot m}{r^2}$$

It is fairly obvious that the force the Earth exerts on a heavy body is more that what it exerts on a light body, so you can not compare the accelerations by comparing just the masses in this case.

Newton's second law gives $$a = \frac{F}{m} = G \frac{M}{r^2}$$

For two bodies an equal distance away from the Earth's center of mass, you can see that the acceleration for both bodies is indeed, the same and independent of the body mass.

Of course, this assumes there is no drag on the falling body, and that the only appreciable force is that due to Newton's Law.

To summarize:

There is no difference in the acceleration of two objects of identical shape but different masses. The difference you observe between a feather and a golf ball, for example, is not because of gravity - it's because of the shape and the air resistance the body encounters.

Clarification This answer assumes you're in an inertial reference frame (attached to the center of mass of the falling object and the Earth), and that "falling" means the motion of the object towards this center of mass.

If you consider falling to mean the motion of the object relative to the Earth, then, because the force exerted on the Earth by a heavier body is more, the Earth will accelerate towards the center of mass ever so slightly, and therefore the object will fall ever so slightly faster.

Answer 3

The difference is very minute, however the difference is there.. the difference is actually the amount that said object pulls on the Earth itself. so the difference between a 1 pound weight pulling on the Earth vs a 12 lbs bowling ball. Considering the Earth weighs in at a massive $1.317 \times 10^{25}$ lbs you could get a rough estimate by saying the Earth exerts 1g, So a 1 pound weight would exert $1/1.317 \times 10^{25}$ of that force, where the 12 pound bowling ball would exert $12/1.317 \times 10^{25}$ g. Hence the 12 pound bowling ball pulls the Earth to itself quicker than the 1 pound weight. So what is the difference? It's so small that it is almost immeasurable, but the difference is there none-the-less.

Answer 4

Although Pranav Hosangadi has explained, I will try to explain where you might be going wrong. I think this will be helpful for you. And I think it will not be waste of time in typing the answer for you.

I was going to ask: if mass is an objects tendency to resist acceleration then why do two objects of different masses fall to the Earth at the same acceleration?

Yes, you are right. If we want to push a feather we need less force and the feather resists the acceleration to less extent, and then if you want to push a iron ball we need more force and the iron ball resists the acceleration to more extent.

So, it is correct that mass is the tendency of object to resist acceleration.

Then, why do feather and iron ball gets accelerated to same extent towards the earth?
Isn't iron ball's mass greater than feather's mass, so iron ball should have greater tendency to resist acceleration than feather, thus iron ball should fall first than feather?

Here, if we would had applied same force on iron ball and feather, iron ball would had greater tendency to resist acceleration than feather, so iron ball would get accelerated to less extent than feather. Here, although iron ball gets accelerated to less extent, its greater mass than feather makes force to be equal to force exerted on feather, where acceleration of feather is more but mass is less.

But, earth doesn't exert same force on feather and iron ball. Force exerted on iron ball is greater than force exerted on feather. As it is confirmed from Pranav Hosangdi's answer that acceleration due to gravity is independent of mass of the body to which earth exerts force.

Then I read those posts and it seems that even though it is very small, the more massive object falls faster. Okay I understand, both objects attract each other.

Seeing your comment, I hope you have understood that because of air resistance iron ball falls first than feather.

If two cars of different masses collide doesn't the car with less mass accelerate more even though both cars received the same Force.

Yes, the car with less mass accelerates to greater extent than the car with greater mass as you stated force to be same on both.

Then that implies you need more force to accelerate a large mass than to accelerate a small mass. Because that is how I see it, the Moon attracts the Earth with the same Force as the Earth attracts the Moon but the Earth accelerates less due to its larger mass.

Yes, you need more force to accelerate a large mass than to accelerate a small mass. That is the reason earth exerts more force on massive objects than on tiny objects. You are also correct in saying about earth moon interaction.

So then how is mass an object's tendency to resist acceleration?

So far there is nothing which has contradicted with how mass can't be tendency to resist acceleration. If there is any thing you can comment.

. So should we not really be able to see the difference in acceleration when dropping a massive object?

Massive object and tiny objects are not exerted with the same force by the earth, you are thinking that on both the object force is same. Earth exerts different force on massive objects and tiny objects. Acceleration due to gravity has constant value at different points in space, just like electric field has constant value at different points in space. Acceleration due to gravity can be considered as gravitational field. So, no matter what mass you keep, it will be accelerated to same extent.

Answer 5

The Earth has to pull harder (exert more force) to attract an object with more mass. The object mass and the force applied by Earth change at the same rate.