Xjyuk

I have no physics background, which is the genesis of my question.

In pop-science, it is frequently mentioned that Newton's apple didn't fall toward his head, but rather that his head came up and smacked the apple. Or, put another way, if you jump out of a window, you don't crash into the Earth, the Earth comes up and crashes into you.

Now, that is difficult to conceptualize since it is so far from daily experience. In other words, if one were sitting far away from Earth, viewing it from outer space, would one see oscillations of the Earth moving around smacking every free-falling object coming toward it? Meaning an apple falls from a tree in China, so Earth moves to the “east,” by some incomparably small number, in order to hit the apple; and an apple falls in the US, so it moves to the “west” to hit that apple.

That obviously isn't what it means, but that is how my non-physics-oriented brain tries to handle the information. How do I justify the Earth smacking something when it can't move in every direction to hit every object?

The earth's gravity attracts the apple with a force of $mg$ where $m$ is the mass of the apple and $g$ is the acceleration due to gravity, which may be considered a constant and equal to 9.81 $fracms^2$ if the separation is not too great.

Newton's third law essentially states that every action has an equal and opposite reaction. So the apple exerts an equal and opposite force of $mg$ on the earth. Although the forces are equal and opposite, the accelerations are not and are determined by Newton's second law, or $F=ma$, applied to each of the apple and the earth..

The acceleration of the apple is given by, where $m$ is the mass of the apple,

$$a_apple=fracFm=fracmgm=g=9.81fracms^2$$

Which is, of course, the acceleration of the apple downward toward the earth that we normally observe. However the earth, of mass $M$ is also accelerating upward, and its acceleration is given by

$$a_earth=fracFM=fracmMg$$

The mass $M$ of the earth is 5.972 x $10^24$ kg. The mass of an apple is about 0.1 kg. This means the acceleration of the earth upwards towards the apple is 1.67 x $10^-26fracms^2$. This is so small that it is essentially impossible to observe it.

Bottom line: While it is true that when an object falls to the earth the earth also rises to the object, if the object's mass is much much less than the mass of the earth, like our apple, the earth's upward acceleration would be too small to observe.

Hope this helps.

Newton's apple didn't fall toward his head, but rather that his head came up and smacked the apple.

This is relative:

1. From the apple's point of view, the Newton's head came up.


2. From the Newton's point of view, the apple fall toward his head.


3. From the center of mass of the system "Earth + apple" point of view, both movements perform.


4. From the Sun's point of view, both movements perform, too, and their trajectories are not linear.
   

The other question is which object attracts the other one. The answer is that both of them attract the other object (with equal force).

I think the important thing to bear in mind is that in classical dynamics, before you can have motion, there has to be acceleration, and before there can be acceleration, there has to be a force acting.

In the case of the apple and the Earth, when the apple is suspended, both bodies exert an equal and opposite force on each other (by Newton's 3rd Law). However, what you have to bear in mind is that there are also forces acting on the Earth from the hundreds (if not thousands) of other apples that are just being dropped in that same instant, at different points above the Earth. Of course, I am exaggerating a bit - there won't be that many apples, but there will be a lot of other objects all over the surface of the Earth, which are all simultaneously imposing gravitational reaction forces on it.

Overall, on average, the sum of all these forces is going to be pretty close to zero. Or, at least, it will be vastly dwarfed by the gravitational forces caused by the Sun and Moon. The Earth isn't going to be reacting to each little force that acts upon it individually and jumping around between them - it will be reacting to the overall resultant force generated by all of those forces at any given time, which will be relatively smooth and steady (on average).

The other thing to bear in mind is that, even if we just consider the Earth and a single apple in isolation, before you can have movement you have to have acceleration. The tiny gravitational force from the apple will cause an even tinier acceleration on the Earth, due to its very much larger mass. So, by the time the apple hits the ground, the Earth will have accelerated by such a tiny amount that any motion will be almost imperceptible and most likely impossible to detect/measure. However, again, this situation is highly unrealistic, because in practice it is not possible to isolate the Earth and a single apple from other nearby cosmic bodies, which will be generating much more significant forces.