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I 'get' Newton's third law of motion, except for one thing. We know that if we let an apple fall to the Earth, the earth will fall to the apple, because the Earth must experience the same force in the opposite direction (the third law).

This I cannot imagine. Is there a possibly intuitive way to explain this?

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All you're really having trouble imagining here is big numbers. Large distances and long times are similarly hard to imagine for the same reason. The mass of the earth is about 25 orders of magnitude larger than an apple, so it's acceleration is 25 orders of magnitude smaller (essentially zero if you were to attempt to measure it). – David H Sep 23 '13 at 7:30

The general idea is according to the webpage "Newton's Third Law" (Tahsiri) is that as Newton's 3rd Law for the apple and the Earth can be stated as:

$$F_{apple} = F_{Earth}$$

and as the 2nd Law states:

$$F = ma$$

Then we have:

$$m_{apple} a_{apple} = m_{Earth}a_{Earth}$$

As the mass of the Earth is substantially larger than that of the apple, therefore the acceleration is proportionally smaller (far smaller, almost zero), as:

$$m_{apple}/m_{earth} << 1$$


$$a_{earth}/a_{apple} << 1$$

An analogy that may help with this, is a collision between a fly and an elephant; the elephant will hardly move (high mass, hence low acceleration), and the fly will...well... go flying off with a high acceleration as it has a considerably lower mass.

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