An apple falls towards Earth, But the Earth does not move towards the apple. Why?

Most of us probably might have seen an apple falling towards the Earth, But how many of us have seen the Earth moving towards the apple? No one, probably.

But According to Newton's law of gravitation, an apple does exert a force of gravitation on the Earth, which is equal to the force that the Earth exerts on the apple. But then, why doesn't the Earth move towards the apple? Does that mean that Newton's theory has some limitations? 

The aforementioned is what we will be discussing here today, so let's begin.

Let's consider the mass of the apple as Ma, the acceleration of the apple as Aa, The mass of the Earth as Me, and the Acceleration of Earth towards the apple as Ae. 

Now, by Newton's second equation of motion:

Force = mass x acceleration

For apple:

Force = Ma x Aa

For Earth:

Force  = Me x Ae

By equating both the equations, we get:

 

MaAa = MeAe

On rearranging it, we get:

Ae/Aa = Ma/Me

Now, you have to appreciate the fact that the mass of the Earth is much much greater than the mass of the apple.

 

Me >> Ma

This means that Ma/Me is much much smaller than 1, and so is Ae/Aa.

Ae/Aa << 1

Let's Understand this with the help of an example.

Let's consider the mass of the apple as 1kg(For our convenience)

Then,

Ma = 1Kg

Aa = 10m/s square

Me = 5.972 × 10^24 kg

Ae = ?

Ae/Aa = Ma/Me

Putting the values:

Ae/10 = 1/5.972 x 10^24

Ae = 10/5.972 x 10^24

As you can see, the above example proves that the Earth does moves towards the apple, but the acceleration is so meager that we don't notice it and see only the apple falling towards the Earth

THANKS FOR READING!

-Arvin Gupta