Who moves when an object falls by gravity? [duplicate]

Question

I'm reading calculus book by Morris Kline. This book treats physics because calculus grew out of physical and geometrical problems. I have no background about physics and got a question while reading the book.

$$F = \frac{GmM}{r^2}$$

When considering a tennis ball and the earth with this Newton's law and assuming the earth's mass is concentrated at its center, they exert the same force according to the equation. Then if I drop the tennis ball onto the ground and it travels 40 feet, which one moves by the force as a result? Does the tennis ball move 40 feet to the earth, the earth move 40 feet to the ball, each one move 20 feet to each other or anything else?

Answer 1

They both move towards each other. But the acceleration of, and thus distance traveled by, the earth is infinitesimal compared to the the ball because the mass of the earth is so much greater than the ball.

From Newton's 2nd law, the acceleration of the ball towards the earth is

$$a_{ball}=\frac{F}{m}$$

The acceleration of the earth towards the ball is $$a_{earth}=\frac{F}{M}$$

Since $$M \gg m \\ a_{earth} \ll a_{ball}$$

Hope this helps.

Answer 2

Both will move to keep the centre of the mass of the system stationary as there is no net external force. Gravitational forces are internal forces when considering as a system. Gravitational forces on each body are equal by magnitude. According to Newton's law of motion, $F=ma$, these forces cause acceleration of the bodies. But earth is a huge mass with respect to the ball. So earth's acceleration will be very small, or we can say neglegible. So the movement of ball only is observable clearly.