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Suppose we had a bowling ball that we took to space. Also suppose we stopped completely and released the ball. At which point would the gravitational pull of Earth be so weak that the ball would not fall towards Earth, but rather some other object?

I realize this might vary based on how close the Moon is, if there are asteroids and such with a stronger pull, but generally? I assume $1\ 00\ 000 \ \mathrm{km}$ would be close enough? What about $\mathrm{1\ 000\ 000 \ km}$? Is this question even solvable? Why / why not?

HoolaBoola
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What you are looking for is known as the Sphere of Influence, and the distance you need depends on which direction you go. Going in the direction of the moon will get you out of the Earth's sphere the fastest. If you get within 66,100km of the moon, it becomes the dominant gravitational body. As the moon is 384,400 km from the earth, give or take a whole ton of assumptions, that says you have to be on the order of 318,300km from earth before another gravitational body takes over and pulls the bowling ball away from Earth.

Cort Ammon
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It depends what direction you're going, since gravity does have infinite range (as far as we know), so you need to know what other object it would be falling toward. If you are asking for the closest point at which an object would not fall toward the Earth, the Lagrange point $L_1$ of the Earth-Moon system is $3.2639\cdot 10^8\ \mathrm m$ away from the centre of mass of the Earth along the line from the Earth to the Moon, so anything just a little past this will fall toward the Moon. This point represents where the gravity of the Earth and Moon cancel out.

Sandejo
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