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nasu

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For a body to escape the pull of earth its kinetic energy ##K## must be greater than ##U## i.e., the total energy ##E=K+U## must be positive ( ##E## is constant).

As the distance increases, ##U## goes to zero and ##K## slowly decreases. The total energy ##E## will always remain constant though.

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##U=mgh## is only an approximate relation ( where g doesn't change appreciably with distance). For your case, where the body wants to escape the earth's attraction, you cannot assume ##g## to be constant. Also, in ##U=mgh##, the potential at the earth's surface is zero and at a very large distance it's infinity. Compare this to ##-\frac {GM_em}{r}##

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jbriggs444

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Only differences in potential energy are physically meaningful. The actual value at a chosen reference point is irrelevant.

When using ##U\, = \, mgh##, one is implicitly assuming a reference point at ground level, a height measured up from ground level and a potential energy of zero at the reference point.

When using ##U \, = \, - \frac{GM_em}{r}##, one is assuming a reference point at infinity, a radius measured out from the center of the earth and a potential energy of zero at the reference point.

The difference between potential energy of a mass at a height of 10 meters above the earth's surface and the potential energy energy of a mass at the earth's surface is the same, either way. Both formulas give the same answer for the difference.

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sophiecentaur

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What's going on is that the gravitational potential on the Earth's surface is Negative (with respect to infinity). Moving upwards is going in the Positive direction (doing work against gravity). It's like climbing up a mineshaft. The whole shaft has negative height but your height is increasing as you move up. I heard the term "Number Line" on the Radio, this morning. Moving right on the number line is always Increasing; going Up through the gravity field is Increasing your Potential.

-GmM/(Greater R) is more than -GmM/(Small R)

As far as gravity is concerned, the absolute potential Everywhere in the Universe is Negative (until they find something with Negative Mass).

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