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I'm sure we will all agree to define "up" as away from the center of the earth.

The force on the object "going up" is:

GMm/(d^2) (Newton's law of universal gravitation)

where d is its distance fromn the earth's center.

M is the mass of the earth

m is the mass of the object

G is the universal gravitational constant

(I'm using the symbol ^ to indicate an exponent: ^2 is squared, ^3 is cubed, etc.)

We'll be using Newton's second law: F = ma (Force = mass * acceleration)

The calculus starts off with the definition of acceleration as the time derivative of velocity:

a = dv/dt

and continues with:

a = dv/dt = (dv/dx) (dx/dt)

and since dx/dt is the definition of velocity:

a = v(dv/dx)

a dx = v dv

Here, we'll substitute for a, using the Law of gravitation and remember that the acceleration is in a direction opposite to that of increasing distance from the earth:

-GM(1/x) dx = v dv

Now we'll integrate both sides using the following limits:

R is the initial distance from the center of the earth

X is the current distance from the center ov the earth

V is the initial upward velocity

v is the current velocity

The result of the integration is (sorry but I don't know how to represent integral signs on my keyboard):

-GM(1/r - 1/R) = (v^2 - V^2)/2

Being very careful to deal with the signs, we can rewrite:

v^2 = V^2 + 2GM(R/r - 1)/R

In the second term, r starts out at R so the term equals zero and the velocity is the initial velocity, as required.

As the value of r increases, the term contributes a negative value to the velocity.

If that negative value ever causes the velocity to become zero, the object stops rising and begins falling, increasing its velocity as it heads downward.

(Notice that when it reaches the starting point its velocity is the same as when it started, as it should be.

But, if r is allowed to get really, really big, the velocity approaches:

V^2 - 2GM/R

So if V starts out larger than the square root of 2GM/r, then the velocity will never drop to zero and the object will continue rising and

IT WON'T COME DOWN!