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## Homework Statement

Suppose that a hole has been drilled through the center of the Earth, and that an object is dropped into this hole. Write a first-order differential equation for the object's velocity,

*v as a function of the distance r*from the Earth's center (i.e., an equation involving

*dv/dr*), and solve it to determine the speed the object achieves as it reaches the center of the Earth. Check this speed with the result you get from simple conservation of energy considerations. Consider the Earth's mass density to be uniform throughout. [Hint: recall Gauss' Law as it applies to the gravitational field of a spherically symmetric mass distribution.]

## Homework Equations

I think the Gauss' Law for gravitation is simply to point out that the force is only effected by the mass enclosed, not the outer mass.

## The Attempt at a Solution

Well, here is what I know...

The mass enclosed is a function of radius so [tex] a = \frac{dv}{dt} = \frac{Fm(r)}{r^2} [/tex]

I thought, to get [tex] \frac{dv}{dr} [/tex] I multiply both sides by [tex] \frac{dt}{dr} [/tex]. This gives me [tex] \frac {dt}{dr} \frac {Gm(r)}{r^2} = \frac {dv}{dr} [/tex]. Now I have a function in terms of dv/dr, but I am not sure if this is right or what to do next.

Thx for any tips or help!

## Homework Statement

## Homework Equations

## The Attempt at a Solution

## Homework Statement

## Homework Equations

## The Attempt at a Solution

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