I am supposed to calculate how fast an object is moving at the center of the earth if thrown down some hole that was drilled through the earth. I am supposed to set up a dr/dv differential equation and solve it. I am supposed to use Gauss's law. I am to assume the Earth is uniformly dense.(adsbygoogle = window.adsbygoogle || []).push({});

Here is what I have so far, and you will see why I am puzzled:

int(g . da) = Menc / G

(g is the force, G is the universal gravitational constant).

So g = Fg/m = GMe/r^2, no problem there. I can take it out of the integral by symmetry:

GMe/r^2 int(da) = Menc / G

Now the da is what is getting me. A = 4 pi r^2, so da = 8pi*r dr, but then I have int (8pi*r dr) and then my dr's vanish. I am supposed to get this in terms of a DE with dr/dv so I can't have my dr's vanishing.

One thing I thought about doing, and I don't know if this is right or not:

int(da) = 4pi* (dr)^2

And Menc = int(k*dv) (v is volume, k is some constant),

Menc = 4/3 (Pi*(dr)^3).

When I collect the Menc and the da, all but one dr cancels, leaving me with.

g = (4/3)Pi*dr / (G)

Is this the right approach?

My other problem, assuming that what I have is true, is that I can rewrite g in terms of Force, which has an acceleration, which has a dv/dt term in it. But now I have 3 differentials; dv/dt and dr. dt is the odd guy out:

[some coefficient here] dv/dt = (4/3)Pi*dr / G.

I can think of no way to "chain rule" dt out. Is there something obvious I'm missing.

Thanks in advance.

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# Speed of an object thrown down a hole to the center of the earth

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