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ActaPhysica

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- The usual approach for solving for the time it takes to fall to the center of the earth neglects air resistance & uses Hooke's Law. But if you solve it "the hard way"...

The usual approach for solving for the time it takes to fall to the center of the earth neglects air resistance & uses Hooke's Law. But if you solve it "the hard way"...

Gme*m/r^2 = ma

me = 4/3 rho pi r^3

G * 4/3 rho pi r = a

separate variables, integrate twice

r*Ln(r) - r = 2/3 G rho pi t^2

The lower limit of integration (0) has Ln(0) getting large and negative much more slowly than r going to 0, so there's no problem here.

But, this answer gives a very large time. What am I doing wrong?

Gme*m/r^2 = ma

me = 4/3 rho pi r^3

G * 4/3 rho pi r = a

separate variables, integrate twice

r*Ln(r) - r = 2/3 G rho pi t^2

The lower limit of integration (0) has Ln(0) getting large and negative much more slowly than r going to 0, so there's no problem here.

But, this answer gives a very large time. What am I doing wrong?