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    Drilled hole through earth - Diff EQ w/ Gauss' Law

    You might be better off trying to find the position, [;r;], as a function of time and then differentiating. To do this, use Gauss's Law to find the force acting on the particle as a function of [;r;] and then use Newton's Law: [F=ma=\frac{d^2r}{dt^2}] to get an ordinary differential equation...
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    Linear momentum of particle

    Just evaluate the integral [;\int_{-\infty}^{\infty} dx;]. What do you get? Also, [; <p>= m\frac{d<x>}{dt} ;], so if you can find [;<x>;], you should also be able to find [;<p>;].
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