1. The problem statement, all variables and given/known data You are a space traveler and you land on the surface of a new planet. You decide to drill a tunnel through the center of the planet to the opposite side of the planet. Your idea is to use this tunnel to easily transport rocks from your location to the opposite side of the planet. So, you drop a rock in the tunnel and it falls down the hole toward the center of the planet, passes through the center of the planet, then makes it to the opposite side of the planet. Assume the planet is a uniform solid sphere of radius 8.50×10^6 m. The rock is dropped from rest into the hole, and it initially has an acceleration of 10.70 m/s2. How long will it take to make it to the other side of the planet? 2. Relevant equations F = (G m1 m2)/r^2 U = -(G m1 m2)/r 3. The attempt at a solution So I guess I am having a hard time starting this, which in essence is pretty much the only hard part of any physics problem. I pretty sure this will be a differential equation since the acceleration due to gravity will be changing as the object gets closer to the center of the sphere. I also believe that this might be able to be solved as a simple harmonic oscillator diff'eq. First ill try and get the mass of the planet. Using a and r I can solve for the escape velocity and then use that to solve for the mass. Using equations: Ve = Sqrt[2*a*r] M = (Ve^2 * R)/(2*G) Now F = m1 a m1 = mass of stone m1*a = (G m1 m2)/r^2 m1 cancels on both sides a = dv/dt dv/dt = (G m2)/r^2 so a will change but so will r, that's two changing variables since r is a function of t also. Would I have to do some Lagrange Equations of Motions? This where I am lost at. Don't know if I am making this more complicated then it needs to be.