Escape velocity from Earth's center using energy conservation

[Airsteve0]

I am having trouble with the following problem:

With what velocity should a projectile be started ar the center of the Earth to reach the surface of the Earth?

When I attempt to solve this using 1/2mv^2 = -(GMm)/R I get an answer of 11.8 km/s. However, the correct answer is 7.9km/s. I know that it has something to do with the potential inside the Earth but I don't even know where to start with incorporating that. Also could someone tell me how the negative on the right hand side is cancelled. Thanks a lot!

 

[rcgldr]

With what velocity should a projectile be started ar the center of the Earth to reach the surface of the Earth?

At any point r < R inside the earth, the net gravitiational force from all the mass from the radius at that point to the surfce (from r to R) cancels out and is zero. I'm assuming you're supposed to figure out the gravitational potential formula for a point r within the Earth (assuming it's a sphere).

Also could someone tell me how the negative on the right hand side is cancelled.

It's supposed to be negative, and absent any other forces, given an initial condition (position and velocity), the sum of gravitational potential energy and kinetic energy will be a constant. Note that GPE becomes less negative as R increases.