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Escape velocity from solar system

Your residual speed at the surface needs to be the escape velocity from the surface.
But you can try using conservation of energy.

 Gravity bellow the surface of the earth is proportional to r, not 1/r^2.
... assuming constant density for the Earth.

Basically you want to know what initial speed you have to be doing to escape a mass distribution - starting from within the distribution. This sort of thing is pretty hand-wavy since, for something like the solar-system, the specific path taken is important (i.e. if all the planets are on the other side of the Sun to the Earth when you start, you can start out slower maybe.)

So the concept of escape velocity is a bit vague in this scale.
Anyway, see: