Question

You throw a stone into some nice soft and gushy mud. It penetrates one inch, slowed by a constant force. If you wanted the stone to penetrate four inches instead of one, you would have to throw the stone into the mud?

I say eight times as fast?

Answer 1

No. If the retarding force is constant (which means independent of speed), and you want it to penetrate four times as far, it will have to do four times as much work against the mud. That will require four times the kinetic energy.

I think this is an unrealistic problem. Fluid resistance forces are not independent of velocity.

Answer 2

twice as fast