Since Merry-Go-Round rides aren't free, you want to get the most for your amusement dollar. Think back to high school physics - think torque: "This force is defined by linear force multiplied by a radius."
So if you are on the outside of the Merry-Go-Round your kid is having more force applied, traveling more distance in the same time (i.e. faster) and just getting more fun overall for your $4.50.
On the other hand, if your kid is prone to motion sickness don't forget your torque. Sit the kid on the inside where he or she will be less ravaged by the vicissitudes of cruel physics, otherwise the Merry-Go-Round might make your kid torque!