A ball rolls along a horizontal frictionless surface without slipping toward and incline.  The ball moves up the incline a certain distance before reversing direction and moving back down the incline.  In which of the following situations does the ball move the greatest distance up the incline before turning around?

To determine in which of the following situations the ball moves the greatest distance up the incline before turning around, we need to consider the factors that affect the ball's motion.

1. Angle of the incline: The steeper the incline, the harder it is for the ball to move up before reversing direction. Therefore, a lower incline angle will allow the ball to move a greater distance up the incline before turning around.

2. Initial velocity: The higher the initial velocity of the ball, the farther it can travel up the incline before reversing. Therefore, a greater initial velocity will result in the ball moving a greater distance up the incline before turning around.

3. Ball's mass: The mass of the ball affects its inertia, which in turn affects how far it can travel up the incline before reversing. A lighter ball will have less inertia and will move a greater distance up the incline before turning around.

Based on these factors, the situation that would result in the ball moving the greatest distance up the incline before turning around is the one with the lowest incline angle, highest initial velocity, and the lightest ball. Therefore, the specific details of the situations mentioned in the question are required to determine which one meets these criteria.