What is the distance acted by the kinetic friction in a slipping ball until it starts rolling?

Question

After reading this question:

Initially at time t=0, a solid sphere slides with velocity v along a horizontal surface. The coefficient of friction is u. Find the required time for the sphere to stop sliding (the sphere will then be rolling).

I decided to solve it using energy methods. I assumed that the work done by the friction force would be $ -F_{f} d $ where $ d $ is the displacement from $ x_{0} $ to the point where the ball starts rolling without slipping.

Some people pointed me out that the distance the kinetic friction force acts is not really the displacement, $ d $. But the translational distance $ x $, minus the total rotational arc done by the ball, $ s $, up to the point of rolling.

$$ d = x - s $$

However, I don't fully understand what do they mean by that.

Answer 1

Well, displacement is the shortest path between final and initial position, it is not necessary that force is always acting in the direction parallel to that of motion it may perpendicular like in a curvilinear motion.