Taking the freight car as the frame of reference: the ball is subjected to friction F and inertia force -ma in the horizontal direction.

For the ball, there is a theorem of motion of the center of mass:

f-ma=ma '

fR=Jβ

β=a'/R

The solution of simultaneous three formulas is: a' = ma/(j/r 2-m) j = 2mr 2/5.

So a'= -5a/3