Energy-preserving change of rolling radius ($v_f/v_i=\sqrt{(m+I/h_i^2)/(m+I/h_f^2)}$)

A rigid body rolling with effective radius $h$ has kinetic energy $(m+I/h^2)v^2/2$. If a transition changes the rolling constraint while preserving this total energy, the translational speed changes according to the title formula. Maintaining the same speed would generally require energy loss or gain. The energy-preserving condition is an assumption about the transition, not a generic property of sudden impacts.