Sphere rolling in a symmetric groove ($a_{\mathrm{cm}}=g\sin \theta /[1+2/(5{\sin }^2(\varphi /2))]$)

A sphere of radius $a$ touching both faces of a symmetric groove of opening angle $\varphi$ rotates instantaneously about the line joining its contacts. The centre is a distance $h=a\sin (\varphi /2)$ from this line, so rolling without slipping means $v=h\omega$. For a homogeneous sphere, $K=m{v}^2/2+I{\omega }^2/2=m{v}^2[1+2/(5{\sin }^2(\varphi /2))]/2$. Differentiating this kinetic energy with respect to distance down an incline gives the displayed acceleration. Both contact forces do no work for fixed faces and no slip.

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.