A wheel of mass $$M$$, radius $$R$$, and moment of inertia $$I$$ rotates with initial angular velocity $$\omega_0$$ about a fixed axle and is brought to rest by a constant frictional torque $$\tau$$. The stopping arc length covered before stopping is $$d_{\text{rotation}} = R\theta_{\text{rotation}}$$. If the same wheel rolls without slipping on a horizontal surface with initial linear speed $$v_0 = \omega_0 R$$ (and therefore initial angular speed $$\omega_0$$), and is slowed to rest by the same constant frictional torque $$\tau$$ (applied at the axle), what is the ratio $$\dfrac{d_{\text{rolling}}}{d_{\text{rotation}}}$$, where $$d_{\text{rolling}}$$ is the stopping distance for rolling and $$d_{\text{rotation}}$$ is for pure rotation?