From the relation between the kinetic friction force acting on the particle, the translational and rotational velocities of the particle rolling in the x-direction may be expressed as follows:
 | (2–1) |
 | (2–2) |
where:
is the particle's position in the horizontal direction. 
is the particle's translational velocity in the horizontal direction. 
is the particle's rotational velocity. 
is the gravity acceleration in the vertical direction. 
is the dynamic friction coefficient between the particle and the surface
materials. 
is the particle's mass. 
is the particle's diameter. 
is the particle's moment of inertia.
By integrating equations Equation 2–1 and Equation 2–2 with the initial conditions 
and 
, the time 
when the rolling friction stops and 
equals 
, can be given by:
 | (2–3) |