8.1.4.31 Geometric and physical applications

Problem: A sandwich, as usual, has fallen down with the buttered side on the horizontal heated surface, having the initial speed of translational motion $V_0$. Assuming that the force of resistance to the motion of the sandwich along the surface was proportional to the momentum, i.e. $F_{\text{res. }}=-kmV$, where $k$ is the coefficient of proportionality, find the space passed by the sandwich until the butter is completely melted. The sandwich has left a fixed trace of melted butter of constant thickness and width on the surface, and the speed of the slide to the indicated moment of the time has decreased twice. The initial mass of the butter on the sandwich is equal to the mass of a slice of bread.