Two friends decide to shove boxes up a rough plank inclined at an angle of . The plank is slightly smoother at the bottom and a bit rougher at the top, such that the coefficient of kinetic friction increases linearly with the distance along the plank:. One of the friends shoves a box up the plank so that it leaves the bottom of the plank at a speed of . Assuming that the coefficients of kinetic and static friction are equal , when the box first comes to rest, it will remain at rest if
For some constant positive integers ; what is the value of ?
Details and Assumptions