Chapter 7 - Potential Energy and Energy Conservation - Problems - Exercises - Page 233: 7.58

$d = \frac{\frac{1}{2}v_0^2+gL~sin(\alpha)}{g~sin(\beta) +g~cos(\beta)~\mu_r}$

$K_2+U_2 = K_1+U_1+W$ $0+mgd~sin(\beta) = \frac{1}{2}mv_0^2+mgL~sin(\alpha)-mg~cos(\beta)~\mu_r~d$ $gd~sin(\beta) +g~cos(\beta)~\mu_r~d = \frac{1}{2}v_0^2+gL~sin(\alpha)$ $d = \frac{\frac{1}{2}v_0^2+gL~sin(\alpha)}{g~sin(\beta) +g~cos(\beta)~\mu_r}$