Trecherous incline

Two friends decide to shove boxes up a rough plank inclined at an angle of \(\theta\). 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 \(s\) along the plank:\[\mu_k=ks\]. One of the friends shoves a box up the plank so that it leaves the bottom of the plank at a speed of \(V_{0}\). Assuming that the coefficients of kinetic and static friction are equal \(\mu_k=\mu_s\), when the box first comes to rest, it will remain at rest if \[(V_{0})^A \geq \frac{B \times g \times sin^C(\theta)}{k \times cos^D(\theta)} \] For some constant positive integers \(A,B,C,D\); what is the value of \(A+B+C+D\)?

Details and Assumptions

• \(g\) is gravitational acceleration