Trecherous incline

Two friends decide to shove boxes up a rough plank inclined at an angle of θ\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 ss along the plank:μk=ks\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 V0V_{0}. Assuming that the coefficients of kinetic and static friction are equal μk=μs\mu_k=\mu_s, when the box first comes to rest, it will remain at rest if (V0)AB×g×sinC(θ)k×cosD(θ)(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,DA,B,C,D; what is the value of A+B+C+DA+B+C+D?

Details and Assumptions

  • gg is gravitational acceleration
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