# 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
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