A block of mass m is at rest at the origin at t = 0. It is pushed with constant force F₀ from x = 0 to x = L across a horizontal surface whose coefficient of kinetic friction is μ_k  = μ₀(1 – x/L). That is, the coefficient of friction decreases from μ₀ at x = 0 to zero at x = L.

a. Use what you’ve learned in calculus to prove that

b. Find an expression for the block’s speed as it reaches position L.

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dvs V. a, = v, dx