Chapter 6, Problem 73EAP

A block of mass m is at rest at the origin at t = 0. It is pushed with constant force F₀from 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 µ₀at x = 0 to zero at x = L

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

$a_x=v_x\frac{d{v}_x}{d_x}$

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