Randall D. Knight — Dynamics I: Motion Along a Line: Problem 74

A block of mass \(m\) is at rest at the origin at \(t = 0\). It is pushed with constant force \(F_0\) from \(x = 0\) to \(x = L\) across a horizontal surface whose coefficient of kinetic friction is \(\mu_k = \mu_0(1 - x/L)\). That is, the coefficient of friction decreases from \(\mu_0\) at \(x = 0\) to zero at \(x = L\). a. Use what you’ve learned in calculus to prove that \(a_x = v_x \frac{dv_x}{dx}\). b. Find an expression for the block’s speed as it reaches position \(L\).