A 1 kg block moving along the x-axis with speed 10 m/s enters a rough region from x = 0.1 m to x = 1.9 m. The retarding force in this region is F = -kx, with k = 10 N/m. The speed of the block when it leaves the rough region is:
Options

Question

A 1 kg block moving along the x-axis with speed 10 m/s enters a rough region from x = 0.1 m to x = 1.9 m. The retarding force in this region is F = -kx, with k = 10 N/m. The speed of the block when it leaves the rough region is:

Options

G1 Physics · Accepted Answer

Use the work-energy theorem. Initial kinetic energy: Kᵢ = ½mvᵢ² = ½(1)(10²) = 50 J. Work done by the retarding force: W = ∫[0.1 to 1.9] (-kx) dx = -(k/2)[x²]₀.₁¹.⁹ = -5(1.9²-0.1²) = -5(3.61-0.01) = -18 J. Therefore, K f K_f K f ​ = Kᵢ + W = 50-18 = 32 J. Thus, ½(1) v f 2 v_f² v f 2 ​ = 32 ⇒ v f 2 v_f² v f 2 ​ = 64 ⇒ v f v_f v f ​ = 8 m/s. Correct option: D.