A block of mass 10kg, moving in x direction with a constant speed of 10ms⁻¹, is subjected to a retarding force F=0.1x J/m during its travel from x= -60 m to 30 m. Its final KE will be?

Work done is equal to change in KE.
Work done W = ∫₃₀⁻₆₀ Fdx = ∫₃₀⁻₆₀ 0.1xdx = 0.1/2[x²]₃₀⁻₆₀ = 0.1/2(900 - 3600) = -135 J
Change in KE = 1/2m(v₁² - v₂²) = -135 J
1/2 x 10 x (10² - v₂²) = -135
5(100 - v₂²) = -135
100 - v₂² = -27
v₂² = 127
KEfinal = 1/2 x 10 x 127 = 635 J
Initial KE = 1/2 * 10 * 10² = 500 J
Final KE = Initial KE + Work done = 500 -135 = 365 J
However, if the limits of integration are from -60 to 30, then:
W = ∫₃₀⁻₆₀ 0.1x dx = (0.1x²/2)|₃₀⁻₆₀ = 0.05(30² - (-60)²) = 0.05(900 - 3600) = -135 J
ΔKE = W
1/2 * 10 * (v_f² - 10²) = -135
5(v_f² - 100) = -135
v_f² = 100 - 27 = 73
KE_final = 1/2 * 10 * 73 = 365 J