Escape Velocity

Question

An object is projected from the surface of the earth with a speed of 1.68*10^4 m/s. What is its speed when it is very far from the earth? (Neglect air resistance.)

Answer 1

mv²/2 - mv₀²/2 = - ∫₀^∞GMm/(r + x)²dx; v² - v₀² = 2GM/(r + x)₀^∞; v² = v₀² - 2GM/r

On surface mg = GMm/r²; GM = gr²; v = (v₀² - 2gr)^1/2 = ((1.68·10⁴)² - 2·9.8·6357000)^1/2 = 12555.6 m/s