Escape Velocity via Separation of Variables

Calculus · Calculus — University

Question

As a body is projected to a high altitude above the Earth's surface, the variation of the acceleration of gravity with respect to altitude $y$ must be taken into account. Neglecting air resistance, the acceleration is given by $$a = -g_0\left[\dfrac{R^2}{(R+y)^2}\right],$$ where $g_0$ is the constant gravitational acceleration at sea level, $R$ is the radius of the Earth, and the positive direction is measured upward. Given $g_0 = 9.81$ m/s² and $R = 6356$ km, determine the minimum initial velocity (escape velocity…

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