Accounting for the variation of gravitational acceleration a with respect to altitude y (see Prob. 12–33), derive an equation that relates the velocity of a freely falling particle to its altitude. Assume that the particle is released from rest at an altitude y₀ from the earth’s surface. With what velocity does the particle strike the earth if it is released from rest at an altitude y₀ = 500 km? Use the numerical data in Prob. 12–33.

Prob. 12–33.

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, this acceleration is determined from the formula a = −g₀[R²/(R + y)²], where g₀ is the constant gravitational acceleration at sea level, R is the radius of the earth, and the positive direction is measured upward. If 80 = 9.81 m/s² and R = 6356 km, determine the minimum initial velocity (escape velocity) at which a projectile should be shot vertically from the earth's surface so that it does not fall back to the earth. This requires that v = 0 as y → ∞.