Suppose that a projectile is fired straight upward from the surface of the earth with initial velocity vo < 2GM/R. Then its height y(t) above the surface satisfies the initial value problem d²y GM y(0) = 0, y'(0) = vo. dr2 y + R)2 Substitute dv/dt = v(dv/dy) and then integrate to obtain v² = v3 - 2GMY R(R+ y) for the velocity v of the projectile at height y. What maxi- mum altitude does it reach if its initial velocity is 1 km/s?

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