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FIND THE MAXIMUM HEIGHT OF A ROCKET Consider a rocket flying vertically with the dynamics mv0 = F m0 = −µ where the mass, m, velocity, v, and fuel flow, µ, are functions of time. The force, F , is modeled by three terms F = c1µ − c2 v2 − 1 representing the reaction force, drag and gravity, in this order. Assume that the fuel flow can be controlled with µ : (0, ∞) → [0, c], for some positive constants c1 , c2, c, and the mass function satisfies m : (0, ∞) → [0.1, 1], with m(0) = 1, i.e. when m = 0.1 there is no fuel left to burn. Find the maximum height the rocket travels for c = c1 = c2 = 1. Solve the problem numerically for one method and formulate both alternatives: the Hamilton-Jacobi PDE setting and the Lagrange ODE principle. Are there more alternatives? Does the optimal fuel flow strategy change substantially for different values of the constants? 1