1 Rescue Mission You Are Near The Shore At Location Xo Yo When You Hear A Person In Deeper Waters X1 Y Shouti 1 (75.83 KiB) Viewed 13 times
1. Rescue Mission. You are near the shore at location (xo, yo) when you hear a person in deeper waters, (x1, y₁), shouting for help. You are a good swimmer, and your initial velocity at your location is vo. However, due to the currents at the deeper waters, you know that your speed will slow down the farther you are from the shore, v(x): The top view of the scenario - is shown in Figure 2 below. Use the calculus of variations to find the path of minimum time to get to the person. (xo, yo) v(x) = vo/ (x¹/2) Figure 1: Rescue Mission ....... Vo (x₁, y₁) X
(a) What quantity is to be minimized? [1 point] Write it down as a definite integral. [4 points] (b) What is the quantity f, i.e. the integrand of the definite integral from the previous item? [1 point] Get its partial derivatives wrt the dependent variable (e.g. wrt y) and its derivative (e.g. wrt y') (2 points each] (c) Directly substitute the obtained partial derivatives to the Euler equation. [2 points] Solve for (i.e. isolate to the left-hand side) the expression for y' (3 points) (d) Integrate to solve for y(x). [4 points) Draw a rough sketch of the path (i.e. at least with the correct concavity). [1 point]
Useful Equations af d of = Euler Equations: 0 dy dx dy Euler Equation subject to algebraic constraints: af d Of მყ +Σ; 1; dy dx dy jy = 0 dy Transformation (Hamiltonian from Lagrangian): H{qi, Pist} = Ei 9iPi - L{qi, ġi; t} Hamilton's (Canonical) Equations of Motion: ġ; - ƏH Opi and pi = 8H 89,
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