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10 Nov 2019

# The glider Consider a glider flying in the xy-plane, where x and y and respectively the horizontal and vertical axis. Let Theta(t) be the inclination of the glider at time t (the angle between the glider body and the positive x-axis), and let v(t) be the speed of the glider along its path at time t. Note that, since the glider body points in indirection of motion, Theta is also the angle between the velocity vector and the positive x-axis. The forces involved are: gravity, the lift provided by the wings (which is a force perpendicular to the velocity vector), and the drag [caused by air resistance. which is a force parallel to the velocity vector but in the opposite direction], using Newton's law F = mu, one can write second-order differential equations containing d2x/dt2 and d2y/dt2, as usual. After several changes of coordinates, and assuming that the drag is proportional to the square of the velocity, one arrives at the following nonlinear system of first-order differential equations: d theta/dt = v - 1/vcos theta, dv/dt = -sin theta - av2. Remarkably, this system contains only one parameter, a. the "drag-to-lift ratio" (the drag divided by the lift). The value of it can tv viewed as a measure of the quality of the design: the designer of a glider should try to maximize lift while minimizing drag, so as to make a as small as possible. Your goal in this lab is to look at solutions of this model and compare them to the flight of the glider. Specifically, your assignment is to: First, consider the case a = 0, a perfect glider with zero drag. Find analytically all equilibrium solutions. Describe and sketch the flight path in the xy-plane corresponding to any equilibrium solutions. Classify any equilibrium solutions by linearizing the system around them and find their stability. Show that the function C(Theta, v) = v3 - 3v cos theta is a conserved quantity for the system.

The glider Consider a glider flying in the xy-plane, where x and y and respectively the horizontal and vertical axis. Let Theta(t) be the inclination of the glider at time t (the angle between the glider body and the positive x-axis), and let v(t) be the speed of the glider along its path at time t. Note that, since the glider body points in indirection of motion, Theta is also the angle between the velocity vector and the positive x-axis. The forces involved are: gravity, the lift provided by the wings (which is a force perpendicular to the velocity vector), and the drag [caused by air resistance. which is a force parallel to the velocity vector but in the opposite direction], using Newton's law F = mu, one can write second-order differential equations containing d2x/dt2 and d2y/dt2, as usual. After several changes of coordinates, and assuming that the drag is proportional to the square of the velocity, one arrives at the following nonlinear system of first-order differential equations: d theta/dt = v - 1/vcos theta, dv/dt = -sin theta - av2. Remarkably, this system contains only one parameter, a. the "drag-to-lift ratio" (the drag divided by the lift). The value of it can tv viewed as a measure of the quality of the design: the designer of a glider should try to maximize lift while minimizing drag, so as to make a as small as possible. Your goal in this lab is to look at solutions of this model and compare them to the flight of the glider. Specifically, your assignment is to: First, consider the case a = 0, a perfect glider with zero drag. Find analytically all equilibrium solutions. Describe and sketch the flight path in the xy-plane corresponding to any equilibrium solutions. Classify any equilibrium solutions by linearizing the system around them and find their stability. Show that the function C(Theta, v) = v3 - 3v cos theta is a conserved quantity for the system.