gibson:teaching:fall-2012:math445:lab11
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gibson:teaching:fall-2012:math445:lab11 [2012/12/03 19:50] gibson |
gibson:teaching:fall-2012:math445:lab11 [2012/12/04 06:03] (current) gibson [Problem 3: projectile motion] |
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| ===== Problem 2: van der Pol oscillator ===== | ===== Problem 2: van der Pol oscillator ===== | ||
| - | ==== (a) ===== | + | (a) The van der Pol oscillator is defined by the second-order differential equation |
| - | The van der Pol oscillator is defined by the second-order differential equation | + | |
| <latex> | <latex> | ||
| Line 32: | Line 31: | ||
| Hint: put the call to //ode45// inside a //for// loop that sets the value of $y_1$ to the five different initial values. | Hint: put the call to //ode45// inside a //for// loop that sets the value of $y_1$ to the five different initial values. | ||
| - | ==== (b) ==== | + | (b) Put your work for part (a) in a script file, and run the script for a range of values of $\mu$ between -1 and 1. How does the character of the solutions of the equation change as $\mu$ passes through 0? Turn in plots for a negative, zero, and a positive value of $\mu$, whichever values you think best illustrate the change as $\mu$ passes through zero. |
| - | + | ||
| - | Put your work for part (a) in a script file, and run the script for a range of values of $\mu$ between -1 and 1. How does the character of the solutions of the equation change as $\mu$ passes through 0? Turn in plots for a negative, zero, and a positive value of $\mu$, whichever values you think best illustrate the change as $\mu$ passes through zero. | + | |
| Hint: To avoid the need to change the value of $\mu$ in both the script and the function file that defines $dy/dt = f(t,y)$, define $dy/dt = f(t,y)$ as an anonymous function within your script file, so that it inherits the $\mu$ of | Hint: To avoid the need to change the value of $\mu$ in both the script and the function file that defines $dy/dt = f(t,y)$, define $dy/dt = f(t,y)$ as an anonymous function within your script file, so that it inherits the $\mu$ of | ||
| the script file. | the script file. | ||
| - | ===== Problem 3: projectile motion ===== | ||
| - | It is easy to determine the trajectory of a projectile subject to gravity if you neglect air resistance. using elementary physics and calculus, you can show that a projectile will travel farthest across a flat surface if you shoot it at an angle $\theta = \pi/4$ above the horizontal. In this problem you will determine the optimal angle and the maximum range of a projectile subject to air resistance. Use the following equations of motion, developed during class | + | |
| - | + | ||
| - | <latex> | + | |
| - | dx/dt = v_x \\ | + | |
| - | dy/dt = v_y \\ | + | |
| - | dv_x/dt = -mu v_x \sqrt{v_x^2 + v_y^2} \\ | + | |
| - | dv_y/dt = -g - mu v_x \sqrt{v_x^2 + v_y^2} | + | |
| - | </latex> | + | |
gibson/teaching/fall-2012/math445/lab11.1354593045.txt.gz · Last modified: 2012/12/03 19:50 (external edit)
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