Matlab Assignment Math 280 Hybrid Spring 2015

Name Matlab Assignmentmath 280 Hybrid Spring 2015

Name: ——————————— MATLAB ASSIGNMENT MATH 280 HYBRID - SPRING 2015 Return these sheets with your work. You need to show all your work! All problems need to be solved using Matlab. Only typed solutions will be accepted. ONLY PRINTED VERSION WILL BE ACCEPTED! (NO DIGITAL FORMATS!) Only two one-sided pages or one double-side page is allowed for each problem.

Deadline for submission: Monday, May 4, 2015, 12:30pm. There will be no extensions! Use at least four-digit accuracy for all the problems. Warning!!! Sharing your work or your solutions is strictly prohibited and will result in NO credit for this assignment.

Problem 1

(10 points) Use Matlab for the improved Euler method to approximate the solution of the differential equation y' - y² cos(πx) = 0, with initial condition y(-2) = 1, on the interval [-2, 0] with step sizes h = 0.1 and h = 0.05. Plot both the exact and approximate solutions on the same graph over the interval [-2, 0]. Compute and compare the maximum errors of the approximations for both step sizes, identifying the maximal error for each case.

Problem 2

(10 points) Write a Matlab function to solve the problem of a projectile moving upward in a medium with resistance proportional to the square root of velocity (with k = 3, g = 9.81, m = 4). With positive velocity downward, initial velocity v0 = -12, find the time t₀ such that v(t₀) = 0. Also, determine the velocity at t = 2 seconds.

Problem 3

(4+3+3 points) Consider planetary motion described by m d²r/dt² = - GM m r̂ / |r|³, where G is the gravitational constant, r is the position vector, and r̂ is the unit vector of r. In the coordinate system with the Sun at the origin, and with m=1, GM=1, the equations become: d²x/dt² = -x (x² + y²)^−3/2, d²y/dt² = -y (x² + y²)^−3/2. Initial conditions: x(0)=0.5, vx(0)=0, y(0)=0, vy(0)=1.63. (a) Write a Matlab function to determine the orbit (plot x versus y). (b) Vary vy(0), keeping other initial conditions fixed, to find initial velocities that produce a circular orbit. (c) Identify initial velocities that result in hyperbolic orbits.

Problem 4

(10 points) Consider a system modeling a spinning tennis ball with equations:

ẋ = vx,

v̇x = -C_D α v · vx + η C_M α v · vz,

ż = vz,

v̇z = -g - C_D α v · vz - η C_M α v · vx,

where v = √(vx² + vz²), α = (π d²)/(8 m), with parameters: δ = (π d²)/(8 m), C_D and C_M as functions of v and w, initial conditions: x(0)=0, z(0)=h=1.0, v₀=20, θ=23°, w=23. Use Matlab ode45 to solve the equations for spinning and non-spinning cases, plot trajectories, and determine flight times and maximum horizontal distances.

Paper For Above instruction

Solving complex differential equations numerically is a fundamental task in computational physics and engineering, and MATLAB offers a powerful suite of tools for such analyses. This assignment involves implementing various numerical methods, including improved Euler, and simulations of physical systems such as projectile motion with resistance, planetary orbits, and a spinning tennis ball, utilizing MATLAB programming. Each problem emphasizes understanding of differential equations, numerical stability, and physical modeling, aligning with advanced coursework in applied mathematics and physics.

Problem 1: Numerical Solution of ODE using Improved Euler Method

The first problem focuses on applying the improved Euler method, also known as Heun's method, to approximate the solution of the differential equation y' - y² cos(πx) = 0. The initial condition is y(-2) = 1, and the goal is to compute approximate solutions over the interval [-2, 0] with step sizes h=0.1 and h=0.05. The exact solution of the differential equation can be obtained analytically or through MATLAB's symbolic toolbox to provide a baseline for error analysis.

The improved Euler method involves predicting an initial slope at the beginning of the interval, then correcting it by averaging the initial and predicted slopes. Its implementation in MATLAB involves creating a function that accepts the differential equation's function handle, initial conditions, interval, and step size, and outputs the numerical solution points.

The MATLAB script can be structured as follows: define the differential equation as a function, set initial conditions, create a loop for advancing the solution, and utilize the improved Euler update formula. Plotting both the exact and approximate solutions on the same figure allows visual comparison, and computing the maximum absolute difference quantifies the numerical error. The maximum errors for h=0.1 and h=0.05 provide insight into the convergence properties of the method.

This analysis demonstrates the importance of step size in numerical accuracy and helps in understanding stability considerations and error propagation in explicit methods.

Problem 2: Projectile Motion in Resistance Medium

In the second problem, the focus is on modeling a vertical projectile motion through a medium exerting resistance proportional to the square root of velocity. The differential equation describing the velocity v(t) involves a nonlinear term proportional to v¹/². The differential form can be written as:

dv/dt = -g - k v¹/², with initial velocity v(0) = -12 m/s. To solve this, a MATLAB function is created that defines the ODE based on the physical parameters, such as resistance coefficient and gravity, and employs ode45 for numerical integration.

The solution provides the velocity profile v(t), from which the time t₀ when v(t) = 0 can be determined by finding the root of v(t). Additionally, evaluating v(2) gives insights into the projectile's dynamics at a specific time point. MATLAB's interpolation functions, such as spline, facilitate estimating the time of zero velocity and other quantities smoothly.

This problem combines physical intuition with numerical methods to analyze real-world projectile motion, essential for understanding drag effects in fluid dynamics applications.

Problem 3: Planetary Orbit Simulation

The third problem models planetary trajectories under gravity, simplifying to planar motion due to angular momentum conservation. The equations are second-order ODEs, which can be transformed into a system of four first-order ODEs. The initial conditions set the starting position and velocity, and MATLAB's ODE solvers, such as ode45, are used to compute the orbit.

To facilitate numerical solutions, the second-order equations are written as:

dz/dt = f(z), where z = [x, vx, y, vy], with the derivatives expressing the velocity and acceleration components. The MATLAB function encodes this system, which is integrated over a sufficiently long time span to allow the orbit to form—be it circular, elliptical, or hyperbolic.

Experimentation with initial conditions, especially velocity components, reveals the nature of the orbit. For example, tuning vy(0) to produce a nearly circular trajectory involves adjusting the initial velocity until the numerical simulation exhibits closed orbit characteristics, which can be confirmed visually and through energy calculations.

Hyperbolic orbits are identified by initial velocities exceeding the escape velocity threshold, resulting in trajectories that diverge from the Sun. These simulations aid in understanding celestial mechanics, energy conservation, and orbital classification.

Problem 4: Dynamics of a Spinning Tennis Ball

The fourth problem models the motion of a spinning tennis ball considering drag and lift forces due to spin. The equations involve complex velocity-dependent drag and lift coefficients, calculated based on parameters like ball diameter, spin rate, and air properties. These coupled, nonlinear ODEs are solved numerically with MATLAB's ode45.

The initial conditions specify the starting position and velocity components, as well as spin and flight parameters. MATLAB functions encode the equations, including calculations of parameters such as v, α, C_D, and C_M, which depend on the current velocity and other parameters.

Simulation results include trajectories under vacuum conditions, no-spin (drag only), and with-spin (lift and drag). Plotting trajectories provides visual comparison, highlighting the effects of spin-induced forces.

Interpolation methods like spline are then employed to identify maximum flight times and horizontal range, by analyzing the interpolated data points for the projectile's landing point. These insights are pertinent in sports physics, aerodynamics, and optimization of projectile design.

Conclusion

Throughout these assignments, MATLAB proves to be an indispensable tool for solving, analyzing, and visualizing complex differential equations arising in physics and astronomy. Implementing numerical methods such as improved Euler, ode45, and spline interpolation enables precise modeling of physical phenomena, ranging from orbital mechanics to fluid resistance. Mastery of these techniques fosters deeper understanding of system dynamics and informs practical applications in science and engineering.

References

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• Giorgilli, A., & Sansone, G. (1994). Numerical methods in celestial mechanics. Springer.
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