Mathematical Methods for Engineers 1 (ENR 114) MATLAB Team Project, Study Period 2, 2016 See unit website for due date This project is designed to help develop Graduate Qualities 2, 3 and 6, namely lifelong learning, effective problem solving and communicating effectively. It should also provide opportunity for varying student Learning Styles. A team report is to be submitted by each team by the due date. Presentation should be according to the guidelines given in the handout Writing Mathematics Reports. Only A4 size paper should be used. The names and UniSA network user names of each participating team member should appear on the cover page of the report. You should view this project not as an exercise set, but as a report you could be writing in a professional setting. Presentation counts. As such, your report should contain certain elements, for example, a description of the problem, the objectives of the analysis, the conclusions you reached, and any explanatory graphs you need to produce, any code you write, with the code including explanatory comments as well. The assessment will take into account both presentation, and your mathematical and computational analysis of the problem, including your Matlab code, figures and outputs. Each team as a result will obtain a team mark for its report. Individual marks will be either the team mark or an adjusted mark based on any Individual Peer Assessment Forms handed in, in confidence. No joint work is allowed between different teams. A Bouncing Ball Consider a ball allowed to fall onto a hard surface and rebound, continuing to bounce until it is at rest. In the analysis undertaken here, the ball is dropped, i.e. its initial velocity is zero, and the surface it lands on is perpendicular to its flight. A ball with high resilience, for example, a new table tennis ball, retains most of its kinetic energy following the collision of ball and surface, and it rebounds to a high proportion of its original height. A simple model of the ball's behaviour views the movement of the ball in one-dimension, and would ignore air resistance. There is loss of energy in the collision; some energy is transformed into sound, some into a slight rise in the heat of the ball following the deformation and compression it experiences during the impact, and some energy may impart a spin to the ball. The dissipation of energy can be accumulated into the term resilience, a characteristic of various types of balls. The main factors governing the height of the ball's subsequent bounce are; its initial height (and thus velocity), the force of gravity, and the ball's resilience. One measure of resilience is called the coefficient of restitution, commonly given the symbol e. 1 Task 1 We assume that the potential energy of the ball at its initial position ( P.E. = mgh , where m is the mass of the ball, g is the gravitational constant (take g = 9.81m/s2 here), and h is the height of the ball) is entirely converted to kinetic energy (K.E. = 1 2 mv2 , where v is the velocity of the ball just before it hits the surface). Note: it may be easier to take the positive direction as being downward for this problem a) Show that the ball's velocity just before it hits the surface is v = √2gh . Denote the maximum height of the ball before each bounce with a subscript matching the number of the following bounce. Let the velocity of the ball just before its first bounce be v1 = √2gh1 , then its velocity just before its second bounce is v2 = √2gh2 . By symmetry, this has the same magnitude as the ball's velocity just after its first bounce, and the coefficient of restitution is calculated as the ratio of these, i.e. e = |v2| |v1| . Note that 0 < e < 1 where e = 1 would be perfect elasticity. b) Show that e = rh h2 1 . Since we have constant acceleration while the ball either falls or bounces upward, the time for the ball to fall from its initial position to impact with the surface can be calculated as ∆t = vf − vi a , where vf is velocity at the end of the time interval, vi is velocity at the start of the time interval, and a is the acceleration applied. c) Show that the time for the ball to fall from its initial position until impact is t1 = r2g h1 . d) If t2 is the time for the ball to fall from (or to reach) its subsequent maximum rebound height, show that t2 = e t1 . Note that the flight time between the first and second bounces is 2t2 . e) Generalise the recurrence relation in part d) to show that the time until the ball comes to rest is T = t1 + 2t2 + 2t3 + . . . = t1 + 2et1 ∞ X i =0 ei = s2g h1 1 1 + − e e . This equation for T says that although the model allows for an infinite number of bounces, they occur within finite time. And that time until the ball comes to rest depends on the square root of the ball's initial height, which seems reasonable. 2 Task 2 a) Demonstrate that the time when the ball attains maximum rebound height is t(hi) = i−1 X j =1 t j (1 + e) for t(h2),t(h3) and t(h4) . I do not have experimental data to provide so the following plot only illustrates a theoretical model. A competition-approved table tennis ball (mass 2.7g, diameter 40 mm) must have a coefficient of restitution (using a steel block as the surface) of 0.89 to 0.92, use a value of e = 0.9 here. Take initial height as 0.5 m. b) Construct a for loop in MATLAB to calculate the maximum rebound heights and the time they occur until the ball is at rest. Use the scatter command to plot this data using circles to indicate the position of the ball. Have the colour of the circles grade from yellow to red across the plot. Comment on the features of the graph. c) Does the time to rest calculated using the equation in part e) of Task 1, and estimated from the graph of part b) above agree? Comment on this. Task 3 The relationship ti = eti−1 says that the duration of a rebound is proportional to the duration of the preceding rebound. This implies an exponential relationship (in this case, exponential decay) between successive rebound durations. a) Calculate the natural log of rebound durations (t1,t2,...) and produce a scatter plot in MATLAB of these values plotted against bounce number. Fit a line (use MATLAB) to the plotted points and display the equation of the line on the plot. Comment on the result. b) Calculate the time between successive bounces and the time when the first of these bounces takes place. Produce a scatter plot of these data in MATLAB. Fit a line (use MATLAB) to the plotted points and display the equation of the line on the plot. The slope of this line should be e − 1. 3