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Model of a Bouncing Ball

Model of a Bouncing Ball. A ball falls from a height, bounces against a surface, and jumps to the air again. This repeats a number of times until the ball rests on the surface. Model of a Bouncing Ball. Objective

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Model of a Bouncing Ball

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  1. Model of a Bouncing Ball A ball falls from a height, bounces against a surface, and jumps to the air again. This repeats a number of times until the ball rests on the surface.

  2. Model of a Bouncing Ball • Objective • Generate data showing the maximum height reached by the ball after each bounce and plot them • Assumptions • Air resistance is negligible (the speed of the ball is not very high) • The surface is a smooth plane, perpendicular to the falling ball and rigidly attached to an object massively larger than the ball Mathematical Model h = The height from which the ball drops h’ = the new height that the ball bounces back to after hitting the surface k = Coefficient of restitution (a measure of the elasticity of the collision) k=1 for perfect elastic collision, k<1 otherwise According to the theory of collisions: Kinetic energy after collision = k× kinetic energy before collision mv2after = k∙mv2before (Eq. 1) mv2after = mghn (Eq. 3) mv2before = mghn-1 (Eq. 2) hn = khn-1 (Eq. 4)

  3. Optimization • What is Optimization • Its objective is to select the best possible decision for a given set of circumstances without having to enumerate all of the possibilities • Involves maximization or minimization as desired • How can a large drug company determine the monthly product mix at their Indianapolis plant that maximizes corporate profitability? • If Microsoft produces Xbox consoles at three locations, how can they minimize the cost of meeting demand for Xbox consoles? • Linear Programming • To optimize a linear objective function, subject to linear equality and inequality constraints • A constraint is a condition that a solution to an optimization problem must satisfy

  4. Maximize z = 15x1+10x2 subject to 0≤x1 ≤2, 0 ≤ x2 ≤ 3, x1+x2 ≤4 The objective function is z = 15x1+10x2 The constraints are: 0≤ x1≤2, 0 ≤ x2≤ 3, x1+x2 ≤4

  5. x1=2 x2=3 Optimal point (x1=x2=2) x1+ x2 = 4 z = 40 z = 30 z = 20 z = 10 x2 Feasible Region x1 zmax = 15*2 + 10*2 = 50

  6. Excel Solver • A Microsoft Excel Add-In • Go to Tools >>Add-Ins , select Solver Add-in, click OK • Originally designed for optimization problems but also useful for root finding and similar mathematical problems • Target cell • The objective or goal • Maximize, minimize or set a specific value to the target cell • Changing cells • Can be adjusted until the constraints in the problem are satisfied and the cell in the Set Target Cell box reaches its target • Constraints • The restrictions placed on the changing cells

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