EAS208_PROJECT_092210 - EAS 208 DYNAMICS Computer Project...

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EAS 208 – DYNAMICS Computer Project – Fall 2010 Due: Monday, December 6, 2010 Flight of a Golf Ball Consider the dynamics of a golf ball as represented by the following model of projectile motion: 8 De mC R D m g π μ =− vv j & (1) where is the mass of the golf ball, is its velocity, m v D C is the dimensionless drag coefficient, is the diameter of the golf ball, D is the viscosity of air, is the gravitational acceleration assumed to act in the minus g y -direction and the dot represents a derivative with respect to time t . Meanwhile, e R is the dimensionless Reynolds number given by: e UD R ρ = (2) with as the air density and U representing the speed of the golf ball. Thus: U = v (3) After dividing Eq. (1) through by m , one obtains the following vector equation: 8 CRD g m = v v j & (4) 1
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This can be rewritten equivalently as a pair of scalar equations: 8 De uC R D m u π μ =− & (5a) 8 vC R D v m g = & (5c) where and represent the u v x and components of the velocity, respectively. Then, if the position of the golf ball at any time t is given by the coordinates y () x t and ( ) yt , then: x u = & (5b) yv = & (5d) Equations (5a-d) represent a set of first order ordinary differential equations that can be solved for , , ut vt x t and ( ) provided that a set of initial conditions are known. If the effect of air resistance is neglected, this reduces to the usual equations for projectile motion which can be solved in closed form by direct integration (i.e., by hand). However, with the inclusion of the drag terms, the problem becomes nonlinear and, in general, the equations must be solved by numerical methods (i.e., by using a computer). There are many different numerical methods that can be used to solve equations (5a-d).
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This document was uploaded on 10/27/2011 for the course EAS 208 at SUNY Buffalo.

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EAS208_PROJECT_092210 - EAS 208 DYNAMICS Computer Project...

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