Conclusion

The goal in mathematical modeling is to devise a model that is as simple as possible, yet is accurate enough to make valid predictions of the underlying physical processes. When problems arise that require developing new mathematical models, simple models are usually developed first. The analysis of these simple models can direct the modeler in deciding what refinements must be made to obtain the desired accuracy.

Most modeling involves both theoretical and experimental components. Our study of a sphere moving through a fluid illustrates this phenomenon. We used theoretical analysis--dimensional analysis--in reducing the number of variables. Then we used experimental results to find the relationship between CD and Re. This relationship was then incorporated into a theoretical model for particle motion, namely Newton's Second Law. We then used techniques from calculus and differential equations to solve for the velocity and position of the sphere. The final step would be to verify our models from Section 4 using experimental measurements of falling spheres.

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