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After some derivation, the formula for the polynomial drag force constant equates to: $$ B = \frac{mg}{v_t^2} $$ Where B is the coefficient, m is the mass, g is gravity, and vt is terminal velocity. Let's say that the drag force on the object increases because the surface area increases. What impact would this have on the coefficient? Would it increase, decrease, or stay the same?

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From the dimensional analysis, you have that $$ \vec{F}_{drag}\propto - \rho_{air} A v^2 \hat{u}_v $$ so you find that $$ B \propto \rho_{air} A $$ where $A$ is the surface

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