The drag force \(f\) is a force of friction that proportional to the velocity of an object and is applied directly opposite to the velocity. We may write such force as a vector simply as,

\[\boldsymbol{f} = -f(v) \boldsymbol{\hat v}\]

A good approximation for the magnitude is the Taylor series noting that the first term goes to zero.

\[ f(v) \approx bv + cv^2 + ... \]

Ideal Drag Approximation

If the object traveling is a perfect sphere and traveling much less than the speed of sound then we may approximate using the first two terms of \(f(v)\),

\[ f(\nu) = 2r\beta + 4r^2\gamma \]

\(r\) : radius of the object