A student drops two spherical balls of different diameters and different densities. She has a stroboscopic photograph showing the positions of each ball as a function of time. However, she wants to express the velocity of each as a function of time in dimensionless form. Develop the dimensionless group. The equation of motion for each ball is

\[ m g-\frac{\mathrm{C}_{\mathbf{D}}}{2} ho A V^{2}=m \frac{d V}{d t} \]

where \(m\) is ball mass, \(g\) is acceleration of gravity, \(\mathrm{C}_{\mathrm{D}}\) is a dimensionless and constant drag coefficient, \(ho\) is air mass density, \(A\) is ball cross-sectional area \(\left(=\pi \mathrm{D}^{2} / 4\right)\) with \(D\) ball diameter, \(V\) is ball velocity, and \(t\) is time.