A spherical body of mass m and radius r is allowed to fall in a medium of viscocity ? the time in which the velocity of the body increases from 0 to 0.63 times the terminal velocity v is called the time constant ?. dimentionallt ? can be represented by

When an object moves through a viscous medium such as air or water, the object experiences a velocity dependent retarding force.                    This retarding force may be proportional to the first (linear) or the second power (quadratic) of the velocity (a case often studied in classical mechanics).

We will model the movement of an object moving through a viscous medium with the following differential equation:

                                     m.dV/dt = mg - kV

where m is the mass of the object,

                 k is a scaling or proportionality factor that accounts for the area experiencing the viscous force, dimensional formula is MT^-1

                 V is the velocity,

=>         integral of [dV/(mg - kV)] limits from 1 to 0.63V_t = integral of [ dt/m ] limits from 0 to T(time constant)

on solving gives T=(m/k)*ln[mg / (mg - 0.63V_t)]

where its dimesnion formula  is T