Question

A spherical raindrop falling through mist builds up mass. The mass accumlates at a rate proportional to its cross-sectional area and invers;ey proportional to its velocity, that is, dm/dt=k?r2/v, where r is the radius of the raindrop at a given time and v is the downard speed at the same time.Assuming the density P of the rain drop is a constant. Ignoring the resistance do to the fog and the air, calculate the instaneous acceleration of the raindrop as a function of v,r,P,g and k.

Answer 1

Here rate of increase of the mass of the rain drop isdm/dt=k?r²/v

Now mg = d/dt (mv)

             =mdv/dt + dm/dt *v

            Thenmg - dm/dt *v = ma

                     mg- (k?r²/v)*v = ma

                   mg - k?r² = ma

then instantaneous acceleration a = g-  k?r² /m

                                                  =g -k?r² /(4/3)?r³*?

                                                a   = g - (3/4r?)*k