A raindrop of mass m and speed v collects water as it falls through a cloud. Show that, neglecting air resistance, mg = mu0076u0307 + mu0307v. Assume that the drop remains spherical and that the rate of accretion is proportional to the cross-sectional area of the drop multiplied by the speed of fall. Show that this implies that mu0307 = cvm^{2/3}, for some constant c. Using this equation, writing dm/dx = (dm/dt)/(dx/dt), and assuming that the drop starts from rest when it is infinitesimally small, find m as a function of x. Hence show that d(v^2)/dx + 6v^2/x - 2g = 0. Finally, multiply through by a particular power of x (i.e. x^n for some value of n) so that the first two terms combine to form a perfect derivative, and show that the acceleration of the raindrop is constant and equal to g/7.