A spherical raindrop is falling through mist builds up mass. The mass accumulate

Question

A spherical raindrop is falling through mist builds up mass. The mass accumulates at a rate proportional to its cross-sectionalarea and velocity, that is, dm/dt=kr2/v, where ris the radius of the raindrop at a given time and v is the velocityat the same time. Assume the density of the raindropis a constant. Ignoring the resistance due to the fog and theair, calculate the instanteous accleration of the raindrop as afunction of v,r,,g, and k. (Hint - in this variable massproblem, similar to a two stage rocket problem in finding dp/dt andsolving for dv/dt.)

Explanation / Answer

Given dm/dt=kr2/v,

mg = d/dt (mv) = mdv/dt + dm/dt *v --> mg - dm/dt *v = ma --> mg - (kr2/v)*v = ma --> mg - kr2 = ma instantaneous acceleration a = g-  kr2 /m = g -kr2 /(4/3)r3*                                                
a    = g - (3/4r)*k

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