A spherical raindrop falling through mist builds up mass. Themass accumulates at

Question

A spherical raindrop falling through mist builds up mass. Themass accumulates at a rate of proportional to its cross-sectionalarea and velocity, that is, where r is the radius of theraindrop at a given time and v is the velocity at the sametime. Assume the density ? of the raindrop is aconstant. Ignoring the resistance due to the fog and the air,calculate the instantaneous acceleration of the raindrop as afunction of v,r,?, g and k. (Hint - in this variablemass problem, proceed similarly to a rocket problem in findingdp/dt and solving 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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