A spherical raindrop falling through mist builds up mass. The mass accumulates a

Question

A spherical raindrop falling through mist builds up mass. The mass accumulates at a rate proportional to its cross-sectional area and velocity, that is, dm/dt=(k**r2 )/v, where r is the radius of the raindrop at a given time and v is the velocity at the same time. Assume the density p of the raindrop is a constant. Ignoring the resistance due to the fog and the air, calculate the instantaneous acceleration of the raindrop as a function of v,r,p,g and k. (Hint - in this variable mass problem, proceed simiraly to a rocket problem in finding dp/dt and solving for dv/dt.)

Explanation / Answer

the given equation is,

dm/dt=kr2/v,

mg = d/dt (mv)

= mdv/dt + dm/dt *v

solve the above equation, as follows:

mg - [dm/dt]v = ma

mg - (kr2/v)*v = ma

therefore, the instantaneous acceleration is,

a = g-  kr2/m

= g -kr2/(4/3)r3

  = g - (3/4r)k

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