A water droplet evaporates before they reach the ground. Figure 1: Water droplets (source) In this situation, a few assumptions are made: a) At initial point, a typical water droplet is in sphere shape with radius r and remain spherical while evaporating. b) The rate of evaporation (when it loses mass (m)) is proportional to the surface area, S. c) There is no air-resistance and downward direction is the positive direction. To describe this problem, given that p is the mass density of water, rois the radius of water before it drops, m is the water mass, V is the water volume and k is the constant of proportionality. QUESTION: (1) From assumption (b), show that the radius of the water droplet at time t is r(t) = () t + ro. (Hint: m a oY Y =T

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Author:Erwin Kreyszig
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Chapter2: Second-order Linear Odes
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Differential equation

A water droplet evaporates before they reach the ground.
Figure 1: Water droplets (source]
In this situation, a few assumptions are made:
a) At initial point, a typical water droplet is in sphere shape with radius r and remain spherical
while evaporating.
b) The rate of evaporation (when it loses mass (m)) is proportional to the surface area, S.
c) There is no air-resistance and downward direction is the positive direction.
To describe this problem, given that p is the mass density of water, rois the radius of water before it
drops, m is the water mass, V is the water volume and k is the constant of proportionality.
QUESTION:
(1) From assumption (b), show that the radius of the water droplet at time t is
r(t) = () t + ro-
(Hint: m = pV,V =ar³, S = 4nr²).
Transcribed Image Text:A water droplet evaporates before they reach the ground. Figure 1: Water droplets (source] In this situation, a few assumptions are made: a) At initial point, a typical water droplet is in sphere shape with radius r and remain spherical while evaporating. b) The rate of evaporation (when it loses mass (m)) is proportional to the surface area, S. c) There is no air-resistance and downward direction is the positive direction. To describe this problem, given that p is the mass density of water, rois the radius of water before it drops, m is the water mass, V is the water volume and k is the constant of proportionality. QUESTION: (1) From assumption (b), show that the radius of the water droplet at time t is r(t) = () t + ro- (Hint: m = pV,V =ar³, S = 4nr²).
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