How Do You Solve a Raindrop's Growth Rate Using Differential Equations?

[gambler84]

the rate of growth of the mass M of a spherical rain drop falling through a particular cloud is given by dM/dt = Cr^3 where M = (rho)(4/3)(pi)r^3 and C is a constant

a) eliminate M from the above equation so that the size of the drop is expressed solely in terms of the radius r.

b) separate the variables and integrate to find an expression for r(t), given an intial radius r0 at time t=0



my attempt at part a consisted of me switching the Mass equations to Volume equations, yielding V = (4/3)(pi)r^3
and getting r = (3V/4(pi))^1/3

i don't think this is right. i haven't had any differential equations course as of yet

 

[elibj123]

You have M in terms of r. Knowing that r changes in time (dr/dt != 0), what would be dM/dt in terms of r and dr/dt?

 

[gambler84]

idk, that's why i posted the question

 

[Mark44]

Use the chain rule. You have M = f(r) and f = g(t), so dM/dt = (?)(?)