Question

A raindrop of mass m0, starting from rest, falls under the influence of gravity. Assume that as the raindrop travels through the clouds, it gains mass at a rate proportional to the momentum of the raindrop, dmr = kmrvr, where mr is the in- dt stantaneous mass of the raindrop, vr is the instantaneous velocity of the raindrop, 5 and k is a constant with unit [m−1]. You may neglect air resistance. (a) Derive a differential equation for the raindrop’s accelerations dvr in terms of dt k, g, dt and the raindrop’s instantaneous velocity vr . Express your answer using some or all of the following variables: k,g for the gravitational acceleration and vr, the raindrop’s instantaneous velocity.

Answer 1

`(dv_(r))/(dt)=g-k_(r)v_(r)^2`

Step-by-step explanation:

Second Newton's Law:

`F=(dp)/(dt)=(d(mv))/(dt)=m(dv)/(dt)+v(dm)/(dt) \\` (1)

m and v are the instantaneous mass and instantaneous velocity

The only force is the weight:

`F=mg` (2)

On the other hand we know:

`(dm)/(dt)=k*m*v` (3)

We replace (2) and (3) in (1), and we solve for dv/dt :

`(dv)/(dt)=g-kv^2`