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.

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jorgezarate jorgezarate · Answer 1 · 2019-09-15T23:20:37+02:00

Answer:

[tex]\frac{dv_{r}}{dt}=g-k_{r}v_{r}^2[/tex]

Explanation:

Second Newton's Law:

[tex]F=\frac{dp}{dt}=\frac{d(mv)}{dt}=m\frac{dv}{dt}+v\frac{dm}{dt} \\[/tex] (1)

m and v are the instantaneous mass and instantaneous velocity

The only force is the weight:

[tex]F=mg[/tex] (2)

On the other hand we know:

[tex]\frac{dm}{dt}=k*m*v[/tex] (3)

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

[tex]\frac{dv}{dt}=g-kv^2[/tex]