Question

4. Falling Drop 

A raindrop of mass \(m_{0}\), 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, \(\frac{d m_{r}}{d t}=k m_{r} v_{r}\), where \(m_{r}\) is the instantaneous mass of the raindrop, \(v_{r}\) is the instantaneous velocity of the raindrop, and \(k\) is a constant with unit \(\left[m^{-1}\right] .\) You may neglect air resistance.

(a) Derive a differential equation for the raindrop's accelerations \(\frac{d v_{r}}{d t}\) in terms of \(k, g\), and the raindrop's instantaneous velocity \(v_{r}\). Express your answer using some or all of the following variables: \(k, g\) for the gravitational acceleration and \(v_{r}\), the raindrop's instantaneous velocity.

(b) What is the terminal speed, \(v_{T}\), of the raindrop? Express your answer using some or all of the following variables: \(k\) and \(g\) for the gravitational acceleration.

Answer 1