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title: "A raindrop of initial mass \\(m_0\\) is dropped from rest at time \\(t = 0\\) in a mist-filled atmosphere. As it falls under the influence of gravity, it absorbs stationary water droplets such that its mass increases at a rate \\(\\dfrac{dm}{dt} = b m\\), where \\(b\\) is a positive constant. Assuming air resistance is negligible compared to the force required to accelerate the absorbed mist, which of the following expressions gives the speed \\(v(t)\\) of the raindrop as a function of time \\(t\\)?"
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url: "https://nerd-notes.com/ubq/117575/"
date_modified: "2026-08-04T07:49:11+00:00"
---

# A raindrop of initial mass \(m_0\) is dropped from rest at time \(t = 0\) in a mist-filled atmosphere. As it falls under the influence of gravity, it absorbs stationary water droplets such that its mass increases at a rate \(\dfrac{dm}{dt} = b m\), where \(b\) is a positive constant. Assuming air resistance is negligible compared to the force required to accelerate the absorbed mist, which of the following expressions gives the speed \(v(t)\) of the raindrop as a function of time \(t\)?

A raindrop of initial mass \(m_0\) is dropped from rest at time \(t = 0\) in a mist-filled atmosphere. As it falls under the influence of gravity, it absorbs stationary water droplets such that its mass increases at a rate \(\dfrac{dm}{dt} = b m\), where \(b\) is a positive constant. Assuming air resistance is negligible compared to the force required to accelerate the absorbed mist, which of the following expressions gives the speed \(v(t)\) of the raindrop as a function of time \(t\)?

- **A.** \(v(t) = g t\)
- **B.** \(v(t) = \dfrac{g}{b} e^{-bt}\)
- **C.** \(v(t) = \dfrac{g}{b} \left(e^{bt} - 1\right)\)
- **D.** \(v(t) = \dfrac{g}{b} \left(1 - e^{-bt}\right)\)

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/117575/*