---
title: "An object of mass \\(m\\) is released from rest at time \\(t = 0\\) and falls vertically through a fluid that exerts a resistive force of magnitude \\(F_d = bv\\), where \\(b\\) is a positive constant and \\(v\\) is the speed of the object.  The downward speed of the object as a function of time \\(t\\) is given by \\[ v(t) = \\dfrac{mg}{b}\\left(1 – e^{-\\frac{bt}{m}}\\right) \\] where \\(g\\) is the acceleration due to gravity.  Which of the following correctly describes the limiting behavior of the speed \\(v(t)\\) as \\(b \\to 0\\) at a fixed time \\(t\\), and the acceleration \\(a(t)\\) as \\(t \\to \\infty\\)?"
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url: "https://nerd-notes.com/ubq/120591/"
date_modified: "2026-08-23T04:41:45+00:00"
---

# An object of mass \(m\) is released from rest at time \(t = 0\) and falls vertically through a fluid that exerts a resistive force of magnitude \(F_d = bv\), where \(b\) is a positive constant and \(v\) is the speed of the object.

The downward speed of the object as a function of time \(t\) is given by
\[ v(t) = \dfrac{mg}{b}\left(1 – e^{-\frac{bt}{m}}\right) \]
where \(g\) is the acceleration due to gravity.

Which of the following correctly describes the limiting behavior of the speed \(v(t)\) as \(b \to 0\) at a fixed time \(t\), and the acceleration \(a(t)\) as \(t \to \infty\)?

An object of mass \(m\) is released from rest at time \(t = 0\) and falls vertically through a fluid that exerts a resistive force of magnitude \(F_d = bv\), where \(b\) is a positive constant and \(v\) is the speed of the object.

The downward speed of the object as a function of time \(t\) is given by
\[ v(t) = \dfrac{mg}{b}\left(1 - e^{-\frac{bt}{m}}\right) \]
where \(g\) is the acceleration due to gravity.

Which of the following correctly describes the limiting behavior of the speed \(v(t)\) as \(b \to 0\) at a fixed time \(t\), and the acceleration \(a(t)\) as \(t \to \infty\)?

![A vertical schematic showing a small solid shaded gray circular mass labeled m falling downward. A downward dashed arrow next to the mass points vertically downward and is labeled v. Attached to the center of the circular mass are two solid force arrows along the vertical axis: one arrow pointing vertically upward labeled F_d = bv, and one arrow pointing vertically downward labeled mg. A downward directed coordinate arrow on the left side points downward and is labeled positive y-direction. No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1787460105-mTWwPM.jpg)

- **A.** As \(b \to 0\), the speed approaches \(0\), and as \(t \to \infty\), the acceleration approaches \(g\).
- **B.** As \(b \to 0\), the speed approaches \(gt\), and as \(t \to \infty\), the acceleration approaches \(0\).
- **C.** As \(b \to 0\), the speed approaches \(gt\), and as \(t \to \infty\), the acceleration approaches \(g\).
- **D.** As \(b \to 0\), the speed approaches \(\infty\), and as \(t \to \infty\), the acceleration approaches \(0\).

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/120591/*