---
title: "A small bead of mass \\(m\\) is threaded onto a vertical guide wire.  The bead is subjected to a downward gravitational force, a constant upward magnetic force of magnitude \\(F_0\\) (where \\(F_0 < mg\\)), and an upward drag force of magnitude \\(bv\\), where \\(v\\) is the bead's downward speed and \\(b\\) is a positive constant. At time \\(t = 0\\), the bead is projected vertically downward with an initial speed \\(v_0\\). Which of the following expressions represents the downward speed \\(v(t)\\) of the bead as a function of time \\(t\\)?"
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url: "https://nerd-notes.com/ubq/124061/"
date_modified: "2026-09-28T13:30:19+00:00"
---

# A small bead of mass \(m\) is threaded onto a vertical guide wire.

The bead is subjected to a downward gravitational force, a constant upward magnetic force of magnitude \(F_0\) (where \(F_0 < mg\)), and an upward drag force of magnitude \(bv\), where \(v\) is the bead's downward speed and \(b\) is a positive constant. At time \(t = 0\), the bead is projected vertically downward with an initial speed \(v_0\). Which of the following expressions represents the downward speed \(v(t)\) of the bead as a function of time \(t\)?

A small bead of mass \(m\) is threaded onto a vertical guide wire.

The bead is subjected to a downward gravitational force, a constant upward magnetic force of magnitude \(F_0\) (where \(F_0 < mg\)), and an upward drag force of magnitude \(bv\), where \(v\) is the bead's downward speed and \(b\) is a positive constant. At time \(t = 0\), the bead is projected vertically downward with an initial speed \(v_0\). Which of the following expressions represents the downward speed \(v(t)\) of the bead as a function of time \(t\)?

![A single thin vertical line represents a guide wire. Centered along the wire is a small shaded circular bead labeled \(m\). Three vertical vector arrows originate from the center of the bead: one solid arrow pointing straight downward labeled \(mg\), one solid arrow pointing straight upward labeled \(F_0\), and a second solid arrow pointing straight upward labeled \(bv\). To the right of the bead, a downward-pointing arrow is labeled \(v_0\). No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1790602218-bmGk9I.jpg)

- **A.** \(v(t) = \left(\dfrac{mg - F_0}{b}\right)\left(1 - e^{-bt/m}\right)\)
- **B.** \(v(t) = v_0 + \left(\dfrac{mg - F_0}{b}\right)\left(1 - e^{-bt/m}\right)\)
- **C.** \(v(t) = v_0 e^{-bt/m} + \left(\dfrac{mg - F_0}{b}\right)\left(1 - e^{-bt/m}\right)\)
- **D.** \(v(t) = v_0 e^{-bt/m} + \left(\dfrac{mg + F_0}{b}\right)\left(1 - e^{-bt/m}\right)\)

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/124061/*