---
title: "A sphere of mass \\(m\\) is released from rest in a viscous fluid and falls vertically under the influence of gravity and a resistive drag force characterized by constant \\(b\\). The graph shows the sphere’s downward acceleration \\(a\\) as a function of its downward speed \\(v\\), with initial acceleration \\(a_0\\) and terminal speed \\(v_T\\). Which of the following correctly identifies the speed dependence of the drag force, the magnitude of the slope of the graph, and the acceleration of the sphere when its speed is \\(\\dfrac{1}{2}v_T\\)?"
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url: "https://nerd-notes.com/ubq/124058/"
date_modified: "2026-09-28T13:30:17+00:00"
---

# A sphere of mass \(m\) is released from rest in a viscous fluid and falls vertically under the influence of gravity and a resistive drag force characterized by constant \(b\). The graph shows the sphere’s downward acceleration \(a\) as a function of its downward speed \(v\), with initial acceleration \(a_0\) and terminal speed \(v_T\). Which of the following correctly identifies the speed dependence of the drag force, the magnitude of the slope of the graph, and the acceleration of the sphere when its speed is \(\dfrac{1}{2}v_T\)?

A sphere of mass \(m\) is released from rest in a viscous fluid and falls vertically under the influence of gravity and a resistive drag force characterized by constant \(b\). The graph shows the sphere's downward acceleration \(a\) as a function of its downward speed \(v\), with initial acceleration \(a_0\) and terminal speed \(v_T\). Which of the following correctly identifies the speed dependence of the drag force, the magnitude of the slope of the graph, and the acceleration of the sphere when its speed is \(\dfrac{1}{2}v_T\)?

![A two-dimensional Cartesian coordinate system shown in grayscale. The horizontal axis is oriented to the right and labeled \(v\). The vertical axis is oriented upward and labeled \(a\). The axes meet at an origin labeled \(0\). Along the vertical axis, a single tick mark above the origin is labeled \(a_0\). Along the horizontal axis, a single tick mark to the right of the origin is labeled \(v_T\). A single straight solid line segment begins exactly at the point \((0, a_0)\) on the vertical axis and slopes downward to the right with a constant negative slope, terminating exactly at the point \((v_T, 0)\) on the horizontal axis. The background is plain white with no gridlines. No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1790602217-biOQY9.jpg)

- **A.** The drag force is proportional to \(v\), the magnitude of the slope is \(\dfrac{b}{m}\), and the acceleration is \(\dfrac{1}{2}a_0\).
- **B.** The drag force is proportional to \(v\), the magnitude of the slope is \(\dfrac{m}{b}\), and the acceleration is \(\dfrac{1}{4}a_0\).
- **C.** The drag force is proportional to \(v^2\), the magnitude of the slope is \(\dfrac{b}{m}\), and the acceleration is \(\dfrac{3}{4}a_0\).
- **D.** The drag force is proportional to \(v^2\), the magnitude of the slope is \(\dfrac{m}{b}\), and the acceleration is \(\dfrac{1}{2}a_0\).

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/124058/*