---
title: "A block of mass \\(m\\) is projected with initial speed \\(v_0\\) across a horizontal surface at time \\(t = 0\\) and position \\(x = 0\\). The only horizontal force acting on the block is a fluid resistive force of magnitude \\(F_{\\text{drag}} = kv^n\\), where \\(k\\) is a positive constant and \\(n\\) is a positive integer. Which of the following statements correctly characterizes the limiting behavior of the block’s motion as \\(t \\to \\infty\\) for linear drag (\\(n = 1\\)) compared to quadratic drag (\\(n = 2\\))?"
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url: "https://nerd-notes.com/ubq/123989/"
date_modified: "2026-09-28T13:29:59+00:00"
---

# A block of mass \(m\) is projected with initial speed \(v_0\) across a horizontal surface at time \(t = 0\) and position \(x = 0\). The only horizontal force acting on the block is a fluid resistive force of magnitude \(F_{\text{drag}} = kv^n\), where \(k\) is a positive constant and \(n\) is a positive integer. Which of the following statements correctly characterizes the limiting behavior of the block’s motion as \(t \to \infty\) for linear drag (\(n = 1\)) compared to quadratic drag (\(n = 2\))?

A block of mass \(m\) is projected with initial speed \(v_0\) across a horizontal surface at time \(t = 0\) and position \(x = 0\). The only horizontal force acting on the block is a fluid resistive force of magnitude \(F_{\text{drag}} = kv^n\), where \(k\) is a positive constant and \(n\) is a positive integer. Which of the following statements correctly characterizes the limiting behavior of the block's motion as \(t \to \infty\) for linear drag (\(n = 1\)) compared to quadratic drag (\(n = 2\))?

![A rectangular block of mass \(m\) rests on a horizontal line representing a flat surface. A horizontal arrow pointing to the right labeled \(v_0\) originates from the right edge of the block. A horizontal arrow pointing to the left labeled \(F_{\text{drag}}\) originates from the center of the block. A horizontal coordinate axis labeled \(x\) runs beneath the surface pointing to the right, with a vertical tick mark labeled \(x = 0\) aligned with the front edge of the block. No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1790602199-AqPMM2.jpg)

- **A.** In both cases, the speed approaches zero and the total distance traveled approaches a finite value.
- **B.** In both cases, the speed approaches zero, but the total distance traveled approaches a finite value for \(n = 1\) and increases without bound for \(n = 2\).
- **C.** In both cases, the total distance traveled increases without bound because the time required for the speed to reach zero is infinite.
- **D.** For \(n = 1\), the block comes to rest in a finite time after traveling a finite distance, whereas for \(n = 2\), the block travels an infinite distance.

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/123989/*