---
title: "A particle moves along the \\(x\\)-axis with an initial velocity \\(v_0 > 0\\) at position \\(x = 0\\) when time \\(t = 0\\). The acceleration of the particle depends on its instantaneous velocity according to the relation \\(a(v) = -kv\\), where \\(k\\) is a positive constant with units of \\(\\text{s}^{-1}\\). Which of the following statements correctly describes the speed of the particle and its total distance traveled in the limit \\(t \\to \\infty\\)?"
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url: "https://nerd-notes.com/ubq/120487/"
date_modified: "2026-08-23T04:39:07+00:00"
---

# A particle moves along the \(x\)-axis with an initial velocity \(v_0 > 0\) at position \(x = 0\) when time \(t = 0\). The acceleration of the particle depends on its instantaneous velocity according to the relation \(a(v) = -kv\), where \(k\) is a positive constant with units of \(\text{s}^{-1}\). Which of the following statements correctly describes the speed of the particle and its total distance traveled in the limit \(t \to \infty\)?

A particle moves along the \(x\)-axis with an initial velocity \(v_0 > 0\) at position \(x = 0\) when time \(t = 0\). The acceleration of the particle depends on its instantaneous velocity according to the relation \(a(v) = -kv\), where \(k\) is a positive constant with units of \(\text{s}^{-1}\). Which of the following statements correctly describes the speed of the particle and its total distance traveled in the limit \(t \to \infty\)?

- **A.** The speed approaches zero asymptotically as \(t \to \infty\), and the total distance traveled approaches the finite value \(\dfrac{v_0}{k}\).
- **B.** The speed approaches zero asymptotically as \(t \to \infty\), and the total distance traveled approaches infinity.
- **C.** The speed reaches zero at the finite time \(t = \dfrac{1}{k}\), and the total distance traveled is \(\dfrac{v_0}{k}\).
- **D.** The speed reaches zero at the finite time \(t = \dfrac{1}{k}\), and the total distance traveled is \(\dfrac{v_0}{2k}\).

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/120487/*