---
title: "A particle moves along the \\(x\\)-axis with a velocity given by \\(v(t) = v_0 e^{-kt}\\), where \\(v_0\\) and \\(k\\) are positive constants. The particle is located at \\(x = 0\\) at time \\(t = 0\\), and as \\(t \\to \\infty\\) it approaches a maximum total displacement \\(x_{\\text{max}}\\). In terms of \\(k\\), at what time \\(t\\) does the particle reach a position of \\(\\dfrac{1}{2}x_{\\text{max}}\\)?"
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url: "https://nerd-notes.com/ubq/117549/"
date_modified: "2026-08-04T07:49:00+00:00"
---

# A particle moves along the \(x\)-axis with a velocity given by \(v(t) = v_0 e^{-kt}\), where \(v_0\) and \(k\) are positive constants. The particle is located at \(x = 0\) at time \(t = 0\), and as \(t \to \infty\) it approaches a maximum total displacement \(x_{\text{max}}\). In terms of \(k\), at what time \(t\) does the particle reach a position of \(\dfrac{1}{2}x_{\text{max}}\)?

A particle moves along the \(x\)-axis with a velocity given by \(v(t) = v_0 e^{-kt}\), where \(v_0\) and \(k\) are positive constants. The particle is located at \(x = 0\) at time \(t = 0\), and as \(t \to \infty\) it approaches a maximum total displacement \(x_{\text{max}}\). In terms of \(k\), at what time \(t\) does the particle reach a position of \(\dfrac{1}{2}x_{\text{max}}\)?

- **A.** \(\dfrac{1}{2k}\)
- **B.** \(\dfrac{\ln 2}{k}\)
- **C.** \(\dfrac{1}{k}\)
- **D.** \(\dfrac{2\ln 2}{k}\)

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/117549/*