---
title: "A particle moves along the x-axis starting from rest at the origin, with \\(x(0) = 0\\) and \\(v(0) = 0\\) at time \\(t = 0\\). For \\(t \\ge 0\\), its acceleration is modeled by \\(a(t) = a_0 e^{-\\beta t}\\), where \\(a_0\\) and \\(\\beta\\) are positive constants. Which of the following statements correctly characterizes the limiting behavior of the particle’s position \\(x(t)\\)?"
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url: "https://nerd-notes.com/ubq/120484/"
date_modified: "2026-08-23T04:39:06+00:00"
---

# A particle moves along the x-axis starting from rest at the origin, with \(x(0) = 0\) and \(v(0) = 0\) at time \(t = 0\). For \(t \ge 0\), its acceleration is modeled by \(a(t) = a_0 e^{-\beta t}\), where \(a_0\) and \(\beta\) are positive constants. Which of the following statements correctly characterizes the limiting behavior of the particle’s position \(x(t)\)?

A particle moves along the x-axis starting from rest at the origin, with \(x(0) = 0\) and \(v(0) = 0\) at time \(t = 0\). For \(t \ge 0\), its acceleration is modeled by \(a(t) = a_0 e^{-\beta t}\), where \(a_0\) and \(\beta\) are positive constants. Which of the following statements correctly characterizes the limiting behavior of the particle's position \(x(t)\)?

- **A.** As \(t \to \infty\), the position approaches a constant finite value \(x = \dfrac{a_0}{\beta^2}\), while for \(t \ll \dfrac{1}{\beta}\), the position is approximated by \(x(t) \approx a_0 t\).
- **B.** As \(t \to \infty\), the position graph approaches the linear asymptote \(x(t) = \dfrac{a_0}{\beta} t\), while for \(t \ll \dfrac{1}{\beta}\), the position is approximated by \(x(t) \approx a_0 t^2\).
- **C.** As \(t \to \infty\), the position graph approaches the linear asymptote \(x(t) = \dfrac{a_0}{\beta}\left(t + \dfrac{1}{\beta}\right)\), while for \(t \ll \dfrac{1}{\beta}\), the position is approximated by \(x(t) \approx \dfrac{1}{2}a_0 t^2\).
- **D.** As \(t \to \infty\), the position graph approaches the linear asymptote \(x(t) = \dfrac{a_0}{\beta}\left(t - \dfrac{1}{\beta}\right)\), while for \(t \ll \dfrac{1}{\beta}\), the position is approximated by \(x(t) \approx \dfrac{1}{2}a_0 t^2\).

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/120484/*