---
title: "A particle of mass \\(m\\) is initially at rest on a horizontal, frictionless surface at time \\(t = 0\\). Starting at \\(t = 0\\), a horizontal force \\(F(t) = F_0 e^{-t/\\tau}\\) is applied to the particle, where \\(F_0\\) and \\(\\tau\\) are positive constants.  | Choice | Final Momentum \\(p_{\\infty}\\) | Fraction of Total Impulse (\\(0 \\le t \\le \\tau\\)) | | :—: | :—: | :—: | | A | \\(\\dfrac{F_0}{\\tau}\\) | \\(e^{-1}\\) | | B | \\(\\dfrac{F_0}{\\tau}\\) | \\(1 – e^{-1}\\) | | C | \\(F_0 \\tau\\) | \\(e^{-1}\\) | | D | \\(F_0 \\tau\\) | \\(1 – e^{-1}\\) |  Which choice correctly identifies the magnitude of the final momentum \\(p_{\\infty}\\) of the particle as \\(t \\to \\infty\\) and the fraction of the total impulse delivered to the particle during the time interval \\(0 \\le t \\le \\tau\\)?"
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url: "https://nerd-notes.com/ubq/117599/"
date_modified: "2026-08-04T07:49:16+00:00"
---

# A particle of mass \(m\) is initially at rest on a horizontal, frictionless surface at time \(t = 0\). Starting at \(t = 0\), a horizontal force \(F(t) = F_0 e^{-t/\tau}\) is applied to the particle, where \(F_0\) and \(\tau\) are positive constants.

| Choice | Final Momentum \(p_{\infty}\) | Fraction of Total Impulse (\(0 \le t \le \tau\)) |
| :—: | :—: | :—: |
| A | \(\dfrac{F_0}{\tau}\) | \(e^{-1}\) |
| B | \(\dfrac{F_0}{\tau}\) | \(1 – e^{-1}\) |
| C | \(F_0 \tau\) | \(e^{-1}\) |
| D | \(F_0 \tau\) | \(1 – e^{-1}\) |

Which choice correctly identifies the magnitude of the final momentum \(p_{\infty}\) of the particle as \(t \to \infty\) and the fraction of the total impulse delivered to the particle during the time interval \(0 \le t \le \tau\)?

A particle of mass \(m\) is initially at rest on a horizontal, frictionless surface at time \(t = 0\). Starting at \(t = 0\), a horizontal force \(F(t) = F_0 e^{-t/\tau}\) is applied to the particle, where \(F_0\) and \(\tau\) are positive constants.

| Choice | Final Momentum \(p_{\infty}\) | Fraction of Total Impulse (\(0 \le t \le \tau\)) |
| :---: | :---: | :---: |
| A | \(\dfrac{F_0}{\tau}\) | \(e^{-1}\) |
| B | \(\dfrac{F_0}{\tau}\) | \(1 - e^{-1}\) |
| C | \(F_0 \tau\) | \(e^{-1}\) |
| D | \(F_0 \tau\) | \(1 - e^{-1}\) |

Which choice correctly identifies the magnitude of the final momentum \(p_{\infty}\) of the particle as \(t \to \infty\) and the fraction of the total impulse delivered to the particle during the time interval \(0 \le t \le \tau\)?

- **A.** \(p_{\infty} = \dfrac{F_0}{\tau}\), Fraction = \(e^{-1}\)
- **B.** \(p_{\infty} = \dfrac{F_0}{\tau}\), Fraction = \(1 - e^{-1}\)
- **C.** \(p_{\infty} = F_0 \tau\), Fraction = \(e^{-1}\)
- **D.** \(p_{\infty} = F_0 \tau\), Fraction = \(1 - e^{-1}\)

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/117599/*