---
title: "A space probe of mass \\(m\\) coasts in deep space with an initial velocity \\(v_0\\,\\hat{\\imath}\\). At time \\(t = 0\\), a braking thruster activates, exerting a net external force \\(\\vec{F}(t) = -F_0 e^{-t/\\tau}\\,\\hat{\\imath}\\) along the line of motion, where \\(F_0\\) and \\(\\tau\\) are positive constants. In the limit \\(t \\to \\infty\\), which of the following statements correctly describes the motion and velocity of the probe?"
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url: "https://nerd-notes.com/ubq/123986/"
date_modified: "2026-09-28T13:29:58+00:00"
---

# A space probe of mass \(m\) coasts in deep space with an initial velocity \(v_0\,\hat{\imath}\). At time \(t = 0\), a braking thruster activates, exerting a net external force \(\vec{F}(t) = -F_0 e^{-t/\tau}\,\hat{\imath}\) along the line of motion, where \(F_0\) and \(\tau\) are positive constants. In the limit \(t \to \infty\), which of the following statements correctly describes the motion and velocity of the probe?

A space probe of mass \(m\) coasts in deep space with an initial velocity \(v_0\,\hat{\imath}\). At time \(t = 0\), a braking thruster activates, exerting a net external force \(\vec{F}(t) = -F_0 e^{-t/\tau}\,\hat{\imath}\) along the line of motion, where \(F_0\) and \(\tau\) are positive constants. In the limit \(t \to \infty\), which of the following statements correctly describes the motion and velocity of the probe?

- **A.** If \(F_0 \tau < m v_0\), the probe never comes to rest and approaches a final velocity of \(\left(v_0 - \dfrac{F_0 \tau}{m}\right)\hat{\imath}\); if \(F_0 \tau = m v_0\), it comes to rest asymptotically as \(t \to \infty\).
- **B.** Because the braking force acts over an infinite time interval, the total impulse exerted on the probe is unbounded, so the probe eventually reverses direction for any positive values of \(F_0\) and \(\tau\).
- **C.** If \(\dfrac{F_0}{\tau} < m v_0\), the probe never comes to rest and approaches a final velocity of \(\left(v_0 - \dfrac{F_0}{m \tau}\right)\hat{\imath}\); if \(\dfrac{F_0}{\tau} = m v_0\), it comes to rest asymptotically as \(t \to \infty\).
- **D.** If \(F_0 \tau = m v_0\), the probe comes to rest at a finite time; if \(F_0 \tau > m v_0\), the probe asymptotically approaches rest as \(t \to \infty\) without ever reversing direction.

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/123986/*