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title: "A space probe of initial mass \\(m_0\\) moves through deep space with an initial speed \\(v_0\\) in a region free of net external forces. At time \\(t = 0\\), the probe enters a stationary dust cloud and begins collecting mass at a rate \\(\\dfrac{dm}{dt} = C v^2\\), where \\(C\\) is a positive constant and \\(v\\) is the instantaneous speed of the probe. Assuming all collected dust is initially at rest relative to the cloud and remains inside the probe, which of the following expressions correctly gives the speed \\(v(t)\\) of the probe as a function of time \\(t\\)?"
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url: "https://nerd-notes.com/ubq/117590/"
date_modified: "2026-08-04T07:49:13+00:00"
---

# A space probe of initial mass \(m_0\) moves through deep space with an initial speed \(v_0\) in a region free of net external forces. At time \(t = 0\), the probe enters a stationary dust cloud and begins collecting mass at a rate \(\dfrac{dm}{dt} = C v^2\), where \(C\) is a positive constant and \(v\) is the instantaneous speed of the probe. Assuming all collected dust is initially at rest relative to the cloud and remains inside the probe, which of the following expressions correctly gives the speed \(v(t)\) of the probe as a function of time \(t\)?

A space probe of initial mass \(m_0\) moves through deep space with an initial speed \(v_0\) in a region free of net external forces. At time \(t = 0\), the probe enters a stationary dust cloud and begins collecting mass at a rate \(\dfrac{dm}{dt} = C v^2\), where \(C\) is a positive constant and \(v\) is the instantaneous speed of the probe. Assuming all collected dust is initially at rest relative to the cloud and remains inside the probe, which of the following expressions correctly gives the speed \(v(t)\) of the probe as a function of time \(t\)?

- **A.** \(v(t) = \dfrac{v_0}{\left(1 + \dfrac{3 C v_0^2 t}{m_0}\right)^{1/3}}\)
- **B.** \(v(t) = \dfrac{v_0}{\left(1 + \dfrac{C v_0^2 t}{m_0}\right)^{1/3}}\)
- **C.** \(v(t) = \dfrac{v_0}{\left(1 + \dfrac{2 C v_0^2 t}{m_0}\right)^{1/2}}\)
- **D.** \(v(t) = v_0 \exp\left(-\dfrac{C v_0^2 t}{m_0}\right)\)

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/117590/*