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title: "A rocket in deep space starts from rest at time \\(t = 0\\) with an initial mass \\(M_0\\). The rocket engine burns fuel at a constant mass rate \\(R = -\\dfrac{dm}{dt}\\) and expels exhaust gas at a constant speed \\(u\\) relative to the rocket. Assuming no external forces act on the system, which of the following expressions correctly gives the speed \\(v(t)\\) of the rocket as a function of time \\(t\\) for \\(t < \\dfrac{M_0}{R}\\)?"
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url: "https://nerd-notes.com/ubq/117570/"
date_modified: "2026-08-04T07:49:09+00:00"
---

# A rocket in deep space starts from rest at time \(t = 0\) with an initial mass \(M_0\). The rocket engine burns fuel at a constant mass rate \(R = -\dfrac{dm}{dt}\) and expels exhaust gas at a constant speed \(u\) relative to the rocket. Assuming no external forces act on the system, which of the following expressions correctly gives the speed \(v(t)\) of the rocket as a function of time \(t\) for \(t < \dfrac{M_0}{R}\)?

A rocket in deep space starts from rest at time \(t = 0\) with an initial mass \(M_0\). The rocket engine burns fuel at a constant mass rate \(R = -\dfrac{dm}{dt}\) and expels exhaust gas at a constant speed \(u\) relative to the rocket. Assuming no external forces act on the system, which of the following expressions correctly gives the speed \(v(t)\) of the rocket as a function of time \(t\) for \(t < \dfrac{M_0}{R}\)?

![A cylindrical rocket of mass m moving to the right along a horizontal axis in deep space. A stream of exhaust gas is expelled to the left behind the rocket with exhaust speed u relative to the rocket. The rocket's velocity vector points to the right, labeled v. A dashed horizontal line indicates the line of motion. No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1785829749-FLtfbL.jpg)

- **A.** \(v(t) = u \left(\dfrac{Rt}{M_0}\right)\)
- **B.** \(v(t) = u \ln\left(\dfrac{M_0}{M_0 - Rt}\right)\)
- **C.** \(v(t) = u \ln\left(\dfrac{M_0 - Rt}{M_0}\right)\)
- **D.** \(v(t) = u \left(1 - e^{-Rt/M_0}\right)\)

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/117570/*