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title: "A research rocket of initial mass \\(M_0\\) is launched vertically upward from rest at time \\(t = 0\\) near Earth’s surface in a uniform gravitational field of magnitude \\(g\\). The rocket engine produces a constant upward thrust force of magnitude \\(F_0\\) while consuming fuel at a constant mass rate \\(b\\), so that the total mass of the rocket varies with time according to \\(m(t) = M_0 – bt\\). Air resistance is negligible, and \\(F_0 > M_0 g\\). Which of the following expressions represents the upward speed \\(v(t)\\) of the rocket as a function of time for \\(0 \\le t < M_0/b\\)?"
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url: "https://nerd-notes.com/ubq/120553/"
date_modified: "2026-08-23T04:41:34+00:00"
---

# A research rocket of initial mass \(M_0\) is launched vertically upward from rest at time \(t = 0\) near Earth’s surface in a uniform gravitational field of magnitude \(g\). The rocket engine produces a constant upward thrust force of magnitude \(F_0\) while consuming fuel at a constant mass rate \(b\), so that the total mass of the rocket varies with time according to \(m(t) = M_0 – bt\). Air resistance is negligible, and \(F_0 > M_0 g\). Which of the following expressions represents the upward speed \(v(t)\) of the rocket as a function of time for \(0 \le t < M_0/b\)?

A research rocket of initial mass \(M_0\) is launched vertically upward from rest at time \(t = 0\) near Earth's surface in a uniform gravitational field of magnitude \(g\). The rocket engine produces a constant upward thrust force of magnitude \(F_0\) while consuming fuel at a constant mass rate \(b\), so that the total mass of the rocket varies with time according to \(m(t) = M_0 - bt\). Air resistance is negligible, and \(F_0 > M_0 g\). Which of the following expressions represents the upward speed \(v(t)\) of the rocket as a function of time for \(0 \le t < M_0/b\)?

- **A.** \(\dfrac{F_0 t}{M_0} - gt\)
- **B.** \(\dfrac{F_0 t}{M_0 - bt} - gt\)
- **C.** \(\dfrac{F_0}{b} \ln\left(\dfrac{M_0 - bt}{M_0}\right) - gt\)
- **D.** \(\dfrac{F_0}{b} \ln\left(\dfrac{M_0}{M_0 - bt}\right) - gt\)

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/120553/*