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title: "A balloon of initial mass \\(m_0\\) falls vertically downward in a uniform gravitational field \\(g\\). As it falls, it leaks gas at a constant rate \\(\\alpha = -\\dfrac{dm}{dt} > 0\\), giving an instantaneous mass \\(m(t) = m_0 – \\alpha t\\). The surrounding air exerts an upward drag force of magnitude \\(F_d = bv\\), where \\(b > \\alpha\\) is a constant and \\(v\\) is the downward speed. Assuming the expelled gas exits with zero velocity relative to the balloon, which of the following expressions gives the balloon’s downward speed \\(v(t)\\) once it reaches a quasi-steady state in which its speed decreases proportionally to its mass?"
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url: "https://nerd-notes.com/ubq/117659/"
date_modified: "2026-08-04T07:49:36+00:00"
---

# A balloon of initial mass \(m_0\) falls vertically downward in a uniform gravitational field \(g\). As it falls, it leaks gas at a constant rate \(\alpha = -\dfrac{dm}{dt} > 0\), giving an instantaneous mass \(m(t) = m_0 – \alpha t\). The surrounding air exerts an upward drag force of magnitude \(F_d = bv\), where \(b > \alpha\) is a constant and \(v\) is the downward speed. Assuming the expelled gas exits with zero velocity relative to the balloon, which of the following expressions gives the balloon’s downward speed \(v(t)\) once it reaches a quasi-steady state in which its speed decreases proportionally to its mass?

A balloon of initial mass \(m_0\) falls vertically downward in a uniform gravitational field \(g\). As it falls, it leaks gas at a constant rate \(\alpha = -\dfrac{dm}{dt} > 0\), giving an instantaneous mass \(m(t) = m_0 - \alpha t\). The surrounding air exerts an upward drag force of magnitude \(F_d = bv\), where \(b > \alpha\) is a constant and \(v\) is the downward speed. Assuming the expelled gas exits with zero velocity relative to the balloon, which of the following expressions gives the balloon's downward speed \(v(t)\) once it reaches a quasi-steady state in which its speed decreases proportionally to its mass?

- **A.** \(v(t) = \dfrac{m(t)g}{b + \alpha}\)
- **B.** \(v(t) = \dfrac{m(t)g}{b}\)
- **C.** \(v(t) = \dfrac{m_0 g}{b - \alpha}\)
- **D.** \(v(t) = \dfrac{m(t)g}{b - \alpha}\)

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/117659/*