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title: "A sled of initial total mass \\(M_0\\) is initially at rest on a frictionless horizontal surface. At time \\(t = 0\\), a constant horizontal force \\(F_0\\) is applied to the sled, and water simultaneously begins to leak out through a hole in the bottom at a rate given by \\(\\dfrac{dm}{dt} = -bm\\), where \\(b\\) is a positive constant and \\(m(t)\\) is the instantaneous mass of the sled and remaining water. The water leaves the sled with zero horizontal velocity relative to the sled. Which of the following expressions correctly gives the speed \\(v(t)\\) of the sled as a function of time \\(t\\)?"
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url: "https://nerd-notes.com/ubq/117595/"
date_modified: "2026-08-04T07:49:14+00:00"
---

# A sled of initial total mass \(M_0\) is initially at rest on a frictionless horizontal surface. At time \(t = 0\), a constant horizontal force \(F_0\) is applied to the sled, and water simultaneously begins to leak out through a hole in the bottom at a rate given by \(\dfrac{dm}{dt} = -bm\), where \(b\) is a positive constant and \(m(t)\) is the instantaneous mass of the sled and remaining water. The water leaves the sled with zero horizontal velocity relative to the sled. Which of the following expressions correctly gives the speed \(v(t)\) of the sled as a function of time \(t\)?

A sled of initial total mass \(M_0\) is initially at rest on a frictionless horizontal surface. At time \(t = 0\), a constant horizontal force \(F_0\) is applied to the sled, and water simultaneously begins to leak out through a hole in the bottom at a rate given by \(\dfrac{dm}{dt} = -bm\), where \(b\) is a positive constant and \(m(t)\) is the instantaneous mass of the sled and remaining water. The water leaves the sled with zero horizontal velocity relative to the sled. Which of the following expressions correctly gives the speed \(v(t)\) of the sled as a function of time \(t\)?

![A rectangular block representing a sled rests on a smooth horizontal line. A single horizontal arrow labeled F_0 points to the right from the center of the right side of the block. Below the center of the block, three vertical downward arrows represent water droplets falling straight down from a small opening in the bottom. A horizontal dashed arrow pointing right above the block is labeled v(t). No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1785829754-8xY6Gy.jpg)

- **A.** \(v(t) = \dfrac{F_0}{b M_0} \left( 1 - e^{-bt} \right)\)
- **B.** \(v(t) = \dfrac{F_0}{b M_0} \left( e^{bt} - 1 \right)\)
- **C.** \(v(t) = \dfrac{F_0 t}{M_0} e^{bt}\)
- **D.** \(v(t) = \dfrac{F_0}{b M_0} e^{bt}\)

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/117595/*