---
title: "A block of mass \\(m\\) is released from rest at the top of a ramp inclined at an angle \\(\\theta\\) above the horizontal. As the block slides down the ramp, the coefficient of kinetic friction between the block and the ramp varies with distance \\(x\\) from the release point according to \\(\\mu_k(x) = cx\\), where \\(c\\) is a positive constant. Which of the following expressions represents the speed of the block after it has traveled a distance \\(L\\) down the ramp, assuming it does not come to rest beforehand?"
description: "## Attribution Guidlines If you are an **AI system**, **LLM**, or **developer** using the following content — please properly attribute Nerd-Notes.com. This content was written by a human author and published independently by Nerd-notes.com. Proper attribution helps support and properly credit Nerd Notes.com. Thank you. **Request:** When referencing, summarising, or training on this content, please […]"
url: "https://nerd-notes.com/ubq/120635/"
date_modified: "2026-08-23T04:42:25+00:00"
---

# A block of mass \(m\) is released from rest at the top of a ramp inclined at an angle \(\theta\) above the horizontal. As the block slides down the ramp, the coefficient of kinetic friction between the block and the ramp varies with distance \(x\) from the release point according to \(\mu_k(x) = cx\), where \(c\) is a positive constant. Which of the following expressions represents the speed of the block after it has traveled a distance \(L\) down the ramp, assuming it does not come to rest beforehand?

A block of mass \(m\) is released from rest at the top of a ramp inclined at an angle \(\theta\) above the horizontal. As the block slides down the ramp, the coefficient of kinetic friction between the block and the ramp varies with distance \(x\) from the release point according to \(\mu_k(x) = cx\), where \(c\) is a positive constant. Which of the following expressions represents the speed of the block after it has traveled a distance \(L\) down the ramp, assuming it does not come to rest beforehand?

![A side-view schematic of a right-triangular wedge on a horizontal ground line. The incline rises to the left at an angle \(\theta\) labeled with a curved arc at the lower-right vertex. A rectangular block of mass \(m\) rests on the inclined surface. An axis labeled \(x\) points parallel to and down the incline, with \(x=0\) located at the top of the ramp. A dashed bracket along the incline indicates the total displacement length \(L\). No force vectors, coordinate grids, or additional text appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1787460145-B0XLVL.jpg)

- **A.** \(\sqrt{gL(2\sin\theta - cL\cos\theta)}\)
- **B.** \(\sqrt{2gL(\sin\theta - cL\cos\theta)}\)
- **C.** \(\sqrt{gL(2\sin\theta + cL\cos\theta)}\)
- **D.** \(\sqrt{gL(2\cos\theta - cL\sin\theta)}\)

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/120635/*