---
title: "A block of mass \\(m\\) is projected along a horizontal surface in the \\(+x\\)-direction with an initial speed \\(v_0\\) at position \\(x = 0\\). The coefficient of kinetic friction between the block and the surface increases linearly with position according to \\(\\mu(x) = \\mu_0 \\dfrac{x}{L}\\), where \\(\\mu_0\\) and \\(L\\) are positive constants, bringing the block to rest at position \\(x_1\\). In a second trial, an identical block is projected from \\(x = 0\\) with an initial speed of \\(2v_0\\) and comes to rest at position \\(x_2\\). What is the ratio \\(\\dfrac{x_2}{x_1}\\)?"
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url: "https://nerd-notes.com/ubq/124084/"
date_modified: "2026-09-28T13:30:21+00:00"
---

# A block of mass \(m\) is projected along a horizontal surface in the \(+x\)-direction with an initial speed \(v_0\) at position \(x = 0\). The coefficient of kinetic friction between the block and the surface increases linearly with position according to \(\mu(x) = \mu_0 \dfrac{x}{L}\), where \(\mu_0\) and \(L\) are positive constants, bringing the block to rest at position \(x_1\). In a second trial, an identical block is projected from \(x = 0\) with an initial speed of \(2v_0\) and comes to rest at position \(x_2\). What is the ratio \(\dfrac{x_2}{x_1}\)?

A block of mass \(m\) is projected along a horizontal surface in the \(+x\)-direction with an initial speed \(v_0\) at position \(x = 0\). The coefficient of kinetic friction between the block and the surface increases linearly with position according to \(\mu(x) = \mu_0 \dfrac{x}{L}\), where \(\mu_0\) and \(L\) are positive constants, bringing the block to rest at position \(x_1\). In a second trial, an identical block is projected from \(x = 0\) with an initial speed of \(2v_0\) and comes to rest at position \(x_2\). What is the ratio \(\dfrac{x_2}{x_1}\)?

![A horizontal surface is represented by a solid horizontal line. A rectangular block of mass \(m\) rests on the horizontal surface with its left edge aligned with a vertical dashed line labeled \(x = 0\). An arrow labeled \(v_0\) points horizontally to the right from the right edge of the block. Below the surface line, a horizontal coordinate axis labeled \(x\) extends to the right, with a tick mark at the vertical dashed line labeled \(0\). Shading under the horizontal surface consists of vertical hatch marks that become progressively denser from left to right as \(x\) increases. A text label \(\mu(x) = \mu_0\dfrac{x}{L}\) is positioned below the hatch marks. No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1790602221-M598yz.jpg)

- **A.** \( \dfrac{1}{2} \)
- **B.** \( \sqrt{2} \)
- **C.** \( 2 \)
- **D.** \( 4 \)

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/124084/*