---
title: "A small block of mass \\(M\\) is projected with an initial speed \\(v_0\\) onto a horizontal test track. The track is engineered so that the coefficient of kinetic friction \\(\\mu_k\\) between the block and the track increases linearly with the horizontal distance \\(x\\) traveled, such that \\(\\mu_k(x) = cx\\), where \\(c\\) is a positive constant. The block starts at \\(x = 0\\) and comes to rest after traveling a stopping distance \\(x = D\\)."
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url: "https://nerd-notes.com/ubq/117785/"
date_modified: "2026-08-04T07:56:49+00:00"
---

# A small block of mass \(M\) is projected with an initial speed \(v_0\) onto a horizontal test track. The track is engineered so that the coefficient of kinetic friction \(\mu_k\) between the block and the track increases linearly with the horizontal distance \(x\) traveled, such that \(\mu_k(x) = cx\), where \(c\) is a positive constant. The block starts at \(x = 0\) and comes to rest after traveling a stopping distance \(x = D\).

A small block of mass \(M\) is projected with an initial speed \(v_0\) onto a horizontal test track. The track is engineered so that the coefficient of kinetic friction \(\mu_k\) between the block and the track increases linearly with the horizontal distance \(x\) traveled, such that \(\mu_k(x) = cx\), where \(c\) is a positive constant. The block starts at \(x = 0\) and comes to rest after traveling a stopping distance \(x = D\).

![A horizontal line representing a track. On the left end of the track, a rectangular block labeled M is drawn. A right-pointing arrow labeled v_0 extends from the block. Below the track, a horizontal axis is shown with a tick mark aligned with the center of the block labeled x = 0. Further to the right along the axis is another tick mark labeled x = D. The surface of the track between x = 0 and x = D is shaded with a dot gradient that becomes denser toward the right, indicating increasing friction, and is labeled \mu_k = cx. No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1785830209-Z0BI2S.jpg)

**Part a)** **Derive** an expression for the work done by friction on the block as it moves from \(x = 0\) to \(x = D\). Express your answer in terms of \(M\), \(c\), \(D\), and fundamental constants, as appropriate. *(2 points)*

**Part b)** Using your result from part (a) or otherwise, **derive** an expression for the stopping distance \(D\) in terms of \(v_0\), \(c\), and fundamental constants. *(2 points)*

**Part c)** Students want to experimentally determine the value of \(c\). They have access to the block, the engineered track, a launcher that can project the block at known variable speeds \(v_0\), and standard physics lab equipment. *(3 points)*

**Part d)** The students perform the experiment and record the following data. | Trial | \(v_0\) (m/s) | \(D\) (m) | | | |-------|---------------|-----------|---|---| | 1 | 1.5 | 0.24 | | | | 2 | 2.5 | 0.40 | | | | 3 | 3.5 | 0.55 | | | | 4 | 4.5 | 0.71 | | | | 5 | 5.5 | 0.87 | | | *(4 points)*

**Part e)** Using the straight line from part (d)(iii), **calculate** an experimental value for the constant \(c\). *(2 points)*

**Part f)** A second track has a traditional uniform surface with a constant coefficient of kinetic friction \(\mu_{k0}\). A student launches the block on this uniform track and measures the new stopping distance \(D_{new}\) for various launch speeds. If the student plots the exact same vertical and horizontal quantities as they did in part (d), **predict** the shape of the resulting graph (e.g., straight line, parabola, square root curve, etc.). **Justify** your answer using physical principles. *(2 points)*


*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/117785/*