---
title: "A sled of mass \\(m\\) is initially at rest at a height \\(H\\) on a frictionless inclined ramp. At the bottom of the ramp, the sled transitions smoothly onto a horizontal track and slides across a rough patch of snow of length \\(L\\). The coefficient of kinetic friction between the sled and the rough snow is \\(\\mu_k\\). After crossing the rough patch, the sled slides along a frictionless horizontal section and collides with an uncompressed ideal spring of spring constant \\(k\\) attached to a rigid wall. The sled compresses the spring a maximum distance \\(x_0\\) before momentarily coming to rest. Assume the sled can be treated as a point mass and that air resistance is negligible."
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url: "https://nerd-notes.com/ubq/109196/"
date_modified: "2026-03-21T10:39:33+00:00"
---

# A sled of mass \(m\) is initially at rest at a height \(H\) on a frictionless inclined ramp. At the bottom of the ramp, the sled transitions smoothly onto a horizontal track and slides across a rough patch of snow of length \(L\). The coefficient of kinetic friction between the sled and the rough snow is \(\mu_k\). After crossing the rough patch, the sled slides along a frictionless horizontal section and collides with an uncompressed ideal spring of spring constant \(k\) attached to a rigid wall. The sled compresses the spring a maximum distance \(x_0\) before momentarily coming to rest. Assume the sled can be treated as a point mass and that air resistance is negligible.

A sled of mass \(m\) is initially at rest at a height \(H\) on a frictionless inclined ramp. At the bottom of the ramp, the sled transitions smoothly onto a horizontal track and slides across a rough patch of snow of length \(L\). The coefficient of kinetic friction between the sled and the rough snow is \(\mu_k\). After crossing the rough patch, the sled slides along a frictionless horizontal section and collides with an uncompressed ideal spring of spring constant \(k\) attached to a rigid wall. The sled compresses the spring a maximum distance \(x_0\) before momentarily coming to rest. Assume the sled can be treated as a point mass and that air resistance is negligible.

![A side-view diagram showing a small block (representing the sled) initially at rest at the top of a curved ramp of vertical height H. The ramp curves smoothly into a flat horizontal track. On the horizontal track, there is a shaded rectangular region labeled 'Rough Patch' with length L and friction coefficient \(\mu_k\). To the right of the rough patch, the horizontal track continues to a vertical wall with a coiled spring (constant k) attached. The block is also shown in dashed lines at the end of the track, fully compressing the spring by a distance labeled \(x_0\).](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1774089573-HjulrZ.jpg)

**Part a)** Two students are analyzing the energy of the sled's motion from its initial release to the moment of maximum spring compression. Student A defines the system as the sled, the spring, and Earth. Student B defines the system as the sled, the spring, Earth, and the rough surface. For the system defined by Student A, **explain** why the total mechanical energy of the system decreases as the sled travels from its initial position to the point of maximum spring compression. For the system defined by Student B, **indicate** whether the *total energy* of the system increases, decreases, or remains the same during the entire motion. **Justify** your answer. *(3 points)*

**Part b)** **Derive** an expression for the maximum compression distance \(x_0\) of the spring. Express your answer in terms of \(m\), \(H\), \(L\), \(\mu_k\), \(k\), and fundamental constants. *(3 points)*

**Part c)** The experiment is repeated on a different hill where the release height is \(2H\) and the length of the rough patch is \(2L\). All other parameters remain the same. A student argues that because both the initial release height and the length of the friction patch are doubled, the new maximum spring compression will be exactly \(2x_0\). **Indicate** whether the new maximum spring compression is greater than, less than, or equal to \(2x_0\). - [ ] Greater than \(2x_0\) - [ ] Less than \(2x_0\) - [ ] Equal to \(2x_0\) **Justify** your answer using physical principles, without explicitly referencing the equation you derived in Part (b). *(4 points)*

**Part d)** **Explain** how your derived equation from Part (b) supports your reasoning in Part (c). *(2 points)*


*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/109196/*