---
title: "A block of mass \\(m\\) is initially sliding to the right with speed \\(v_0\\) on a horizontal, frictionless track. At point A, the block enters a rough section of track of length \\(L\\) where the coefficient of kinetic friction between the block and the track is \\(\\mu_k\\). After passing point B, the track becomes frictionless again and transitions into a ramp inclined at an angle \\(\\theta\\) to the horizontal. The block travels up the ramp and reaches a maximum vertical height \\(h\\) above the horizontal track before momentarily coming to rest. Assume the block-Earth system has zero gravitational potential energy when the block is on the horizontal track."
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url: "https://nerd-notes.com/ubq/110142/"
date_modified: "2026-04-01T07:59:37+00:00"
---

# A block of mass \(m\) is initially sliding to the right with speed \(v_0\) on a horizontal, frictionless track. At point A, the block enters a rough section of track of length \(L\) where the coefficient of kinetic friction between the block and the track is \(\mu_k\). After passing point B, the track becomes frictionless again and transitions into a ramp inclined at an angle \(\theta\) to the horizontal. The block travels up the ramp and reaches a maximum vertical height \(h\) above the horizontal track before momentarily coming to rest. Assume the block-Earth system has zero gravitational potential energy when the block is on the horizontal track.

A block of mass \(m\) is initially sliding to the right with speed \(v_0\) on a horizontal, frictionless track. At point A, the block enters a rough section of track of length \(L\) where the coefficient of kinetic friction between the block and the track is \(\mu_k\). After passing point B, the track becomes frictionless again and transitions into a ramp inclined at an angle \(\theta\) to the horizontal. The block travels up the ramp and reaches a maximum vertical height \(h\) above the horizontal track before momentarily coming to rest. Assume the block-Earth system has zero gravitational potential energy when the block is on the horizontal track.

![A horizontal line representing a track that transitions into an inclined plane on the right. On the far left of the horizontal track, a square block labeled 'm' has a right-pointing velocity vector labeled 'v_0'. Further right on the horizontal track, a rectangular section is shaded to indicate roughness. The start of the shaded section is labeled point A and the end is labeled point B. A dimension line below the shaded section indicates its length is 'L'. To the right of point B, the track angles upward at an angle 'theta' relative to the horizontal dashed line. A dashed outline of the block is shown at its highest point on the incline, with a vertical dimension line labeled 'h' indicating its height above the horizontal track level.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1774589590-Ht8RRC.jpg)

**Part a)** **Derive** an expression for the change in total mechanical energy of the block-Earth system as the block moves from point A to point B. Express your answer in terms of \(m\), \(v_0\), \(L\), \(\mu_k\), \(\theta\), and fundamental constants, as appropriate. *(2 points)*

**Part b)** **Derive** an expression for the maximum vertical height \(h\) the block reaches on the incline. Express your answer in terms of \(m\), \(v_0\), \(L\), \(\mu_k\), \(\theta\), and fundamental constants, as appropriate. *(3 points)*

**Part c)** A second block of mass \(2m\) is now launched with the same initial speed \(v_0\) along the same track. **Indicate** whether the maximum vertical height reached by the new block is greater than, less than, or equal to the maximum vertical height \(h\) reached by the original block. - [ ] Greater than - [ ] Less than - [ ] Equal to **Justify** your answer using physical principles. *(3 points)*

**Part d)** Consider the block-Earth system. On the axes provided, **sketch** a graph of the total mechanical energy \(E\) of the system as a function of the block's horizontal position \(x\) from \(x = 0\) (point A) to \(x = L\) (point B). *(2 points)*


*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/110142/*