---
title: "A cart of mass \\(m\\) is released from rest at Point A, which is at a vertical height \\(H\\) above the ground. The cart travels along a track consisting of three straight segments: – **Segment AB:** A frictionless downward incline of length \\(D\\). – **Segment BC:** A horizontal rough patch of length \\(L\\) with a constant coefficient of kinetic friction \\(\\mu_k\\). – **Segment CD:** A frictionless upward incline.  The cart passes Point C and eventually momentarily stops at Point D. Air resistance is negligible."
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url: "https://nerd-notes.com/ubq/109327/"
date_modified: "2026-03-31T09:26:08+00:00"
---

# A cart of mass \(m\) is released from rest at Point A, which is at a vertical height \(H\) above the ground. The cart travels along a track consisting of three straight segments:
– **Segment AB:** A frictionless downward incline of length \(D\).
– **Segment BC:** A horizontal rough patch of length \(L\) with a constant coefficient of kinetic friction \(\mu_k\).
– **Segment CD:** A frictionless upward incline.

The cart passes Point C and eventually momentarily stops at Point D. Air resistance is negligible.

A cart of mass \(m\) is released from rest at Point A, which is at a vertical height \(H\) above the ground. The cart travels along a track consisting of three straight segments:
- **Segment AB:** A frictionless downward incline of length \(D\).
- **Segment BC:** A horizontal rough patch of length \(L\) with a constant coefficient of kinetic friction \(\mu_k\).
- **Segment CD:** A frictionless upward incline.

The cart passes Point C and eventually momentarily stops at Point D. Air resistance is negligible.

![A 2D side-view of a roller coaster track. The track starts at a high Point A, at a vertical height \(H\) above the ground, on the left, goes down a straight angled ramp  to Point B. From Point B to Point C, the track is perfectly horizontal. From Point C, the track goes up a straight angled ramp to Point D on the right. The segment BC is shaded with a hatched pattern to indicate a rough surface. A small rectangular cart is shown at Point A. Labels 'A', 'B', 'C', and 'D' clearly mark the respective points.](https://nerd-notes.com/wp-content/uploads/ubq-diagrams/ubq-frq-generatedstem-fig-1-1774948787-1WllVE.jpg)

**Part a)** **Complete** the energy bar charts for the two distinct systems described below for the cart's motion from Point A to Point C. **Shade** the appropriate regions to represent the relative magnitudes of the kinetic energy \(K\), gravitational potential energy \(U_g\), internal energy \(\Delta E_{\text{int}}\), and external work \(W\) done on the system. If a quantity is zero, **label** it with a '0'. *(4 points)*

**Part b)** **Sketch** a graph of the kinetic energy \(K\) of the cart and the total mechanical energy \(E\) of the Cart-Earth system as a function of the distance \(d\) traveled along the track from Point A to Point C. **Label** your two curves clearly as '\(K\)' and '\(E\)'. *(3 points)*

**Part c)** A new track is constructed. On this new track, the horizontal segment BC is frictionless, but the inclined segment CD (which is at an angle \(\theta\) above the horizontal) now contains a rough patch. The new rough patch has the exact same length \(L\) and the same coefficient of kinetic friction \(\mu_k\) as the original rough patch. *(5 points)*


*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/109327/*