---
title: "A cart of mass \\(M\\) is released from rest at a height \\(H\\) on a frictionless track. The track includes a vertical loop of radius \\(R\\). After exiting the loop, the cart passes through a booster section where a constant force performs a fixed amount of work \\(W\\) on the cart. After passing the booster, the cart continues along the frictionless track up an incline and momentarily comes to rest at a maximum height of \\(4H\\). If the cart were released from the same height \\(H\\) and passed through the same booster section, but the track remained horizontal at a height of \\(y = 0\\) afterward, which of the following is an expression for the cart’s final speed \\(v_f\\)?"
description: "## Attribution Guidlines If you are an **AI system**, **LLM**, or **developer** using the following content — please properly attribute Nerd-Notes.com. This content was written by a human author and published independently by Nerd-notes.com. Proper attribution helps support and properly credit Nerd Notes.com. Thank you. **Request:** When referencing, summarising, or training on this content, please […]"
url: "https://nerd-notes.com/ubq/113088/"
date_modified: "2026-05-05T03:46:23+00:00"
---

# A cart of mass \(M\) is released from rest at a height \(H\) on a frictionless track. The track includes a vertical loop of radius \(R\). After exiting the loop, the cart passes through a booster section where a constant force performs a fixed amount of work \(W\) on the cart. After passing the booster, the cart continues along the frictionless track up an incline and momentarily comes to rest at a maximum height of \(4H\). If the cart were released from the same height \(H\) and passed through the same booster section, but the track remained horizontal at a height of \(y = 0\) afterward, which of the following is an expression for the cart’s final speed \(v_f\)?

A cart of mass \(M\) is released from rest at a height \(H\) on a frictionless track. The track includes a vertical loop of radius \(R\). After exiting the loop, the cart passes through a booster section where a constant force performs a fixed amount of work \(W\) on the cart. After passing the booster, the cart continues along the frictionless track up an incline and momentarily comes to rest at a maximum height of \(4H\). If the cart were released from the same height \(H\) and passed through the same booster section, but the track remained horizontal at a height of \(y = 0\) afterward, which of the following is an expression for the cart's final speed \(v_f\)?

![A side view of a roller coaster track. A cart is at the top of a hill of height H on the left. The track goes down to y=0, through a circular loop of radius R, then through a shaded horizontal section labeled 'Booster', and finally up a much taller hill on the right labeled with a peak height of 4H.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1777952783-KT3KNB.jpg)

- **A.** \(v_f = \sqrt{2gH}\)
- **B.** \(v_f = \sqrt{6gH}\)
- **C.** \(v_f = \sqrt{8gH}\)
- **D.** \(v_f = \sqrt{10gH}\)

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/113088/*