---
title: "A cart of mass \\(m\\) accelerates from rest along a straight, horizontal track. The cart’s electric motor delivers a constant mechanical power \\(P\\), and resistive forces are negligible. A motion sensor records the cart’s speed \\(v\\) as a function of its position \\(x\\) along the track. To determine the power-to-mass ratio \\(\\dfrac{P}{m}\\) from the slope \\(S\\) of a linear graph, which quantities should be plotted on the vertical and horizontal axes, and what is the value of \\(S\\) in terms of \\(P\\) and \\(m\\)?"
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url: "https://nerd-notes.com/ubq/124209/"
date_modified: "2026-09-28T13:58:53+00:00"
---

# A cart of mass \(m\) accelerates from rest along a straight, horizontal track. The cart’s electric motor delivers a constant mechanical power \(P\), and resistive forces are negligible. A motion sensor records the cart’s speed \(v\) as a function of its position \(x\) along the track. To determine the power-to-mass ratio \(\dfrac{P}{m}\) from the slope \(S\) of a linear graph, which quantities should be plotted on the vertical and horizontal axes, and what is the value of \(S\) in terms of \(P\) and \(m\)?

A cart of mass \(m\) accelerates from rest along a straight, horizontal track. The cart's electric motor delivers a constant mechanical power \(P\), and resistive forces are negligible. A motion sensor records the cart's speed \(v\) as a function of its position \(x\) along the track. To determine the power-to-mass ratio \(\dfrac{P}{m}\) from the slope \(S\) of a linear graph, which quantities should be plotted on the vertical and horizontal axes, and what is the value of \(S\) in terms of \(P\) and \(m\)?

![A side-view diagram of an experimental setup on a horizontal track. A horizontal track is represented by a single horizontal line. Resting on the track is a rectangular cart labeled m, with two small circular wheels touching the track. A horizontal arrow pointing to the right is attached to the front of the cart and labeled v. At the far left end of the track, positioned behind the cart, is a small rectangular motion sensor mounted on a vertical post. A horizontal dashed line extends from the sensor toward the rear of the cart, indicating the sensor's line of detection. Below the track, a horizontal double-headed arrow parallel to the track extends from the starting position of the cart to its current position, labeled x. No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1790603933-Vn22Ki.jpg)

- **A.** Vertical axis: \(v^2\); Horizontal axis: \(x\); Slope: \(S = \dfrac{P}{m}\)
- **B.** Vertical axis: \(v^2\); Horizontal axis: \(x\); Slope: \(S = \dfrac{2P}{m}\)
- **C.** Vertical axis: \(v^3\); Horizontal axis: \(x\); Slope: \(S = \dfrac{3P}{m}\)
- **D.** Vertical axis: \(v^3\); Horizontal axis: \(x\); Slope: \(S = \dfrac{6P}{m}\)

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/124209/*