---
title: "An experiment is designed to determine the unknown spring constant \\(k\\) of an ideal horizontal spring. A cart of known mass \\(m\\) is placed against the spring, compressing it by a distance \\(x\\) from its equilibrium length along a horizontal track of negligible friction. The cart is released from rest, and a motion sensor records its final speed \\(v\\) after it completely separates from the spring. The trial is repeated for several different initial compression distances \\(x\\). Which quantities should be plotted on the vertical and horizontal axes to yield a linear graph, and what is the resulting expression for \\(k\\) in terms of the slope \\(S\\) of the best-fit line?"
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url: "https://nerd-notes.com/ubq/120648/"
date_modified: "2026-08-23T04:42:29+00:00"
---

# An experiment is designed to determine the unknown spring constant \(k\) of an ideal horizontal spring. A cart of known mass \(m\) is placed against the spring, compressing it by a distance \(x\) from its equilibrium length along a horizontal track of negligible friction. The cart is released from rest, and a motion sensor records its final speed \(v\) after it completely separates from the spring. The trial is repeated for several different initial compression distances \(x\). Which quantities should be plotted on the vertical and horizontal axes to yield a linear graph, and what is the resulting expression for \(k\) in terms of the slope \(S\) of the best-fit line?

An experiment is designed to determine the unknown spring constant \(k\) of an ideal horizontal spring. A cart of known mass \(m\) is placed against the spring, compressing it by a distance \(x\) from its equilibrium length along a horizontal track of negligible friction. The cart is released from rest, and a motion sensor records its final speed \(v\) after it completely separates from the spring. The trial is repeated for several different initial compression distances \(x\). Which quantities should be plotted on the vertical and horizontal axes to yield a linear graph, and what is the resulting expression for \(k\) in terms of the slope \(S\) of the best-fit line?

![A horizontal line represents a frictionless track. On the left, a vertical wall is attached to the left end of a horizontal coil spring labeled k. The right end of the spring is in contact with a rectangular cart labeled mass m. A vertical dashed line indicates the equilibrium position of the spring. A horizontal double-headed dimension arrow between the vertical dashed line and the compressed spring position is labeled x. To the right of the cart, a small rectangular box represents a motion sensor pointing horizontally toward the cart. A single horizontal arrow labeled v points to the right above the cart. No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1787460149-pCc5Sx.jpg)

- **A.** Vertical axis: \(v^2\); Horizontal axis: \(x^2\); \(k = mS\)
- **B.** Vertical axis: \(v^2\); Horizontal axis: \(x^2\); \(k = \dfrac{1}{2}mS\)
- **C.** Vertical axis: \(v\); Horizontal axis: \(x\); \(k = mS\)
- **D.** Vertical axis: \(v^2\); Horizontal axis: \(x\); \(k = mS\)

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/120648/*