---
title: "A cart moves along a straight horizontal track from rest at the origin at time \\(t = 0\\). An optical sensor records the cart’s position \\(x\\) and speed \\(v\\), and the position data are well modeled by the equation \\(x(t) = c t^{3/2}\\), where \\(c\\) is an unknown positive constant. A student wants to linearize the relationship between \\(v\\) and \\(x\\) by plotting a function of \\(v\\) on the vertical axis and \\(x\\) on the horizontal axis to determine \\(c\\) from the slope \\(S\\) of the best-fit line. Which of the following quantities should be plotted on the vertical axis, and what is the resulting expression for \\(c\\) in terms of \\(S\\)?"
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url: "https://nerd-notes.com/ubq/124214/"
date_modified: "2026-09-28T13:59:26+00:00"
---

# A cart moves along a straight horizontal track from rest at the origin at time \(t = 0\). An optical sensor records the cart’s position \(x\) and speed \(v\), and the position data are well modeled by the equation \(x(t) = c t^{3/2}\), where \(c\) is an unknown positive constant. A student wants to linearize the relationship between \(v\) and \(x\) by plotting a function of \(v\) on the vertical axis and \(x\) on the horizontal axis to determine \(c\) from the slope \(S\) of the best-fit line. Which of the following quantities should be plotted on the vertical axis, and what is the resulting expression for \(c\) in terms of \(S\)?

A cart moves along a straight horizontal track from rest at the origin at time \(t = 0\). An optical sensor records the cart's position \(x\) and speed \(v\), and the position data are well modeled by the equation \(x(t) = c t^{3/2}\), where \(c\) is an unknown positive constant. A student wants to linearize the relationship between \(v\) and \(x\) by plotting a function of \(v\) on the vertical axis and \(x\) on the horizontal axis to determine \(c\) from the slope \(S\) of the best-fit line. Which of the following quantities should be plotted on the vertical axis, and what is the resulting expression for \(c\) in terms of \(S\)?

![A horizontal track is represented by a thick horizontal line resting on a flat surface. At the far left end of the track, a small upright rectangular block represents a motion sensor, labeled Sensor. Situated to the right of the sensor is a rectangular cart supported by two small circular wheels resting on the track. A horizontal arrow pointing to the right is centered above the cart and is labeled v. Directly below the track, a horizontal coordinate axis extends parallel to the track, beginning with a vertical tick mark aligned with the sensor labeled x = 0 and terminating on the right with an arrowhead labeled +x. No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1790603966-6g5MLz.jpg)

- **A.** Plot \(v^2\) on the vertical axis, and \(c = \left(\dfrac{4}{9}S\right)^{3/4}\)
- **B.** Plot \(v^3\) on the vertical axis, and \(c = \sqrt{S}\)
- **C.** Plot \(v^3\) on the vertical axis, and \(c = \sqrt{\dfrac{27}{8}S}\)
- **D.** Plot \(v^3\) on the vertical axis, and \(c = \sqrt{\dfrac{8}{27}S}\)

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/124214/*