---
title: "A student designs an experiment to determine the local gravitational acceleration \\(g\\). A launcher at ground level fires a small projectile with a constant initial speed \\(v_0\\) at various launch angles \\(\\theta\\) above the horizontal. A photogate sensor positioned at a fixed height \\(h\\) above the ground records the time interval \\(\\Delta t\\) between the projectile passing the sensor on its upward path and passing the sensor on its downward path. Which quantities should the student plot on the vertical and horizontal axes to produce a linear graph, and how is \\(g\\) determined from the slope \\(M\\) of the best-fit line?"
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url: "https://nerd-notes.com/ubq/120507/"
date_modified: "2026-08-23T04:39:16+00:00"
---

# A student designs an experiment to determine the local gravitational acceleration \(g\). A launcher at ground level fires a small projectile with a constant initial speed \(v_0\) at various launch angles \(\theta\) above the horizontal. A photogate sensor positioned at a fixed height \(h\) above the ground records the time interval \(\Delta t\) between the projectile passing the sensor on its upward path and passing the sensor on its downward path. Which quantities should the student plot on the vertical and horizontal axes to produce a linear graph, and how is \(g\) determined from the slope \(M\) of the best-fit line?

A student designs an experiment to determine the local gravitational acceleration \(g\). A launcher at ground level fires a small projectile with a constant initial speed \(v_0\) at various launch angles \(\theta\) above the horizontal. A photogate sensor positioned at a fixed height \(h\) above the ground records the time interval \(\Delta t\) between the projectile passing the sensor on its upward path and passing the sensor on its downward path. Which quantities should the student plot on the vertical and horizontal axes to produce a linear graph, and how is \(g\) determined from the slope \(M\) of the best-fit line?

![A schematic showing a horizontal ground line. At the left end of the ground line, a launcher barrel is inclined at angle \(\theta\) to the horizontal ground. A dashed parabolic curve extends upward and to the right from the launcher, peaks, and descends to the ground. A horizontal dashed line at height \(h\) above the ground intersects both the ascending and descending segments of the parabolic curve. A vertical double-headed dimension arrow extends from the ground line to the horizontal dashed line and is labeled \(h\). Two circular dots mark where the trajectory crosses the dashed line at height \(h\), with a horizontal dimension bracket between the two dots labeled \(\Delta t\). No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1787459956-Eu4DfF.jpg)

- **A.** Plot \(\Delta t\) on the vertical axis versus \(\sin\theta\) on the horizontal axis, and calculate \(g = \dfrac{2v_0}{M}\).
- **B.** Plot \((\Delta t)^2\) on the vertical axis versus \(\sin\theta\) on the horizontal axis, and calculate \(g = \dfrac{2v_0}{\sqrt{M}}\).
- **C.** Plot \((\Delta t)^2\) on the vertical axis versus \(\sin^2\theta\) on the horizontal axis, and calculate \(g = \dfrac{4v_0^2}{M}\).
- **D.** Plot \((\Delta t)^2\) on the vertical axis versus \(\sin^2\theta\) on the horizontal axis, and calculate \(g = \dfrac{2v_0}{\sqrt{M}}\).

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/120507/*