---
title: "A student is conducting an experiment on projectile motion. A launcher is placed on level ground a horizontal distance \\(D\\) from a vertical building. The building has a small, open window whose center is at a height \\(H\\) above the launcher. The student wants to launch a small sphere of mass \\(m\\) with an initial speed \\(v_0\\) at an angle \\(\\theta\\) above the horizontal so that the sphere enters the window exactly at the peak of its trajectory. Air resistance is negligible."
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url: "https://nerd-notes.com/ubq/113971/"
date_modified: "2026-06-22T07:39:54+00:00"
---

# A student is conducting an experiment on projectile motion. A launcher is placed on level ground a horizontal distance \(D\) from a vertical building. The building has a small, open window whose center is at a height \(H\) above the launcher. The student wants to launch a small sphere of mass \(m\) with an initial speed \(v_0\) at an angle \(\theta\) above the horizontal so that the sphere enters the window exactly at the peak of its trajectory. Air resistance is negligible.

A student is conducting an experiment on projectile motion. A launcher is placed on level ground a horizontal distance \(D\) from a vertical building. The building has a small, open window whose center is at a height \(H\) above the launcher. The student wants to launch a small sphere of mass \(m\) with an initial speed \(v_0\) at an angle \(\theta\) above the horizontal so that the sphere enters the window exactly at the peak of its trajectory. Air resistance is negligible.

![A horizontal line representing the ground. On the right, a vertical rectangle representing a building. A gap in the left wall of the building at height H is labeled 'Window'. On the ground, on the left, a small circular object represents the sphere. A dashed parabolic arc starts at the object, curves up and right, and reaches its vertex exactly at the gap ('Window'). A solid arrow points up and right from the object, labeled v_0. An angle arc between the ground and the arrow is labeled \theta. A horizontal dimension line between the object and the building is labeled D. A vertical dimension line between the ground and the window is labeled H.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1782113994-JdGuBY.jpg)

**Part a)** **Derive** an expression for the horizontal distance \(D\) in terms of \(H\), \(\theta\), and fundamental constants, as appropriate. *(3 points)*

**Part b)** Assume the window is located at a height \(H = 5.0 \text{ m}\) and the sphere is launched at an angle \(\theta = 60^\circ\). **Calculate** the speed of the sphere at the instant it passes through the window. *(2 points)*

**Part c)** The student moves the launcher to a new position that is closer to the building, such that the new horizontal distance is \(D_{\text{new}}\), where \(D_{\text{new}} < D\). The student wishes to launch a new sphere so that it still enters the window exactly at the peak of its trajectory. *(3 points)*

**Part d)** Suppose the student returns the launcher to the original horizontal distance \(D\) but wants the sphere to reach its peak exactly at a new window located at a height of \(4H\). *(3 points)*


*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/113971/*