---
title: "Two identical projectiles are launched from level horizontal ground with identical initial launch speeds \\(v_0\\) but at complementary launch angles \\(\\theta\\) and \\(90^\\circ – \\theta\\), where \\(0 < \\theta < 45^\\circ\\). Both projectiles experience negligible air resistance and travel the same horizontal range \\(R\\) before returning to the ground. If \\(H_1\\) is the maximum vertical height attained by the projectile launched at angle \\(\\theta\\) and \\(H_2\\) is the maximum vertical height attained by the projectile launched at angle \\(90^\\circ – \\theta\\), what is the ratio \\(\\dfrac{H_2}{H_1}\\)?"
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url: "https://nerd-notes.com/ubq/120509/"
date_modified: "2026-08-23T04:39:17+00:00"
---

# Two identical projectiles are launched from level horizontal ground with identical initial launch speeds \(v_0\) but at complementary launch angles \(\theta\) and \(90^\circ – \theta\), where \(0 < \theta < 45^\circ\). Both projectiles experience negligible air resistance and travel the same horizontal range \(R\) before returning to the ground. If \(H_1\) is the maximum vertical height attained by the projectile launched at angle \(\theta\) and \(H_2\) is the maximum vertical height attained by the projectile launched at angle \(90^\circ – \theta\), what is the ratio \(\dfrac{H_2}{H_1}\)?

Two identical projectiles are launched from level horizontal ground with identical initial launch speeds \(v_0\) but at complementary launch angles \(\theta\) and \(90^\circ - \theta\), where \(0 < \theta < 45^\circ\). Both projectiles experience negligible air resistance and travel the same horizontal range \(R\) before returning to the ground. If \(H_1\) is the maximum vertical height attained by the projectile launched at angle \(\theta\) and \(H_2\) is the maximum vertical height attained by the projectile launched at angle \(90^\circ - \theta\), what is the ratio \(\dfrac{H_2}{H_1}\)?

![A schematic diagram showing two parabolic trajectories launched from the same origin on a horizontal ground line. A horizontal solid line at the bottom represents the ground. At the origin on the left, two launch velocity vectors of equal length extend upward and to the right: one vector is tilted at angle \(\theta\) to the horizontal, and the second is tilted at a steeper angle \(90^\circ - \theta\). A low, wide parabolic curve labeled 1 starts at the origin, reaches a maximum height indicated by a vertical dashed dimension line labeled \(H_1\), and lands on the ground at distance \(R\). A tall, narrow parabolic curve labeled 2 starts at the origin, reaches a higher maximum height indicated by a vertical dashed dimension line labeled \(H_2\), and lands at the exact same distance \(R\) on the ground line. A horizontal dimension arrow below the ground line spans from the origin to the common landing point, labeled \(R\). No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1787459957-5PyBP7.jpg)

- **A.** \(1\)
- **B.** \(\cot\theta\)
- **C.** \(\tan^2\theta\)
- **D.** \(\cot^2\theta\)

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/120509/*