---
title: "A projectile is launched from the base of a fixed incline that makes an angle \\(\\phi\\) with the horizontal, where \\(0 \\le \\phi < \\dfrac{\\pi}{2}\\). The projectile is fired with an initial speed \\(v_0\\) at an angle \\(\\theta\\) above the horizontal, where \\(\\phi < \\theta < \\dfrac{\\pi}{2}\\). Neglecting air resistance, which of the following expressions represents the maximum distance \\(R_{\\text{max}}\\) the projectile can travel along the inclined surface, optimized over all possible launch angles \\(\\theta\\)?"
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url: "https://nerd-notes.com/ubq/120493/"
date_modified: "2026-08-23T04:39:10+00:00"
---

# A projectile is launched from the base of a fixed incline that makes an angle \(\phi\) with the horizontal, where \(0 \le \phi < \dfrac{\pi}{2}\). The projectile is fired with an initial speed \(v_0\) at an angle \(\theta\) above the horizontal, where \(\phi < \theta < \dfrac{\pi}{2}\). Neglecting air resistance, which of the following expressions represents the maximum distance \(R_{\text{max}}\) the projectile can travel along the inclined surface, optimized over all possible launch angles \(\theta\)?

A projectile is launched from the base of a fixed incline that makes an angle \(\phi\) with the horizontal, where \(0 \le \phi < \dfrac{\pi}{2}\). The projectile is fired with an initial speed \(v_0\) at an angle \(\theta\) above the horizontal, where \(\phi < \theta < \dfrac{\pi}{2}\). Neglecting air resistance, which of the following expressions represents the maximum distance \(R_{\text{max}}\) the projectile can travel along the inclined surface, optimized over all possible launch angles \(\theta\)?

![A side view schematic shows a wedge-shaped incline rising from the lower left to the upper right. A horizontal baseline extends to the right from the bottom corner, with an angle arc labeled \(\phi\) between the baseline and the incline surface. An initial velocity vector arrow labeled \(v_0\) starts at the bottom corner and points upward and to the right at an angle \(\theta\) relative to the horizontal baseline. A dashed parabolic curve begins at the launch point, rises above the incline, and lands on the inclined surface. A straight dimension line parallel to the incline spans from the launch point to the landing impact point, labeled \(R\). No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1787459949-03RyJE.jpg)

- **A.** \(\dfrac{v_0^2}{g(1 + \sin\phi)}\)
- **B.** \(\dfrac{v_0^2}{g(1 + \cos\phi)}\)
- **C.** \(\dfrac{v_0^2}{g(1 - \sin\phi)}\)
- **D.** \(\dfrac{v_0^2 \cos\phi}{g}\)

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/120493/*