---
title: "A ball of mass \\(m\\) is launched from a flat, horizontal surface with an initial speed \\(v_0\\) at an angle \\(\\theta_0\\) above the horizontal, where \\(0^\\circ < \\theta_0 < 90^\\circ\\). The ball reaches a maximum height and then lands back on the horizontal surface. Air resistance is negligible."
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url: "https://nerd-notes.com/ubq/113963/"
date_modified: "2026-06-22T09:27:03+00:00"
---

# A ball of mass \(m\) is launched from a flat, horizontal surface with an initial speed \(v_0\) at an angle \(\theta_0\) above the horizontal, where \(0^\circ < \theta_0 < 90^\circ\). The ball reaches a maximum height and then lands back on the horizontal surface. Air resistance is negligible.

A ball of mass \(m\) is launched from a flat, horizontal surface with an initial speed \(v_0\) at an angle \(\theta_0\) above the horizontal, where \(0^\circ < \theta_0 < 90^\circ\). The ball reaches a maximum height and then lands back on the horizontal surface. Air resistance is negligible.

**Part a)** Two students are discussing the motion of the ball at the exact instant it reaches its highest point. **Student 1:** "At the highest point, the vertical velocity is zero. Because the velocity has stopped changing momentarily to turn around, the acceleration must also be zero at that instant." **Student 2:** "The ball is still moving horizontally at the highest point, so it must have an acceleration in the horizontal direction." *(5 points)*

**Part b)** Let \(\vec{r}_1\) be the displacement vector from the launch point to the highest point of the trajectory, and let \(\vec{r}_2\) be the displacement vector from the highest point to the landing point. On the blank grid provided, **draw** and **label** the vectors \(\vec{r}_1\) and \(\vec{r}_2\) such that they accurately represent the ball's displacements. Then, **draw** and **label** the total displacement vector \(\vec{r}_{total}\) for the entire flight. *(2 points)*

**Part c)** **Derive** an expression for the magnitude of the displacement vector \(\vec{r}_1\) from the launch point to the highest point. Express your answer in terms of \(v_0\), \(\theta_0\), and fundamental constants. An unsimplified expression is acceptable. *(3 points)*

**Part d)** Student 3 claims that the magnitude of the average velocity of the ball during the trip from the launch point to the highest point is exactly half the launch speed, \(\dfrac{v_0}{2}\). **Determine** if Student 3's claim is correct or incorrect. - [ ] Correct - [ ] Incorrect **Justify** your answer mathematically or conceptually. *(2 points)*


*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/113963/*