---
title: "Two particles of identical mass \\(m\\) travel in the \\(xy\\)-plane with speed \\(v_0\\) at symmetric angles \\(\\theta\\) relative to the \\(+x\\)-axis, having velocity vectors \\(\\vec{v}_1 = v_0\\cos\\theta\\,\\hat{i} + v_0\\sin\\theta\\,\\hat{j}\\) and \\(\\vec{v}_2 = v_0\\cos\\theta\\,\\hat{i} – v_0\\sin\\theta\\,\\hat{j}\\), where \\(0 \\le \\theta \\le \\dfrac{\\pi}{2}\\). The particles collide and stick together to form a single combined object of mass \\(2m\\). Which of the following graphs best represents the fraction of the initial translational kinetic energy dissipated during the collision, \\(\\dfrac{\\Delta K}{K_i}\\), as a function of \\(\\theta\\)?"
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url: "https://nerd-notes.com/ubq/124367/"
date_modified: "2026-09-28T14:04:59+00:00"
---

# Two particles of identical mass \(m\) travel in the \(xy\)-plane with speed \(v_0\) at symmetric angles \(\theta\) relative to the \(+x\)-axis, having velocity vectors \(\vec{v}_1 = v_0\cos\theta\,\hat{i} + v_0\sin\theta\,\hat{j}\) and \(\vec{v}_2 = v_0\cos\theta\,\hat{i} – v_0\sin\theta\,\hat{j}\), where \(0 \le \theta \le \dfrac{\pi}{2}\). The particles collide and stick together to form a single combined object of mass \(2m\). Which of the following graphs best represents the fraction of the initial translational kinetic energy dissipated during the collision, \(\dfrac{\Delta K}{K_i}\), as a function of \(\theta\)?

Two particles of identical mass \(m\) travel in the \(xy\)-plane with speed \(v_0\) at symmetric angles \(\theta\) relative to the \(+x\)-axis, having velocity vectors \(\vec{v}_1 = v_0\cos\theta\,\hat{i} + v_0\sin\theta\,\hat{j}\) and \(\vec{v}_2 = v_0\cos\theta\,\hat{i} - v_0\sin\theta\,\hat{j}\), where \(0 \le \theta \le \dfrac{\pi}{2}\). The particles collide and stick together to form a single combined object of mass \(2m\). Which of the following graphs best represents the fraction of the initial translational kinetic energy dissipated during the collision, \(\dfrac{\Delta K}{K_i}\), as a function of \(\theta\)?

![A two-dimensional Cartesian plane showing the positive x-axis as a horizontal dashed line directed to the right, labeled +x. Two small solid gray circles of identical size represent particles of mass m. The upper particle is located in the first quadrant with an arrow extending from its center toward the origin, labeled \vec{v}_1; a small curved arc indicates the angle \theta between the velocity vector and a horizontal dashed reference line. The lower particle is located symmetrically in the fourth quadrant with an arrow extending from its center toward the origin, labeled \vec{v}_2; a small curved arc indicates the angle \theta between this vector and a horizontal dashed reference line. Both arrows terminate at the origin where the collision occurs. No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1790604298-tSdaN9.jpg)

- **A.** Graph A
- **B.** Graph B
- **C.** Graph C
- **D.** Graph D

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/124367/*