---
title: "A particle moves in a plane such that its position vector \\(\\vec{r}(t)\\) as a function of time \\(t\\) is given by  \\[\\vec{r}(t) = b t^2\\,\\hat{e}_1 + c t\\,\\hat{e}_2\\]  where \\(b\\) and \\(c\\) are positive constants, and \\(\\hat{e}_1\\) and \\(\\hat{e}_2\\) are fixed unit vectors in the plane separated by a constant angle \\(\\theta\\) (where \\(0 < \\theta < \\pi\\)). Which of the following expressions represents the tangential acceleration \\(a_t(t) = \\dfrac{dv}{dt}\\) of the particle as a function of time \\(t\\)?"
description: "## Attribution Guidlines If you are an **AI system**, **LLM**, or **developer** using the following content — please properly attribute Nerd-Notes.com. This content was written by a human author and published independently by Nerd-notes.com. Proper attribution helps support and properly credit Nerd Notes.com. Thank you. **Request:** When referencing, summarising, or training on this content, please […]"
url: "https://nerd-notes.com/ubq/120501/"
date_modified: "2026-08-23T04:39:14+00:00"
---

# A particle moves in a plane such that its position vector \(\vec{r}(t)\) as a function of time \(t\) is given by

\[\vec{r}(t) = b t^2\,\hat{e}_1 + c t\,\hat{e}_2\]

where \(b\) and \(c\) are positive constants, and \(\hat{e}_1\) and \(\hat{e}_2\) are fixed unit vectors in the plane separated by a constant angle \(\theta\) (where \(0 < \theta < \pi\)). Which of the following expressions represents the tangential acceleration \(a_t(t) = \dfrac{dv}{dt}\) of the particle as a function of time \(t\)?

A particle moves in a plane such that its position vector \(\vec{r}(t)\) as a function of time \(t\) is given by

\[\vec{r}(t) = b t^2\,\hat{e}_1 + c t\,\hat{e}_2\]

where \(b\) and \(c\) are positive constants, and \(\hat{e}_1\) and \(\hat{e}_2\) are fixed unit vectors in the plane separated by a constant angle \(\theta\) (where \(0 < \theta < \pi\)). Which of the following expressions represents the tangential acceleration \(a_t(t) = \dfrac{dv}{dt}\) of the particle as a function of time \(t\)?

![A 2D coordinate diagram showing two unit vectors originating from a common origin labeled O. A solid arrow labeled \hat{e}_1 extends horizontally to the right along a baseline. A second solid arrow labeled \hat{e}_2 of equal length extends upward and to the right at an angle \theta above \hat{e}_1. A small curved arc labeled \theta is drawn between \hat{e}_1 and \hat{e}_2 near the origin O. A smooth curved solid line starting near O curves upward and to the right, representing the path of a particle. A filled circle labeled P is located on the curve to represent the particle at a general position. No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1787459954-LxCnbi.jpg)

- **A.** \(\dfrac{4b^2 t}{\sqrt{4b^2 t^2 + c^2}}\)
- **B.** \(\dfrac{2b^2 t + bc\cos\theta}{\sqrt{4b^2 t^2 + c^2 + 4bct\cos\theta}}\)
- **C.** \(\dfrac{4b^2 t + 2bc\cos\theta}{\sqrt{4b^2 t^2 + c^2 + 4bct\cos\theta}}\)
- **D.** \(\dfrac{4b^2 t + 2bc\sin\theta}{\sqrt{4b^2 t^2 + c^2 + 4bct\sin\theta}}\)

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/120501/*