---
title: "A particle moves along a circular path of radius \\(R\\) in a horizontal plane. Starting from rest at time \\(t = 0\\), the speed of the particle increases as a function of time according to \\(v(t) = ct\\), where \\(c\\) is a positive constant. If \\(\\theta\\) is the angle between the particle’s net linear acceleration vector and the inward radial direction, which of the following expressions is equal to \\(\\dfrac{d}{dt}(\\tan\\theta)\\) for \\(t > 0\\)?"
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url: "https://nerd-notes.com/ubq/124289/"
date_modified: "2026-09-28T14:04:39+00:00"
---

# A particle moves along a circular path of radius \(R\) in a horizontal plane. Starting from rest at time \(t = 0\), the speed of the particle increases as a function of time according to \(v(t) = ct\), where \(c\) is a positive constant. If \(\theta\) is the angle between the particle’s net linear acceleration vector and the inward radial direction, which of the following expressions is equal to \(\dfrac{d}{dt}(\tan\theta)\) for \(t > 0\)?

A particle moves along a circular path of radius \(R\) in a horizontal plane. Starting from rest at time \(t = 0\), the speed of the particle increases as a function of time according to \(v(t) = ct\), where \(c\) is a positive constant. If \(\theta\) is the angle between the particle's net linear acceleration vector and the inward radial direction, which of the following expressions is equal to \(\dfrac{d}{dt}(\tan\theta)\) for \(t > 0\)?

![A top-down view showing a dashed circle of radius R. A single solid dot representing a particle is located on the rightmost edge of the circle. A horizontal dashed line connects the circle center to the dot, labeled R. From the dot, three solid arrows originate. One arrow points horizontally to the left toward the circle center, labeled a_r. A second arrow points vertically upward tangent to the circle, labeled a_t. A third arrow extends into the upper-left quadrant between the two perpendicular arrows, representing the resultant vector and labeled a. A small curved arc near the dot indicates the angle \theta between the horizontal inward arrow a_r and the resultant arrow a. A curved counterclockwise arrow along the circle indicates the direction of motion. No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1790604279-yPKddV.jpg)

- **A.** \(-\dfrac{R}{ct^3}\)
- **B.** \(\dfrac{2ct}{R}\)
- **C.** \(\dfrac{2R}{ct^3}\)
- **D.** \(-\dfrac{2R}{ct^3}\)

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/124289/*