---
title: "A particle in the \\(xy\\)-plane moves from the origin \\((0,0)\\) to a final position \\((x_0, y_0)\\) along the parabolic path \\(y = kx^2\\), where \\(k\\) is a positive constant. During this displacement, the particle is acted upon by a two-dimensional force \\(\\vec{F} = ay\\hat{i} + bx\\hat{j}\\), where \\(a\\) and \\(b\\) are constants. Which of the following expressions represents the work done by the force on the particle?"
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url: "https://nerd-notes.com/ubq/117569/"
date_modified: "2026-08-04T07:49:09+00:00"
---

# A particle in the \(xy\)-plane moves from the origin \((0,0)\) to a final position \((x_0, y_0)\) along the parabolic path \(y = kx^2\), where \(k\) is a positive constant. During this displacement, the particle is acted upon by a two-dimensional force \(\vec{F} = ay\hat{i} + bx\hat{j}\), where \(a\) and \(b\) are constants. Which of the following expressions represents the work done by the force on the particle?

A particle in the \(xy\)-plane moves from the origin \((0,0)\) to a final position \((x_0, y_0)\) along the parabolic path \(y = kx^2\), where \(k\) is a positive constant. During this displacement, the particle is acted upon by a two-dimensional force \(\vec{F} = ay\hat{i} + bx\hat{j}\), where \(a\) and \(b\) are constants. Which of the following expressions represents the work done by the force on the particle?

![A Cartesian coordinate system in the xy-plane showing a thin black curve starting at the origin (0,0) and curving upward into the first quadrant, ending at a point labeled (x_0, y_0). An arrowhead along the curve indicates movement toward (x_0, y_0). The curve is labeled y = kx^2. No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1785829749-q7l1NC.jpg)

- **A.** \(\dfrac{k(a + 2b)x_0^3}{3}\)
- **B.** \(\dfrac{k(a + b)x_0^3}{3}\)
- **C.** \(\dfrac{akx_0^3}{3} + \dfrac{bx_0^2}{2}\)
- **D.** \(\dfrac{k(2a + b)x_0^3}{3}\)

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/117569/*