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title: "A particle moves along the positive \\(x\\)-axis with a constant speed \\(v_0\\). Starting at time \\(t = 0\\) at position \\(x = 0\\), the particle experiences a force parallel to its velocity given by \\(F(x,t) = k_0 e^{-\\alpha t} x\\), where \\(k_0\\) and \\(\\alpha\\) are positive constants. Which of the following expressions represents the total work done on the particle by this force from \\(t = 0\\) to \\(t \\to \\infty\\)?"
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url: "https://nerd-notes.com/ubq/117581/"
date_modified: "2026-08-04T07:49:11+00:00"
---

# A particle moves along the positive \(x\)-axis with a constant speed \(v_0\). Starting at time \(t = 0\) at position \(x = 0\), the particle experiences a force parallel to its velocity given by \(F(x,t) = k_0 e^{-\alpha t} x\), where \(k_0\) and \(\alpha\) are positive constants. Which of the following expressions represents the total work done on the particle by this force from \(t = 0\) to \(t \to \infty\)?

A particle moves along the positive \(x\)-axis with a constant speed \(v_0\). Starting at time \(t = 0\) at position \(x = 0\), the particle experiences a force parallel to its velocity given by \(F(x,t) = k_0 e^{-\alpha t} x\), where \(k_0\) and \(\alpha\) are positive constants. Which of the following expressions represents the total work done on the particle by this force from \(t = 0\) to \(t \to \infty\)?

- **A.** \(\dfrac{k_0 v_0^2}{2\alpha^2}\)
- **B.** \(\dfrac{k_0 v_0^2}{\alpha^2}\)
- **C.** \(\dfrac{2 k_0 v_0^2}{\alpha^2}\)
- **D.** \(\dfrac{k_0 v_0^2}{4\alpha^2}\)

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/117581/*