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title: "A test vehicle of mass \\(m\\) is initially at rest at position \\(x = 0\\) on a horizontal, frictionless track. For the interval \\(0 \\le x \\le L\\), a propulsion system exerts a horizontal force on the vehicle given by \\(F(x) = F_{\\max} \\sin\\left(\\dfrac{\\pi x}{L}\\right)\\) in the \\(+x\\)-direction, where \\(F_{\\max}\\) and \\(L\\) are positive constants. For \\(x > L\\), the propulsion system is deactivated and no horizontal forces act on the vehicle. Which of the following integral expressions correctly determines the maximum speed \\(v_{\\max}\\) attained by the vehicle?"
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url: "https://nerd-notes.com/ubq/124154/"
date_modified: "2026-09-28T13:30:47+00:00"
---

# A test vehicle of mass \(m\) is initially at rest at position \(x = 0\) on a horizontal, frictionless track. For the interval \(0 \le x \le L\), a propulsion system exerts a horizontal force on the vehicle given by \(F(x) = F_{\max} \sin\left(\dfrac{\pi x}{L}\right)\) in the \(+x\)-direction, where \(F_{\max}\) and \(L\) are positive constants. For \(x > L\), the propulsion system is deactivated and no horizontal forces act on the vehicle. Which of the following integral expressions correctly determines the maximum speed \(v_{\max}\) attained by the vehicle?

A test vehicle of mass \(m\) is initially at rest at position \(x = 0\) on a horizontal, frictionless track. For the interval \(0 \le x \le L\), a propulsion system exerts a horizontal force on the vehicle given by \(F(x) = F_{\max} \sin\left(\dfrac{\pi x}{L}\right)\) in the \(+x\)-direction, where \(F_{\max}\) and \(L\) are positive constants. For \(x > L\), the propulsion system is deactivated and no horizontal forces act on the vehicle. Which of the following integral expressions correctly determines the maximum speed \(v_{\max}\) attained by the vehicle?

![A Cartesian coordinate graph with a horizontal axis labeled x and a vertical axis labeled F. The axes intersect at the origin (0,0). A single solid curve begins at the origin (0,0), forms a smooth concave-down sinusoidal arch that reaches a maximum at a horizontal position marked L/2, and returns to the horizontal axis at a horizontal position marked L. A vertical dashed line extends downward from the curve peak to the tick label L/2 on the horizontal axis. A horizontal dashed line extends leftward from the curve peak to a tick mark labeled F_{\max} on the vertical axis. No other curves, labels, gridlines, or shading appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1790602246-h74jmo.jpg)

- **A.** \(\sqrt{\dfrac{2}{m} \int_0^L F_{\max} \sin\left(\dfrac{\pi x}{L}\right) dx}\)
- **B.** \(\sqrt{\dfrac{2}{m} \int_0^{L/2} F_{\max} \sin\left(\dfrac{\pi x}{L}\right) dx}\)
- **C.** \(\sqrt{\dfrac{1}{m} \int_0^L F_{\max} \sin\left(\dfrac{\pi x}{L}\right) dx}\)
- **D.** \(\sqrt{\dfrac{1}{m} \int_0^{L/2} F_{\max} \sin\left(\dfrac{\pi x}{L}\right) dx}\)

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/124154/*