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title: "An idealized cart of mass \\(M\\) accelerates from rest at position \\(x = 0\\) along a frictionless horizontal track under the action of an engine delivering a constant mechanical power \\(P_0\\). A student notes that because \\(P_0 = Fv\\), the driving force \\(F(v) = P_0/v\\) diverges toward infinity as the speed approaches zero at the starting point. However, when the cart reaches a displacement \\(x = D\\), its measured kinetic energy is finite. Which of the following provides the correct physical explanation for why the work done by the engine over the displacement \\(D\\) is finite?"
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url: "https://nerd-notes.com/ubq/124146/"
date_modified: "2026-09-28T13:30:45+00:00"
---

# An idealized cart of mass \(M\) accelerates from rest at position \(x = 0\) along a frictionless horizontal track under the action of an engine delivering a constant mechanical power \(P_0\). A student notes that because \(P_0 = Fv\), the driving force \(F(v) = P_0/v\) diverges toward infinity as the speed approaches zero at the starting point. However, when the cart reaches a displacement \(x = D\), its measured kinetic energy is finite. Which of the following provides the correct physical explanation for why the work done by the engine over the displacement \(D\) is finite?

An idealized cart of mass \(M\) accelerates from rest at position \(x = 0\) along a frictionless horizontal track under the action of an engine delivering a constant mechanical power \(P_0\). A student notes that because \(P_0 = Fv\), the driving force \(F(v) = P_0/v\) diverges toward infinity as the speed approaches zero at the starting point. However, when the cart reaches a displacement \(x = D\), its measured kinetic energy is finite. Which of the following provides the correct physical explanation for why the work done by the engine over the displacement \(D\) is finite?

![A schematic showing a horizontal line representing a frictionless track. On the line sits a rectangular cart labeled M with two small circular wheels. A vertical dashed line passes through the front of the cart at its initial position and is labeled x = 0. A second vertical dashed line to the right is labeled x = D. A single horizontal arrow points to the right from the center of the cart and is labeled F. A horizontal arrow above the cart points to the right and is labeled v. A horizontal coordinate axis below the track is labeled x. No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1790602244-5897j9.jpg)

- **A.** The cart undergoes motion where speed scales as the square root of displacement (\(v \propto x^{1/2}\)) by standard kinematics, which causes the driving force \(F = P_0/v\) to diverge as \(x^{-1/2}\); integrating this force over displacement yields a convergent improper integral that produces a finite work proportional to \(D^{1/2}\).
- **B.** Because constant power requires \(P_0 = Mv^2(dv/dx)\), separation of variables reveals that speed scales as the cube root of displacement (\(v \propto x^{1/3}\)), meaning the force diverges as \(x^{-1/3}\); integrating this force over displacement yields a convergent improper integral that evaluates to a finite work proportional to \(D^{2/3}\).
- **C.** The work done is finite because mechanical power represents the rate of energy transferred per unit distance (\(P_0 = dW/dx\)), which directly requires the kinetic energy to increase linearly with displacement (\(K = P_0 D\)), rendering the velocity dependence of the driving force irrelevant to the total work.
- **D.** The work is finite because the cart's speed increases linearly with time (\(v \propto t\)), which allows the net work to be computed using the average force evaluated at the average speed (\(F_{\text{avg}} = P_0/v_{\text{avg}}\)), thereby replacing the singular initial force with a well-behaved constant value.

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/124146/*