---
title: "Two carts, Cart 1 and Cart 2, each of mass \\(m\\), move in the positive \\(x\\)-direction along a horizontal frictionless track with initial speeds \\(v_0\\) and \\(3v_0\\), respectively. Starting at time \\(t = 0\\), both carts are subjected to the same time-dependent horizontal force \\(F(t) = c t\\) in the positive \\(x\\)-direction until time \\(t = T\\), where \\(c = \\dfrac{4mv_0}{T^2}\\). What is the ratio \\(\\dfrac{\\Delta K_1}{\\Delta K_2}\\) of the change in kinetic energy of Cart 1 to the change in kinetic energy of Cart 2 over this time interval?"
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url: "https://nerd-notes.com/ubq/124181/"
date_modified: "2026-09-28T13:30:59+00:00"
---

# Two carts, Cart 1 and Cart 2, each of mass \(m\), move in the positive \(x\)-direction along a horizontal frictionless track with initial speeds \(v_0\) and \(3v_0\), respectively. Starting at time \(t = 0\), both carts are subjected to the same time-dependent horizontal force \(F(t) = c t\) in the positive \(x\)-direction until time \(t = T\), where \(c = \dfrac{4mv_0}{T^2}\). What is the ratio \(\dfrac{\Delta K_1}{\Delta K_2}\) of the change in kinetic energy of Cart 1 to the change in kinetic energy of Cart 2 over this time interval?

Two carts, Cart 1 and Cart 2, each of mass \(m\), move in the positive \(x\)-direction along a horizontal frictionless track with initial speeds \(v_0\) and \(3v_0\), respectively. Starting at time \(t = 0\), both carts are subjected to the same time-dependent horizontal force \(F(t) = c t\) in the positive \(x\)-direction until time \(t = T\), where \(c = \dfrac{4mv_0}{T^2}\). What is the ratio \(\dfrac{\Delta K_1}{\Delta K_2}\) of the change in kinetic energy of Cart 1 to the change in kinetic energy of Cart 2 over this time interval?

![A horizontal track is shown as a solid line. Two identical rectangular carts, labeled Cart 1 and Cart 2, sit on the surface, one vertically above the other in separate horizontal lanes. For Cart 1, a horizontal arrow labeled v_0 points to the right from the center of the cart, and a horizontal arrow labeled F(t) points to the right from its front edge. For Cart 2, a horizontal arrow labeled 3v_0 points to the right from the center of the cart, and a horizontal arrow labeled F(t) of the same length as on Cart 1 points to the right from its front edge. A horizontal coordinate axis below shows +x pointing to the right. No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1790602259-zMjAIC.jpg)

- **A.** \(\dfrac{1}{9}\)
- **B.** \(\dfrac{1}{3}\)
- **C.** \(\dfrac{9}{25}\)
- **D.** \(\dfrac{1}{2}\)

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/124181/*