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title: "Cart 1 of mass \\(m\\) and Cart 2 of mass \\(2m\\) travel along a horizontal frictionless track with the same initial linear momentum of magnitude \\(p_0\\). Each cart is brought to rest by a bumper that exerts a retarding force of magnitude \\(F_1(t) = F_{\\text{max}, 1} \\sin\\left(\\dfrac{\\pi t}{\\Delta t}\\right)\\) over the interval \\(0 \\le t \\le \\Delta t\\) on Cart 1, and \\(F_2(t) = F_{\\text{max}, 2} \\sin\\left(\\dfrac{\\pi t}{2\\Delta t}\\right)\\) over the interval \\(0 \\le t \\le 2\\Delta t\\) on Cart 2. Which of the following correctly compares the peak retarding forces, \\(F_{\\text{max}, 1}\\) and \\(F_{\\text{max}, 2}\\), and the magnitudes of the net work done on the carts, \\(|W_1|\\) and \\(|W_2|\\), during their respective stopping processes?"
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url: "https://nerd-notes.com/ubq/124217/"
date_modified: "2026-09-28T13:59:29+00:00"
---

# Cart 1 of mass \(m\) and Cart 2 of mass \(2m\) travel along a horizontal frictionless track with the same initial linear momentum of magnitude \(p_0\). Each cart is brought to rest by a bumper that exerts a retarding force of magnitude \(F_1(t) = F_{\text{max}, 1} \sin\left(\dfrac{\pi t}{\Delta t}\right)\) over the interval \(0 \le t \le \Delta t\) on Cart 1, and \(F_2(t) = F_{\text{max}, 2} \sin\left(\dfrac{\pi t}{2\Delta t}\right)\) over the interval \(0 \le t \le 2\Delta t\) on Cart 2. Which of the following correctly compares the peak retarding forces, \(F_{\text{max}, 1}\) and \(F_{\text{max}, 2}\), and the magnitudes of the net work done on the carts, \(|W_1|\) and \(|W_2|\), during their respective stopping processes?

Cart 1 of mass \(m\) and Cart 2 of mass \(2m\) travel along a horizontal frictionless track with the same initial linear momentum of magnitude \(p_0\). Each cart is brought to rest by a bumper that exerts a retarding force of magnitude \(F_1(t) = F_{\text{max}, 1} \sin\left(\dfrac{\pi t}{\Delta t}\right)\) over the interval \(0 \le t \le \Delta t\) on Cart 1, and \(F_2(t) = F_{\text{max}, 2} \sin\left(\dfrac{\pi t}{2\Delta t}\right)\) over the interval \(0 \le t \le 2\Delta t\) on Cart 2. Which of the following correctly compares the peak retarding forces, \(F_{\text{max}, 1}\) and \(F_{\text{max}, 2}\), and the magnitudes of the net work done on the carts, \(|W_1|\) and \(|W_2|\), during their respective stopping processes?

- **A.** Peak force: \(F_{\text{max}, 1} < F_{\text{max}, 2}\) ; Magnitude of work: \(|W_1| < |W_2|\)
- **B.** Peak force: \(F_{\text{max}, 1} < F_{\text{max}, 2}\) ; Magnitude of work: \(|W_1| > |W_2|\)
- **C.** Peak force: \(F_{\text{max}, 1} > F_{\text{max}, 2}\) ; Magnitude of work: \(|W_1| < |W_2|\)
- **D.** Peak force: \(F_{\text{max}, 1} > F_{\text{max}, 2}\) ; Magnitude of work: \(|W_1| > |W_2|\)

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/124217/*