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title: "A cart of initial mass \\(m_0\\) is attached to an ideal spring with spring constant \\(k_0\\) on a frictionless horizontal track. At time \\(t = 0\\), the cart is displaced from its equilibrium position \\(x = 0\\) and released from rest. As the cart oscillates, sand drops vertically into the cart from a stationary funnel above at a rate such that the total mass of the cart increases according to \\(m(t) = m_0 e^{bt}\\), where \\(b\\) is a positive constant. The falling sand has zero horizontal velocity immediately before entering the cart. Which of the following differential equations correctly governs the horizontal position \\(x(t)\\) of the cart as a function of time \\(t\\)?"
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url: "https://nerd-notes.com/ubq/117649/"
date_modified: "2026-08-04T07:49:29+00:00"
---

# A cart of initial mass \(m_0\) is attached to an ideal spring with spring constant \(k_0\) on a frictionless horizontal track. At time \(t = 0\), the cart is displaced from its equilibrium position \(x = 0\) and released from rest. As the cart oscillates, sand drops vertically into the cart from a stationary funnel above at a rate such that the total mass of the cart increases according to \(m(t) = m_0 e^{bt}\), where \(b\) is a positive constant. The falling sand has zero horizontal velocity immediately before entering the cart. Which of the following differential equations correctly governs the horizontal position \(x(t)\) of the cart as a function of time \(t\)?

A cart of initial mass \(m_0\) is attached to an ideal spring with spring constant \(k_0\) on a frictionless horizontal track. At time \(t = 0\), the cart is displaced from its equilibrium position \(x = 0\) and released from rest. As the cart oscillates, sand drops vertically into the cart from a stationary funnel above at a rate such that the total mass of the cart increases according to \(m(t) = m_0 e^{bt}\), where \(b\) is a positive constant. The falling sand has zero horizontal velocity immediately before entering the cart. Which of the following differential equations correctly governs the horizontal position \(x(t)\) of the cart as a function of time \(t\)?

![A horizontal track with a vertical wall anchored at the left end. An ideal spring with spring constant k_0 extends horizontally from the wall to a rectangular cart of mass m_0. Above the cart, a funnel dispenses sand vertically downward into the cart. A horizontal coordinate axis labeled x points to the right with its origin x = 0 marked at the cart's equilibrium position. No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1785829769-p3AOXO.jpg)

- **A.** \(\dfrac{d^2x}{dt^2} + b\dfrac{dx}{dt} + \dfrac{k_0}{m_0}e^{-bt}x = 0\)
- **B.** \(\dfrac{d^2x}{dt^2} + \dfrac{k_0}{m_0}e^{-bt}x = 0\)
- **C.** \(\dfrac{d^2x}{dt^2} - b\dfrac{dx}{dt} + \dfrac{k_0}{m_0}e^{-bt}x = 0\)
- **D.** \(\dfrac{d^2x}{dt^2} + b\dfrac{dx}{dt} + \dfrac{k_0}{m_0}e^{bt}x = 0\)

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/117649/*