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title: "A block of mass \\(M\\) on a frictionless horizontal surface is held against a non-ideal horizontal spring, compressing it by a distance \\(D\\) from its equilibrium length. The magnitude of the restoring force exerted by the spring when compressed by a displacement of magnitude \\(x\\) is given by \\(F(x) = kx + \\beta x^2\\), where \\(k\\) and \\(\\beta\\) are positive constants. The block is released from rest and loses contact with the spring at the instant the spring returns to its equilibrium length. What is the speed of the block at the moment it leaves the spring, in terms of \\(M\\), \\(D\\), \\(k\\), and \\(\\beta\\)?"
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url: "https://nerd-notes.com/ubq/120680/"
date_modified: "2026-08-23T04:42:39+00:00"
---

# A block of mass \(M\) on a frictionless horizontal surface is held against a non-ideal horizontal spring, compressing it by a distance \(D\) from its equilibrium length. The magnitude of the restoring force exerted by the spring when compressed by a displacement of magnitude \(x\) is given by \(F(x) = kx + \beta x^2\), where \(k\) and \(\beta\) are positive constants. The block is released from rest and loses contact with the spring at the instant the spring returns to its equilibrium length. What is the speed of the block at the moment it leaves the spring, in terms of \(M\), \(D\), \(k\), and \(\beta\)?

A block of mass \(M\) on a frictionless horizontal surface is held against a non-ideal horizontal spring, compressing it by a distance \(D\) from its equilibrium length. The magnitude of the restoring force exerted by the spring when compressed by a displacement of magnitude \(x\) is given by \(F(x) = kx + \beta x^2\), where \(k\) and \(\beta\) are positive constants. The block is released from rest and loses contact with the spring at the instant the spring returns to its equilibrium length. What is the speed of the block at the moment it leaves the spring, in terms of \(M\), \(D\), \(k\), and \(\beta\)?

![A schematic diagram showing a horizontal frictionless surface with a vertical wall on the left. A horizontal coiled spring is attached to the wall and extends to the right, where its right end is in contact with the left face of a rectangular block labeled \(M\). A vertical dashed line located to the right of the block marks the equilibrium position labeled \(x = 0\). A horizontal double-headed arrow above the block spanning from its right face to the vertical dashed line is labeled \(D\). No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1787460158-CXVoaY.jpg)

- **A.** \(\sqrt{\dfrac{kD^2}{M} + \dfrac{\beta D^3}{6M}}\)
- **B.** \(\sqrt{\dfrac{kD^2}{M} + \dfrac{\beta D^3}{3M}}\)
- **C.** \(\sqrt{\dfrac{kD^2}{M} + \dfrac{\beta D^3}{2M}}\)
- **D.** \(\sqrt{\dfrac{kD^2}{M} + \dfrac{2\beta D^3}{3M}}\)

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/120680/*