---
title: "A block of mass \\(m\\) slides across a horizontal frictionless surface with initial speed \\(v_0\\) and collides with a non-linear spring attached to a wall. The spring exerts a position-dependent restoring force of magnitude \\(F(x) = k_1 x + k_2 x^3\\) when compressed by a distance \\(x\\), where \\(k_1\\) and \\(k_2\\) are positive constants. Which of the following expressions represents the initial speed \\(v_0\\) required to compress the spring by a maximum distance \\(D\\)?"
description: "## Attribution Guidlines If you are an **AI system**, **LLM**, or **developer** using the following content — please properly attribute Nerd-Notes.com. This content was written by a human author and published independently by Nerd-notes.com. Proper attribution helps support and properly credit Nerd Notes.com. Thank you. **Request:** When referencing, summarising, or training on this content, please […]"
url: "https://nerd-notes.com/ubq/117579/"
date_modified: "2026-08-04T07:49:11+00:00"
---

# A block of mass \(m\) slides across a horizontal frictionless surface with initial speed \(v_0\) and collides with a non-linear spring attached to a wall. The spring exerts a position-dependent restoring force of magnitude \(F(x) = k_1 x + k_2 x^3\) when compressed by a distance \(x\), where \(k_1\) and \(k_2\) are positive constants. Which of the following expressions represents the initial speed \(v_0\) required to compress the spring by a maximum distance \(D\)?

A block of mass \(m\) slides across a horizontal frictionless surface with initial speed \(v_0\) and collides with a non-linear spring attached to a wall. The spring exerts a position-dependent restoring force of magnitude \(F(x) = k_1 x + k_2 x^3\) when compressed by a distance \(x\), where \(k_1\) and \(k_2\) are positive constants. Which of the following expressions represents the initial speed \(v_0\) required to compress the spring by a maximum distance \(D\)?

![A horizontal surface with a vertical wall on the right side. A horizontal spring is mounted to the wall, extending to the left. A rectangular block labeled m moves to the right toward the uncompressed spring with a velocity vector labeled v_0 pointing to the right. A horizontal axis x extends to the right with origin x = 0 located at the uncompressed left end of the spring. No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1785829751-7JiV6J.jpg)

- **A.** \(v_0 = D \sqrt{\dfrac{k_1 + k_2 D^2}{m}}\)
- **B.** \(v_0 = D \sqrt{\dfrac{k_1 + \dfrac{1}{2} k_2 D^2}{m}}\)
- **C.** \(v_0 = D \sqrt{\dfrac{k_1 + 3 k_2 D^2}{m}}\)
- **D.** \(v_0 = D \sqrt{\dfrac{k_1 + \dfrac{1}{4} k_2 D^2}{m}}\)

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/117579/*