---
title: "Two identical horizontal spring-mass systems, each consisting of a block of mass \\(m\\) attached to an ideal spring of force constant \\(k\\), rest on a frictionless surface.  System 1 is displaced to position \\(x = A\\) and released from rest at time \\(t = 0\\), oscillating with total mechanical energy \\(E\\) and maximum speed \\(v_{1,\\max}\\).  System 2 is initially held at rest at position \\(x = \\dfrac{1}{2}A\\), and at time \\(t = 0\\) receives an instantaneous horizontal impulse that sets it into oscillation with total mechanical energy \\(2E\\).  What is the ratio of the speed of the block in System 2 immediately after the impulse to the maximum speed of the block in System 1?"
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url: "https://nerd-notes.com/ubq/124694/"
date_modified: "2026-09-28T14:09:08+00:00"
---

# Two identical horizontal spring-mass systems, each consisting of a block of mass \(m\) attached to an ideal spring of force constant \(k\), rest on a frictionless surface.

System 1 is displaced to position \(x = A\) and released from rest at time \(t = 0\), oscillating with total mechanical energy \(E\) and maximum speed \(v_{1,\max}\).

System 2 is initially held at rest at position \(x = \dfrac{1}{2}A\), and at time \(t = 0\) receives an instantaneous horizontal impulse that sets it into oscillation with total mechanical energy \(2E\).

What is the ratio of the speed of the block in System 2 immediately after the impulse to the maximum speed of the block in System 1?

Two identical horizontal spring-mass systems, each consisting of a block of mass \(m\) attached to an ideal spring of force constant \(k\), rest on a frictionless surface.

System 1 is displaced to position \(x = A\) and released from rest at time \(t = 0\), oscillating with total mechanical energy \(E\) and maximum speed \(v_{1,\max}\).

System 2 is initially held at rest at position \(x = \dfrac{1}{2}A\), and at time \(t = 0\) receives an instantaneous horizontal impulse that sets it into oscillation with total mechanical energy \(2E\).

What is the ratio of the speed of the block in System 2 immediately after the impulse to the maximum speed of the block in System 1?

![A schematic diagram showing two identical horizontal spring-mass systems arranged one above the other, labeled System 1 and System 2. A single vertical dashed line extends downward through both systems to indicate the equilibrium position, labeled x = 0. For System 1, a horizontal line represents a frictionless floor and a vertical wall is at the left. A horizontal coil spring of force constant k connects the wall to a rectangular block of mass m. The right edge of the block is aligned with a vertical tick mark labeled x = A. For System 2, identical floor, wall, spring, and block are shown directly below System 1. The right edge of this block is aligned with a vertical tick mark labeled x = A/2. A thick horizontal arrow pointing to the right contacts the left face of the block in System 2, labeled J. No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1790604547-O8W0iV.jpg)

- **A.** \(\dfrac{\sqrt{3}}{2}\)
- **B.** \(\dfrac{\sqrt{7}}{2}\)
- **C.** \(\sqrt{2}\)
- **D.** \(\dfrac{7}{4}\)

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/124694/*