---
title: "A projectile of mass \\(M\\) is launched over horizontal ground. At the apex of its trajectory, at a height \\(H\\) and horizontal distance \\(D\\) from the launch point, the projectile has speed \\(v_0\\) and explodes into two fragments of masses \\(m_1\\) and \\(m_2\\), where \\(m_1 + m_2 = M\\). The explosion releases energy \\(E\\), propelling both fragments horizontally along the original line of flight such that fragment \\(2\\) lands at horizontal position:  \\[ x_2 = 2D + \\sqrt{\\dfrac{2H}{g}}\\sqrt{\\dfrac{2m_1 E}{m_2 M}} \\]  In the limiting case where \\(m_2 \\ll m_1\\), which of the following correctly describes the landing position \\(x_2\\) of fragment \\(2\\) and the landing position \\(x_{\\text{cm}}\\) of the system’s center of mass at the instant of impact?"
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url: "https://nerd-notes.com/ubq/124277/"
date_modified: "2026-09-28T14:04:38+00:00"
---

# A projectile of mass \(M\) is launched over horizontal ground. At the apex of its trajectory, at a height \(H\) and horizontal distance \(D\) from the launch point, the projectile has speed \(v_0\) and explodes into two fragments of masses \(m_1\) and \(m_2\), where \(m_1 + m_2 = M\). The explosion releases energy \(E\), propelling both fragments horizontally along the original line of flight such that fragment \(2\) lands at horizontal position:

\[ x_2 = 2D + \sqrt{\dfrac{2H}{g}}\sqrt{\dfrac{2m_1 E}{m_2 M}} \]

In the limiting case where \(m_2 \ll m_1\), which of the following correctly describes the landing position \(x_2\) of fragment \(2\) and the landing position \(x_{\text{cm}}\) of the system’s center of mass at the instant of impact?

A projectile of mass \(M\) is launched over horizontal ground. At the apex of its trajectory, at a height \(H\) and horizontal distance \(D\) from the launch point, the projectile has speed \(v_0\) and explodes into two fragments of masses \(m_1\) and \(m_2\), where \(m_1 + m_2 = M\). The explosion releases energy \(E\), propelling both fragments horizontally along the original line of flight such that fragment \(2\) lands at horizontal position:

\[ x_2 = 2D + \sqrt{\dfrac{2H}{g}}\sqrt{\dfrac{2m_1 E}{m_2 M}} \]

In the limiting case where \(m_2 \ll m_1\), which of the following correctly describes the landing position \(x_2\) of fragment \(2\) and the landing position \(x_{\text{cm}}\) of the system's center of mass at the instant of impact?

- **A.** \(x_2 \to \infty\) and \(x_{\text{cm}} \to \infty\), because fragment \(2\) travels infinitely far downrange and its unbounded displacement dominates the weighted average position of the system.
- **B.** \(x_2 \to 2D\) and \(x_{\text{cm}} = 2D\), because as the mass of fragment \(2\) approaches zero, it carries negligible momentum and kinetic energy from the explosion, causing both fragments to land at the unperturbed location.
- **C.** \(x_2 \to \infty\) and \(x_{\text{cm}} = 2D\), because fragment \(2\) receives nearly all of the explosion's kinetic energy and its velocity diverges as \(m_2^{-1/2}\), but its contribution \(m_2 x_2\) to the center of mass approaches zero as \(m_2^{1/2}\).
- **D.** \(x_2 \to \infty\) and \(x_{\text{cm}} \to D\), because the recoil from ejecting fragment \(2\) at divergent speed brings fragment \(1\) to rest at the apex, causing it to fall vertically at \(x = D\) while carrying virtually all the mass.

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/124277/*