---
title: "A small particle of mass \\(m\\) and positive charge \\(+q\\) is launched from a very large distance (assume \\(r \\to \\infty\\)) with initial speed \\(v_0\\) directly toward the center of a fixed, solid insulating sphere. The fixed sphere has radius \\(R\\) and a uniform positive charge \\(+Q\\). The particle momentarily comes to rest at a distance \\(r_1\\) from the center of the fixed sphere, where \\(r_1 > R\\). The effects of gravity are negligible."
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url: "https://nerd-notes.com/ubq/117871/"
date_modified: "2026-08-04T08:00:09+00:00"
---

# A small particle of mass \(m\) and positive charge \(+q\) is launched from a very large distance (assume \(r \to \infty\)) with initial speed \(v_0\) directly toward the center of a fixed, solid insulating sphere. The fixed sphere has radius \(R\) and a uniform positive charge \(+Q\). The particle momentarily comes to rest at a distance \(r_1\) from the center of the fixed sphere, where \(r_1 > R\). The effects of gravity are negligible.

A small particle of mass \(m\) and positive charge \(+q\) is launched from a very large distance (assume \(r \to \infty\)) with initial speed \(v_0\) directly toward the center of a fixed, solid insulating sphere. The fixed sphere has radius \(R\) and a uniform positive charge \(+Q\). The particle momentarily comes to rest at a distance \(r_1\) from the center of the fixed sphere, where \(r_1 > R\). The effects of gravity are negligible.

![A fixed solid sphere on the left with radius R, labeled with a central plus sign and '+Q'. A horizontal dashed axis passes through the center of the sphere and extends to the right. On this axis, far to the right, is a small dot labeled '+q'. A vector arrow points from the small dot to the left, labeled v_0. A label 'r_1' marks a point on the axis between the sphere and the small dot, with r_1 > R. No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1785830408-hfU3LF.jpg)

**Part a)** **Draw** and shade a bar chart on the provided grid to represent the kinetic energy \(K\), electric potential energy \(U_E\), and total mechanical energy \(E\) of the particle-sphere system when the particle is at \(r = \infty\) and when it is at \(r = r_1\).

**Part b)** **Derive** an expression for the distance of closest approach \(r_1\). Express your answer in terms of \(m\), \(q\), \(Q\), \(v_0\), and physical constants, as appropriate.

**Part c)** **Sketch** a graph of the particle's speed \(v\) as a function of distance \(r\) from the center of the fixed sphere. The sketch should clearly show the behavior of the particle from \(r = r_1\) to very large values of \(r\).

**Part d)** The particle is now launched from \(r = \infty\) with an initial speed of \(2v_0\). Assume the particle still does not reach the surface of the fixed sphere.


*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/117871/*