---
title: "A small particle of mass \\(m\\) and charge \\(+q\\) is launched from a large distance with initial speed \\(v_0\\) toward a fixed point charge \\(+Q\\). The initial line of motion is offset from \\(+Q\\) by an impact parameter \\(b\\). Let \\(d_0 = \\dfrac{q Q}{2\\pi\\varepsilon_0 m v_0^2}\\) be the distance of closest approach for a head-on collision (\\(b = 0\\)). Which of the following expressions gives the distance of closest approach \\(r_{\\text{min}}\\) for a trajectory with a non-zero impact parameter \\(b\\)?"
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/118145/"
date_modified: "2026-08-04T08:05:31+00:00"
---

# A small particle of mass \(m\) and charge \(+q\) is launched from a large distance with initial speed \(v_0\) toward a fixed point charge \(+Q\). The initial line of motion is offset from \(+Q\) by an impact parameter \(b\). Let \(d_0 = \dfrac{q Q}{2\pi\varepsilon_0 m v_0^2}\) be the distance of closest approach for a head-on collision (\(b = 0\)). Which of the following expressions gives the distance of closest approach \(r_{\text{min}}\) for a trajectory with a non-zero impact parameter \(b\)?

A small particle of mass \(m\) and charge \(+q\) is launched from a large distance with initial speed \(v_0\) toward a fixed point charge \(+Q\). The initial line of motion is offset from \(+Q\) by an impact parameter \(b\). Let \(d_0 = \dfrac{q Q}{2\pi\varepsilon_0 m v_0^2}\) be the distance of closest approach for a head-on collision (\(b = 0\)). Which of the following expressions gives the distance of closest approach \(r_{\text{min}}\) for a trajectory with a non-zero impact parameter \(b\)?

![A horizontal axis passing through a central fixed dot labeled +Q. At a large distance to the left, a small dot labeled +q moves to the right with an initial velocity vector arrow labeled v_0. The line of initial motion is parallel to the horizontal axis and separated from it by a vertical distance labeled impact parameter b. A curved hyperbolic trajectory path shows the particle +q deflecting away from +Q. At the point on the curve closest to +Q, a dashed line segment connects +Q to the trajectory and is labeled r_min. A right-angle indicator shows the velocity vector at this closest point is perpendicular to the dashed line segment r_min. No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1785830730-a4WZVX.jpg)

- **A.** \(r_{\text{min}} = \sqrt{d_0^2 + b^2}\)
- **B.** \(r_{\text{min}} = \dfrac{d_0 + \sqrt{d_0^2 + 4b^2}}{2}\)
- **C.** \(r_{\text{min}} = \dfrac{d_0}{2} + b\)
- **D.** \(r_{\text{min}} = \dfrac{-d_0 + \sqrt{d_0^2 + 4b^2}}{2}\)

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/118145/*