---
title: "An asteroid of mass \\(m\\) orbits the Sun of mass \\(M\\) in an elliptical path. At perihelion, the asteroid is at a distance \\(r_p\\) from the Sun with speed \\(v_p\\), where its velocity vector is perpendicular to its position vector relative to the Sun. The asteroid travels along an outbound arc to aphelion at a distance \\(r_a\\) from the Sun, and then returns to perihelion along an inbound arc. Which of the following correctly pairs the speed of the asteroid at aphelion, \\(v_a\\), with the relationship between the work done by the gravitational force on the asteroid along the outbound arc, \\(W_{\\text{out}}\\), and along the inbound arc, \\(W_{\\text{in}}\\)?"
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url: "https://nerd-notes.com/ubq/124437/"
date_modified: "2026-09-28T14:05:31+00:00"
---

# An asteroid of mass \(m\) orbits the Sun of mass \(M\) in an elliptical path. At perihelion, the asteroid is at a distance \(r_p\) from the Sun with speed \(v_p\), where its velocity vector is perpendicular to its position vector relative to the Sun. The asteroid travels along an outbound arc to aphelion at a distance \(r_a\) from the Sun, and then returns to perihelion along an inbound arc. Which of the following correctly pairs the speed of the asteroid at aphelion, \(v_a\), with the relationship between the work done by the gravitational force on the asteroid along the outbound arc, \(W_{\text{out}}\), and along the inbound arc, \(W_{\text{in}}\)?

An asteroid of mass \(m\) orbits the Sun of mass \(M\) in an elliptical path. At perihelion, the asteroid is at a distance \(r_p\) from the Sun with speed \(v_p\), where its velocity vector is perpendicular to its position vector relative to the Sun. The asteroid travels along an outbound arc to aphelion at a distance \(r_a\) from the Sun, and then returns to perihelion along an inbound arc. Which of the following correctly pairs the speed of the asteroid at aphelion, \(v_a\), with the relationship between the work done by the gravitational force on the asteroid along the outbound arc, \(W_{\text{out}}\), and along the inbound arc, \(W_{\text{in}}\)?

![An elliptical orbit drawn with a solid line, elongated horizontally. A filled circle labeled Sun sits at the left focal point. The leftmost point on the ellipse is labeled perihelion, located at a horizontal dashed line of length \(r_p\) from the focal point, with a small solid circle representing the asteroid. A straight velocity vector arrow labeled \(\vec{v}_p\) points vertically upward from the asteroid, perpendicular to the horizontal dashed line. The rightmost point of the ellipse is labeled aphelion, located at a horizontal dashed line of length \(r_a\) from the focal point. Along the top curved half of the ellipse, a directional arrow points counterclockwise from perihelion to aphelion to indicate the outbound path. Along the bottom curved half of the ellipse, a directional arrow points counterclockwise from aphelion to perihelion to indicate the inbound path. No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1790604330-uWbi8u.jpg)

- **A.** Speed: \(v_a = \sqrt{\dfrac{r_p}{r_a}}\,v_p\) ; Work: \(W_{\text{out}} = W_{\text{in}} = 0\)
- **B.** Speed: \(v_a = \sqrt{\dfrac{r_p}{r_a}}\,v_p\) ; Work: \(W_{\text{out}} = -W_{\text{in}} < 0\)
- **C.** Speed: \(v_a = \left(\dfrac{r_p}{r_a}\right) v_p\) ; Work: \(W_{\text{out}} = -W_{\text{in}} < 0\)
- **D.** Speed: \(v_a = \left(\dfrac{r_p}{r_a}\right) v_p\) ; Work: \(W_{\text{out}} = W_{\text{in}} = 0\)

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/124437/*