A spacecraft is in a circular orbit about the moon, 1.22...

A spacecraft is in a circular orbit about the moon, $1.22 \times 10^{5} \mathrm{~m}$ above its surface. The speed of the spacecraft is $1620 \mathrm{~m} / \mathrm{s}$, and the radius of the moon is $1.74 \times 10^{6} \mathrm{~m}$. If the moon were a smooth, reflective sphere, (a) how far below the moon's surface would the image of the spacecraft appear, and (b) what would be the apparent speed of the spacecraft's image? (Hint: Both the spacecraft and its image have the same angular speed about the center of the moon.)

Key Concepts

Virtual Image Formation

In mirror systems, a virtual image is produced when the reflected rays diverge, so that the image appears to be located behind the mirror. In the case of an object outside a spherical mirror (with the object along the radius), using the mirror equation one finds that the image distance is negative, indicating that the image is virtual and located inside the sphere. This concept is crucial for determining how far below the moon’s surface the image appears.

Relating Angular Speed to Tangential Speed of an Image

Because the mirror (or, in this problem, the moon) is a sphere and the object and its image lie along a line through the center, the image will have the same angular speed about the center as the actual object. This means that if the object’s angular speed is known, one can compute the apparent tangential speed of the image by multiplying the angular speed by the radius at which the image appears. This connection is used to determine the apparent speed of the image as seen in the reflection.

Circular Orbits and Angular Velocity

An object in a circular orbit around a central body (such as a spacecraft around the moon) moves with a constant angular speed ?, which is given by the ratio of its tangential (orbital) speed to the orbital radius, ? = v / r. This connection is fundamental in orbital mechanics and allows one to relate the motion of an object in its actual orbit to the apparent motion (angular displacement) of its image when reflected from a spherical surface.

Spherical Mirror Equation and Focal Length

A spherical mirror (or a reflective sphere) obeys the mirror equation, 1/do + 1/di = 1/f, where do is the object distance from the mirror’s surface, di is the image distance (with sign conventions indicating whether the image is real or virtual), and f is the focal length of the mirror. For a sphere of radius R, the focal length is f = R/2. This relationship is key to finding where the image of the object appears relative to the mirror (such as the moon’s surface).

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