Suppose a moon that was 100 kilometers in diameter broke up...

Suppose a moon that was 100 kilometers in diameter broke up into particles just 1 millimeter in diameter. a. What was the surface area of the moon before it broke up? b. How many particles 1millimeter in diameter could be made from the moon? c. What is the total surface area of all of the 1 -mm particles you found in part b? d. What is the ratio of the surface area in part $c$ to the surface area in part a? Explain why this ratio also represents how much sunlight would be reflected by the particles versus the original moon.

Suppose a moon that was 100 kilometers in diameter broke up into particles just 1 millimeter in diameter.
a. What was the surface area of the moon before it broke up?
b. How many particles 1millimeter in diameter could be made from the moon?
c. What is the total surface area of all of the 1 -mm particles you found in part b?
d. What is the ratio of the surface area in part $c$ to the surface area in part a? Explain why this ratio also represents how much sunlight would be reflected by the particles versus the original moon.

Key Concepts

Surface Area of a Sphere

This concept involves calculating the area that covers the outer surface of a sphere. The formula, 4?r², where r is the radius, is fundamental in many applications, including determining the amount of material needed to cover spherical objects or assessing exposure to environmental phenomena such as sunlight.

Volume of a Sphere

This concept is used to determine the amount of three?dimensional space occupied by a sphere. The volume is calculated by the formula (4/3)?r³. Understanding the volume is key when comparing the mass or content of spherical objects, especially when they are broken into smaller parts.

Scaling Laws and Dimensional Analysis

Scaling laws explain how changes in dimensions, such as reducing the diameter of an object, affect other properties like volume and surface area. When an object is divided into smaller parts, volume (which scales with the cube of the linear dimension) determines the number of parts, while each part’s surface area (scaling with the square of the dimension) contributes to the total exposed area.

Reflectivity Related to Surface Area

The amount of sunlight an object reflects is largely dependent on its total surface area. Increasing the total surface area by breaking an object into smaller particles can result in a greater overall reflection of sunlight, as each particle contributes its own reflective area. This concept links geometric properties with physical phenomena such as light reflection.

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