Question

The horizon is the line separating sky from the Earth's surface. Consider the shape of the Earth. How does the distance from your eyes to the horizon change as your elevation above the ground increases? To answer this question, draw a semicircle to represent part of the Earth's surface; then draw a vertical tower up from the surface. With your ruler, draw a line from various elevations on the tower to where the line is tangent to the surface of the Earth. (A tangent is a line that touches a circle at one point and is perpendicular to a radius.)

Short answer

The short answer to the question is that as the observer's elevation above the ground increases, the distance from the observer's eyes to the horizon also increases. This relationship is illustrated by drawing tangent lines from increasing elevations on a vertical tower to points on a semicircle representing the Earth's surface and measuring the respective distances.

Step-by-step solution

Draw a Semicircle Representing the Earth's Surface

Draw the Earth's surface by creating a semicircle. This semicircle will represent a part of the Earth's surface.

Draw a Vertical Tower

Draw a vertical line within the semicircle to represent the tower from which the observer is looking toward the horizon. This line should begin from the circular edge of the semicircle and extend upwards.

Identify Points of Elevation on the Tower

Mark various points on the vertical line representing the tower to represent different elevations. There is no specific number of elevations you have to use but make sure to have at least 2-3 marks for visibility.

Draw Tangent Lines from Elevation Points to the Earth's Surface

From each point of elevation, draw a line that is tangent to the semicircle (Earth's surface). A tangent line touches the semicircle at only one point and is perpendicular to the radius at that point. Repeat this for each marked elevation on the tower.

Measure the Distance from Each Elevation to the Tangent Point on the Earth's Surface

Use a ruler to measure the distance from each elevation point to the point where the tangent line meets the Earth's surface. This distance represents the distance from the observer's eyes to the horizon at each elevation. Record the measurements.

Analyze the Relationship between Elevation and Distance to the Horizon

By comparing the measurements for each elevation and their corresponding distances to the horizon, analyze the relationship between elevation and distance to the horizon. Based on this analysis, one can observe that as the elevation increases, the distance to the horizon also increases.

Key concepts

Tangent Lines

Imagine you're standing at the edge of a vast circle, and stretching out straight from your feet is a line that just grazes the edge without cutting into the circle itself. This is what mathematicians call a tangent line. In the context of our beautiful planet Earth, when you gaze out towards the horizon, you're actually looking along a tangent line that extends from your eyes and touches the Earth's curved surface at a single point.

This simple geometric construct is fundamental in understanding how we perceive the horizon. If you were to move higher, say climbing a tower, that tangent line—the edge of your visual limit—would touch the Earth further away, extending your horizon. It's like standing on your tiptoes to see over a fence; the higher you go, the further you can see. Geometry thus plays a critical role in our daily observations of the world.

Geometry in Earth Science

Geometry isn't just for mathematicians; it's crucial in Earth science too. The Earth's roundness means that geometric principles are key to understanding the planet's surface features. For instance, the concept of the horizon is grounded in geometry—specifically, the properties of circles.

By using geometry, we can reason about distances and angles on the Earth's surface with surprising accuracy. The drawing of a tangent line from a tower to the horizon is not just an abstract exercise; it is grounded in the same geometric rules that guide sailors and pilots. They use the Earth's geometry to navigate, measuring angles with the sky and the horizon to find their way across the ocean or through the clouds.

Measuring Distances on Earth

Measuring distances on Earth can be quite the challenge due to its curvature. It's not like measuring a straight line on a flat piece of paper. On the grand scale, Earth's surface is a vast curve, and distances across it are arcs, not straight lines. Sophisticated tools and methods, such as triangulation and satellite technology, are used to measure these complex distances.

For everyday purposes, like determining the distance to the horizon, simpler geometric principles apply. As you rise above the surface, the arc distance to the point where your sight line—a tangent to that huge 'circle' of Earth—increases. This could be measured, in principle, with a ruler and some elementary trigonometry. So, as our observational point ascends, our ability to see wider reaches of the Earth's surface grows accordingly. This interplay between geometry and distance measurement is both practically important and theoretically enlightening.

Earth's Surface Curvature

The curvature of Earth's surface is the central theme that ties together the concepts of tangent lines, geometry, and measuring distances. It's the reason we have horizons at all! The Earth's curvature bends away from us as we look out, creating that line where the sky appears to meet the ground.

Understanding this curvature is crucial for many areas of science and technology, from designing airplane flight paths to predicting satellite orbits. Even your daily weather forecast uses principles of Earth's curvature to model how storm systems will move over its surface. By visualizing Earth's surface as a series of tangent points touched by lines from varying elevations, we can grasp conceptually how curvature impacts what we can see and measure. From a practical standpoint, this understanding is instrumental in everything from erecting skyscrapers to setting up communication towers—they're all optimized based on the curve of the Earth.