Problem 31 - Physics for Scientists and Engineers with Modern Physics

Question

(III) You are in a hot air balloon, 200 \(\mathrm{m}\) above the flat Texas plains. You look out toward the horizon. How far out can you sec-that is, how far is your horizon? The Earth's radius is about 6400 \(\mathrm{km}\) .

Step-by-Step Solution

Answer

The horizon is approximately 50.6 km away.

Step 1: Convert Balloon Height to Kilometers

First, we need to convert the height of the balloon from meters to kilometers to match the units of the Earth's radius. Since there are 1000 meters in a kilometer, we have:\[200 \text{ meters} = 0.2 \text{ kilometers}\]

Step 2: Apply the Horizon Distance Formula

The formula to calculate the distance to the horizon is given by:\[d = \sqrt{2 \cdot R \cdot h + h^2}\]where \(R\) is the Earth's radius (6400 km) and \(h\) is the height above the Earth's surface in kilometers (0.2 km).

Step 3: Substitute the Values

Substitute the given values into the formula:\[d = \sqrt{2 \times 6400 \times 0.2 + 0.2^2}\]

Step 4: Calculate the Expression

First, calculate the expression inside the square root:\[2 \times 6400 \times 0.2 = 2560 \]and\[0.2^2 = 0.04\]Then, add these results:\[2560 + 0.04 = 2560.04\]Now, calculate the square root:\[d = \sqrt{2560.04} \approx 50.6\]

Step 5: Conclude the Calculation

Finally, the distance to the horizon from a balloon 200 meters above the ground is approximately 50.6 kilometers.

Key Concepts

Horizon Distance Formula

The horizon distance formula is a mathematical equation used to calculate how far we can see over the Earth's surface from a certain height. This formula is particularly handy for applications such as aviation, maritime navigation, and even for people like you in a hot air balloon! The formula helps you figure out the farthest point on the horizon that you can spot, depending on your height above the ground.
The formula is:

• \[ d = \sqrt{2 \cdot R \cdot h + h^2} \] where:
  • \( d \) is the distance to the horizon,
  • \( R \) represents the Earth's radius, and
  • \( h \) is the height above the Earth's surface.

This square root-containing equation combines two components:

• The term \( 2 \cdot R \cdot h\) accounts for the Earth's curvature, and
• \( h^2 \) is an additional correction that becomes significant at greater heights.

Using this formula, you can quickly calculate the horizon distance, giving you an insight into your visibility range based on your elevation.

Balloon Height Conversion

Before using the horizon distance formula, it's important to make sure your measurements are in the same unit system. For instance, if the Earth's radius is given in kilometers, your height should also be in kilometers.
For our balloon example, we need to convert a height of 200 meters into kilometers to match the unit for Earth's radius.
Remember:

• 1 kilometer equals 1000 meters.

Therefore, converting the balloon's height from meters to kilometers is straightforward:

• \[ 200 \text{ meters} = 0.2 \text{ kilometers} \]

This conversion is crucial because the formula requires compatible units to ensure an accurate calculation. This initial step lays the groundwork for applying the mathematical equation without any mismatch in measurement systems.

Earth Radius

The Earth's radius is a fundamental part of the horizon distance formula. The average radius of Earth is about 6400 kilometers. This value is derived from the roughly spherical shape of the Earth, although this is an approximation since Earth is not a perfect sphere.
Knowing the Earth's radius is critical as it impacts the calculation of how far one can see to the horizon.
When we talk about the Earth's radius in the context of horizon distance:

• We're considering the distance from the Earth's center to its surface,
• Which impacts the curvature and thus the apparent distance to the horizon.

In practice, understanding the radius allows one to calculate the extent of their view from a given height above the Earth's surface, making the concept of Earth’s radius a key element in these types of calculations.