Problem 31 - Physics for Scientists and Engineers with Modern Physics

Question

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 see-that is, how far is your horizon? The Earth's radius is about \(6400 \mathrm{~km}\).

Step-by-Step Solution

Answer

You can see approximately 50.4 kilometers away to the horizon.

Step 1: Identify the Variables

Identify the given information and variables of the problem. The height of the balloon above the Earth is given as \( h = 200 \) meters. The radius of the Earth is given as \( R = 6400 \) kilometers, which needs to be converted into meters: \( R = 6400 \times 1000 = 6,400,000 \) meters.

Step 2: Visualize the Problem

Visualize the scenario by imagining a right triangle. The Earth's radius \( R \) forms one leg, the distance to the horizon \( d \) forms the other leg, and \( R + h \) (the radius plus the height of the balloon) forms the hypotenuse of the right triangle.

Step 3: Apply the Pythagorean Theorem

Apply the Pythagorean theorem to find the distance to the horizon: \((R + h)^2 = R^2 + d^2\). Rearrange for \(d\): \(d^2 = (R + h)^2 - R^2\).

Step 4: Calculate Using the Provided Values

Substitute the known values into the rearranged Pythagorean equation: \(d^2 = (6,400,000 + 200)^2 - (6,400,000)^2\). Calculate \((6,400,200)^2\) and subtract \((6,400,000)^2\).

Step 5: Simplify the Calculation

Recognize that \((a + b)^2 = a^2 + 2ab + b^2\). Here, calculate only the extra terms using \(2ab + b^2\): \(2 \times 6,400,000 \times 200 + 200^2\), then compute.

Step 6: Solve for the Distance to the Horizon

Compute the expression: \(d^2 = 2 \times 6,400,000 \times 200 + 200^2 = 2,560,000,000 + 40,000\). Find \(d\) by taking the square root: \(d = \sqrt{2,560,040,000}\).

Step 7: Determine the Final Answer

Calculate \(d\), which gives the final answer for the distance to the horizon. After computing the square root, you will find \(d \approx 50,398 \) meters or roughly \(50.4\) kilometers.

Key Concepts

Pythagorean Theorem

The Pythagorean Theorem is a fundamental concept in mathematics, particularly in geometry. It relates the lengths of the sides of a right triangle. A right triangle has one angle of 90 degrees, and the theorem provides a relationship between the lengths of the two legs and the hypotenuse (the side opposite the right angle). This relationship is given by the formula:\[ c^2 = a^2 + b^2 \]Where:

• \(c\) is the length of the hypotenuse,
• \(a\) and \(b\) are the lengths of the other two sides.

In the context of our problem, the Pythagorean Theorem helps us find the distance to the horizon. By considering the Earth’s radius as one side and the height of our viewpoint plus the Earth’s radius as the hypotenuse, we can calculate the horizontal distance to the horizon. This conversion from a real-world situation into a geometric problem showcases the power and versatility of the Pythagorean Theorem even in physics applications. Understanding and mastering this theorem will make solving many geometric problems much easier.

Earth's Radius

The Earth's radius is a crucial measurement needed for calculations involving our planet. In simple terms, it's the distance from the center of the Earth to its surface. Knowing this value is essential for many physics calculations, including those involving gravity and distance. For our balloon problem, the Earth's radius is used as part of the triangle formed by the Earth, the balloon, and the horizon. In our scenario, we assume the Earth is a perfect sphere, which simplifies the calculation. The Earth's average radius is about 6,400 kilometers, and converting it to meters gives us 6,400,000 meters. Key points to remember about the Earth's radius:

• It helps in understanding distances and scaling maps.
• It's fundamental in calculations related to planetary science.
• It's necessary for solving geometry-related questions on a planetary scale.

With the Earth’s radius as one of the sides of our right triangle, we can apply geometry to find out how far we can see from a given height above the Earth’s surface, like from a balloon.

Distance to Horizon

The distance to the horizon is a fascinating concept that reflects how far you can see from a certain height before the Earth's curvature makes objects disappear from view. This calculation depends on the height of the observer (or object, like a hot air balloon) above the Earth's surface.To find this distance, we use the geometry of a triangle where the hypotenuse includes the observer's height. In our problem, the observer is the hot air balloon at 200 meters high. This practical application of the Pythagorean Theorem involves rearranging the formula for the hypotenuse to solve for the distance to the horizon:\[ d^2 = (R + h)^2 - R^2 \]Where:

• \(d\) is the distance to the horizon,
• \(R\) is the Earth's radius,
• \(h\) is the height of the observer above the Earth's surface.

Solving this equation gives us the perspective view limit for an observer at a height \(h\). This specific scenario demonstrates how understanding geometry allows us to quantify our visual limits based on physical parameters, bringing real-world application and mathematical theory together.