Problem 15 - Physics

Question

Two astronauts are \(1.5 \mathrm{~m}\) apart in their spaceship. One speaks to the other. The conversation is transmitted to earth via electromagnetic waves. The time it takes for sound waves to travel at \(343 \mathrm{~m} / \mathrm{s}\) through the air between the astronauts equals the time it takes for the electromagnetic waves to travel to the earth. How far away from the earth is the spaceship?

Step-by-Step Solution

Answer

The spaceship is 1.31 million meters away from Earth.

Step 1: Calculate the time for sound to travel

First, calculate the time it takes for the sound waves to travel between the astronauts. Using the formula for time, \( t = \frac{d}{v} \), where \( d \) is the distance and \( v \) is the velocity, we have \( t = \frac{1.5 \, \text{m}}{343 \, \text{m/s}} \). Thus, \( t = 4.37 \times 10^{-3} \, \text{seconds} \).

Step 2: Calculate the distance for electromagnetic waves

The time calculated in Step 1 is the same time it takes for electromagnetic waves to travel from the spaceship to Earth. Since electromagnetic waves travel at the speed of light (approximately \( 3 \times 10^8 \, \text{m/s} \)), use the formula \( d = vt \). Thus, the distance to Earth \( d = 3 \times 10^8 \, \text{m/s} \times 4.37 \times 10^{-3} \, \text{s} = 1.31 \times 10^6 \, \text{m} \).

Key Concepts

Electromagnetic Waves

Electromagnetic waves are a type of wave that can travel through the vacuum of space. Unlike sound waves, which need a medium like air or water to travel, electromagnetic waves do not require any material to carry their energy from one place to another. They are created by the vibration of electric and magnetic fields and include a wide range of types, such as radio waves, microwaves, visible light, and gamma rays.

The ability of these waves to travel through space is why astronauts can communicate with people on Earth, even from the vast distances of outer space. When they speak, the sound waves are converted into electromagnetic waves, which can be transmitted over great distances very rapidly, at the speed of light.

Speed of Light

The speed of light is a fundamental constant in physics, symbolized by the letter "c." It represents the ultimate speed limit of the universe, at which all electromagnetic waves, including light itself, travel. The speed of light in a vacuum is approximately \(3\times 10^8\, \text{m/s}\), making it incredibly fast. This immense speed allows light to travel from the Sun to Earth, covering 93 million miles, in just about 8 minutes!

In practical applications, this speed is crucial for calculating distances, as seen when measuring how far a spaceship is from Earth based on the time taken for electromagnetic waves to travel between them. Knowing the speed of light allows scientists and engineers to pinpoint locations with astonishing precision from vast distances.

Acoustic Velocity

Acoustic velocity refers to the speed at which sound waves travel through a medium. Sound waves move at different speeds depending on the material they are passing through. For example, in air at room temperature, the speed of sound is approximately \(343\, \text{m/s}\). In water, sound travels faster, reaching speeds of about \(1500\, \text{m/s}\), and even faster in solids.

The concept of acoustic velocity is vital for understanding how humans communicate in different environments. For astronauts inside a space station or spaceship, sound typically travels through the provided atmosphere or air within the capsule. By knowing the speed of sound in air, we can calculate how long it takes for a conversation to travel between astronauts separated by a certain distance, as demonstrated in the exercise problem.

Distance Calculation

Calculating distances using the travel time of waves involves basic physics formulas. To find out how far a spaceship is from Earth using the travel time of electromagnetic waves, you would use the formula \(d = vt\), where \(d\) is distance, \(v\) is velocity, and \(t\) is time.

In the given exercise, once we know that the time taken by the sound waves to travel between astronauts is the same as the time taken by electromagnetic waves to reach Earth, we use this common time to calculate distance. For electromagnetic waves traveling at the speed of light, multiply this time by the speed of light to get the distance \(d = 3 \times 10^8\, \text{m/s} \times 4.37 \times 10^{-3}\, \text{s} = 1.31 \times 10^6\, \text{m}\). This gives us the distance that electromagnetic waves cover, which is how we find out how far the spaceship is from Earth.