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Chapter 1: Problem 42

Be sure to show all calculations clearly and state your final answers in complete sentences. Spacecraft Communication. We use radio waves, which travel at the speed of light, to communicate with robotic spacecraft. How long does it take a message to travel from Earth to a spacecraft at a. Mars at its closest to Earth (about 56 million km)? b. Mars at its farthest from Earth (about 400 million \(\mathrm{km}\) )? c. Pluto at its average distance from Earth (about 5.9 billion km)?

Short Answer

Expert verified
At closest, Mars takes ~3.11 minutes; at farthest, ~22.24 minutes; Pluto ~327.90 minutes (5.47 hours).

Step by step solution

01

Identify the Speed of Light

The speed of light in a vacuum is a constant value, which is approximately 299,792 kilometers per second.
02

Calculate Time for Mars at Closest Distance

Use the formula for time, which is \( t = \frac{d}{s} \), where \( d \) is distance and \( s \) is speed. Substitute \( d = 56,000,000 \) km and \( s = 299,792 \) km/s. \[t = \frac{56,000,000}{299,792} \approx 186.85 \text{ seconds.}\]
03

Calculate Time for Mars at Farthest Distance

Again, use the formula \( t = \frac{d}{s} \) with \( d = 400,000,000 \) km.\[t = \frac{400,000,000}{299,792} \approx 1334.56 \text{ seconds.}\]
04

Calculate Time for Pluto at Average Distance

Finally, apply the same formula with \( d = 5,900,000,000 \) km.\[t = \frac{5,900,000,000}{299,792} \approx 19,674.27 \text{ seconds.}\]
05

Convert Seconds to Minutes for Context

To make these numbers more understandable, convert the times calculated from seconds to minutes. For 186.85 seconds: \[ 186.85 \div 60 \approx 3.11 \text{ minutes.}\]For 1334.56 seconds: \[ 1334.56 \div 60 \approx 22.24 \text{ minutes.}\]For 19,674.27 seconds: \[ 19,674.27 \div 60 \approx 327.90 \text{ minutes or roughly 5.47 \text{ hours.}}\]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Radio Waves
Radio waves play a crucial role in space communication. These waves form part of the electromagnetic spectrum, similar to light waves but with longer wavelengths. That gives them certain advantages, such as the ability to travel vast distances through the vacuum of space.
To communicate with spacecraft, scientists on Earth send signals through radio waves. These signals can be used for various purposes, including data transmission and remote control instructions to the spacecraft. The benefits of radio waves include:
  • Long-distance capabilities: Radio waves can carry signals across millions or even billions of kilometers.
  • Reliability: They can penetrate through clouds, dust, and atmospheric gases without losing strength.
Understanding how these waves work helps us appreciate their effectiveness in maintaining communication with distant robotic explorers.
Speed of Light
The speed of light is an important constant in physics and is vitally important for space communication. Light, radio waves, and all electromagnetic waves travel at this constant speed, approximately 299,792 kilometers per second (km/s) in a vacuum. This speed is denoted by the symbol "c".
The remarkable speed of light enables us to send and receive messages over staggering distances in relatively short amounts of time. For instance:
  • A journey to the Moon and back for a signal takes only about 2.5 seconds.
  • Communicating with Mars takes longer, depending on the planet's position relative to Earth.
Understanding the speed of light is key in calculating how long it takes signals to travel to our farthest satellites and robotic explorers.
Distance Calculation
Calculating distance is a fundamental aspect of determining how long it takes a radio signal to travel from Earth to another celestial body. The formula to calculate the time it takes a signal to reach its destination involves distance (\(d\)), speed (\(s\)), and time (\(t\).
The formula is given by:
\(t = \frac{d}{s}\).
To apply this formula, we know that the speed is the speed of light, \(s \approx 299,792 \text{ km/s}\), and distance is the specific measure between Earth and the spacecraft. For instance, consider Mars:
  • At its closest, Mars is approximately 56 million km from Earth.
  • At its farthest, this distance grows to about 400 million km.
This straightforward calculation provides us with the time it takes for signals to traverse these tremendous distances.
Time Conversion
Time conversion is essential when turning the computation results into understandable units, particularly in practical applications like space communication. When computing the time taken for radio waves to travel across space, the result is often in seconds. These durations need to be converted into minutes or hours for easier understanding.
For example:
  • A computed travel time of 186.85 seconds from Earth to Mars (at its closest) can be converted to minutes as follows:
    \(186.85 \div 60 \approx 3.11 \text{ minutes}\).
  • In a similar way, 1334.56 seconds become approximately 22.24 minutes.
Such conversions allow people working with spacecraft communication to better comprehend the time scale they are dealing with, enabling them to plan and execute operations more efficiently.

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Most popular questions from this chapter

Be sure to show all calculations clearly and state your final answers in complete sentences. Distances by Light. Just as a light-year is the distance that light can travel in 1 year, we define a light-second as the distance that light can travel in 1 second, a light-minute as the distance that light can travel in 1 minute, and so on. Calculate the distance in both kilometers and miles represented by each of the following: a. 1 light-second. b. 1 light-minute. c. 1 light-hour. d. 1 light-day.

A Human Adventure. Astronomical discoveries clearly are important to science, but are they also important to our personal lives? Defend your opinion.

Choose the best answer to each of the following. Explain your reasoning with one or more complete sentences. Which of the following correctly lists our "cosmic address" from small to large? (a) Earth, solar system, Milky Way Galaxy, Local Group, Local Supercluster, universe (b) Earth, solar system, Local Group, Local Supercluster, Milky Way Galaxy, universe (c) Earth, Milky Way Galaxy, solar system, Local Group, Local Supercluster, universe.

NASA Missions. Visit the NASA website to learn about upcoming astronomy missions. Write a one-page summary of the mission you believe is most likely to give us new astronomical information before the end of your astronomy course.

Thinking About Scale. One key to success in science is finding simple ways to evaluate new ideas, and making a simple scale model is often helpful. Suppose someone tells you that the reason it is warmer during the day than at night is that the day side of Earth is closer to the Sun than the night side. Evaluate this idea by thinking about the size of Earth and its distance from the Sun in a scale model of the solar system.
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