Problem 54 - Chemistry Matter and Change

Question

Spacecraft How long does it take a radio signal from the Voyager spacecraft to reach Earth if the distance between Voyager and Earth is \(2.72 \times 10^{9} \mathrm{km}\) ?

Step-by-Step Solution

Answer

It takes approximately 9067 seconds for the radio signal to reach Earth.

Step 1: Understand the Speed of Light

Radio signals travel at the speed of light. The speed of light in a vacuum is approximately \[3.00 \times 10^{5} \text{ km/s}\]. This constant is crucial for calculating the time it takes for the signal to travel from Voyager to Earth.

Step 2: Use the Formula for Time

To find the time it takes for the radio signal to reach Earth, use the formula:\[ \text{Time} = \frac{\text{Distance}}{\text{Speed}} \]This formula will allow us to calculate the time by dividing the distance the signal travels by the speed at which it travels.

Step 3: Substitute the Values

Substitute the given values into the formula:\[ \text{Time} = \frac{2.72 \times 10^{9} \text{ km}}{3.00 \times 10^{5} \text{ km/s}} \]This substitution sets up the equation with the known distance and the speed of light.

Step 4: Perform the Calculation

Now, perform the division to calculate the time:\[ \text{Time} = \frac{2.72 \times 10^{9}}{3.00 \times 10^{5}} \]Calculate this to find the time in seconds it takes for the signal to reach Earth.

Step 5: Final Result

The result of the calculation is:\[ \text{Time} = 9066.67 \text{ seconds} \]Therefore, it takes approximately 9067 seconds for the radio signal to travel from the Voyager spacecraft to Earth.

Key Concepts

Radio Signals

Radio signals are an essential tool used for communication between spacecraft and Earth. These signals are a type of electromagnetic wave, similar to light waves but at a much lower frequency. They allow us to receive information across vast distances in space almost instantly. This is crucial for space exploration as it enables communication with spacecraft like the Voyager.
To understand why radio signals are so effective over long distances, it's important to know they travel at the speed of light. In a vacuum, this speed is approximately \(3.00 \times 10^{5} \text{ km/s}\). This impressive speed allows signals to cover vast stretches of space in relatively short time periods, making them perfect for these kinds of missions.

• Electromagnetic wave
• Travel at speed of light
• Effective for long-distance communication

The use of radio signals in space communication ensures data transmitted from spacecraft reach Earth's receiving stations efficiently.

Distance Calculation

In problems involving the distance between Earth and a spacecraft like Voyager, it's vital to know the exact distance that the radio signals need to travel. This distance is generally given in kilometers or meters. In the specific case with Voyager, the distance to Earth is given as \(2.72 \times 10^{9} \text{ km}\).
Knowing the exact distance helps calculate how long the radio signal takes to travel from the spacecraft to Earth. This measurement forms the numerator in the time calculation equation, which is essential for figuring out the precise time interval involved.

• Given in kilometers or meters
• Important for time calculation
• Forms numerator in the equation

Thus, having an accurate measurement of distance is crucial for precise time calculation of radio signals traveling from space to Earth.

Time Calculation

Calculating the time it takes for a radio signal to travel from a spacecraft to Earth involves using a straightforward formula:
\[ \text{Time} = \frac{\text{Distance}}{\text{Speed}} \]This formula divides the known distance traveled by the speed at which the signal travels, which in the case of radio signals is the speed of light.
In our specific example, we substitute the given values:\[ \text{Time} = \frac{2.72 \times 10^{9} \text{ km}}{3.00 \times 10^{5} \text{ km/s}} \]Performing this calculation gives the time in seconds it takes for the radio signal to reach Earth, providing a crucial understanding of the dynamics involved in space communication.

• Uses speed of light for calculation
• Provides time in seconds
• Central part of space communication

Understanding and performing these calculations helps ensure effective communication across the vastness of space.