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 \(9.067 \times 10^3 \, \mathrm{s}\), or 9067 seconds, for a radio signal from the Voyager spacecraft to reach Earth.

Step 1: Convert distance to meters

We are given the distance between Voyager and Earth in kilometers, so we should first convert it to meters to match the units of the speed of light. To do this, we will multiply it by 1000: \(2.72 \times 10^{9} \, \mathrm{km} = 2.72 \times 10^{9} \times 1000 \, \mathrm{m} = 2.72 \times 10^{12} \, \mathrm{m}\)

Step 2: Calculate the time taken for the radio signal to travel

Now that we have the distance in meters, we can use the formula for time to calculate the time it takes for the radio signal to travel: \(time = \frac{distance}{speed}\) \(time = \frac{2.72 \times 10^{12} \, \mathrm{m}}{3.0 \times 10^8 \, \mathrm{m/s}}\) \[\approx \frac{2.72}{3.0} \times 10^{4} \mathrm{s}\]

Step 3: Simplify the result

Let's do the math and simplify the result: \[\approx 0.9067 \times 10^{4} \mathrm{s}\] \(time \approx 9.067 \times 10^3 \, \mathrm{s}\) So, it takes approximately \(9.067 \times 10^3 \, \mathrm{s}\), or 9067 seconds, for a radio signal from the Voyager spacecraft to reach Earth.

Key Concepts

Speed of Light

The speed of light is a fundamental constant in physics and plays a critical role in understanding radio signal propagation. It defines how fast light—and all electromagnetic waves—travels in a vacuum, which is approximately 299,792,458 meters per second (or roughly \(3.0 \times 10^8\, \mathrm{m/s}\)). This speed is incredibly fast, and it means that light—and radio signals—can travel significant distances in very short periods.

On Earth, we accept this speed as a constant when conducting various calculations involving electromagnetic waves. It influences how we time phenomena, synchronize systems, and understand the universe. In the case of signals from spacecraft, such as Voyager, knowing this speed allows crucial calculations in determining how long signals take to reach Earth from different points in our solar system.

Distance Conversion

When dealing with cosmic distances, it's important to convert kilometers to meters for consistency with the speed of light's unit in meters per second. This conversion is a straightforward multiplication: for every kilometer, there are 1000 meters, so you multiply the distance in kilometers by 1000 to convert.

In the calculation for the Voyager spacecraft radio signal, we start with a distance of \(2.72 \times 10^9\) kilometers. To find the distance in meters:

• Multiply \(2.72 \times 10^9\) km by 1000.
• The result is \(2.72 \times 10^{12}\) meters.

Converting units is essential for ensuring accuracy in physics, particularly in calculations involving extensive distances like those found in space.

Time Calculation

Calculating how long it takes for a signal to travel involves using the basic formula for time, given as \(\text{time} = \frac{\text{distance}}{\text{speed}}\). This formula is straightforward and hinges on knowing both the speed and the distance to determine the time.

For signals traveling from Voyager to Earth, plug in the distance in meters (\(2.72 \times 10^{12}\) m) and the speed of light (\(3.0 \times 10^8\) m/s) into the formula:

• \(\text{time} = \frac{2.72 \times 10^{12}}{3.0 \times 10^8}\)
• Approximate the division to get \(9.067 \times 10^3\) seconds, or 9067 seconds.

This time calculation tells us how long a radio signal from the Voyager spacecraft will take to travel to Earth, showcasing the vastness of space and the practicality of using physics to solve real-world astronomical problems.