The distance from Earth to Mars varies from 56 million km to 400 million \(\mathrm{km} .\) How long does it take a radio signal traveling at the speed of light to reach a spacecraft on Mars when Mars is closest and when Mars is farthest away?

Understanding the speed of light is essential in this exercise. The speed of light in a vacuum is approximately \(3 \times 10^8\) meters per second (m/s). This value is constant and is represented by the symbol \(c\). Light and radio signals travel at this incredible speed, allowing them to cover vast distances in relatively short times. Knowing this speed helps us calculate how long it takes for signals to travel between planets, like Earth and Mars.

When working with astronomical distances, we often need to convert units. In this exercise, distances are given in kilometers (km). However, since the speed of light is in meters per second (m/s), we must convert kilometers to meters.
1 kilometer equals 1000 meters. For example, to convert 56 million kilometers to meters, we multiply: \(56 \times 10^6 \text{ km} \times 10^3 \text{ m/km} = 56 \times 10^9 \text{ m}\). Practicing this conversion is crucial for accurate calculations.

To find how long a radio signal takes to travel a given distance, we use the formula: \( t = \frac{d}{v} \), where \(t\) is time, \(d\) is distance, and \(v\) is velocity (speed of light).
Let’s apply this to the given example for the closest distance to Mars (56 million km). We converted that distance to meters: \(56 \times 10^9 \text{ m}\). Plugging this into the formula, we get:
\( t = \frac{56 \times 10^9 \text{ m}}{3 \times 10^8 \text{ m/s}} = 186.67 \text{ seconds} \). This tells us that it takes about 186.67 seconds for a signal to travel the closest distance to Mars.
For the farthest distance (400 million km converted to meters), the calculation is: \( t = \frac{400 \times 10^9 \text{ m}}{3 \times 10^8 \text{ m/s}} = 1333.33 \text{ seconds} \).

The distance between Earth and Mars changes because both planets have elliptical orbits around the Sun. The closest possible distance is about 56 million kilometers, while the farthest can be around 400 million kilometers. This variation affects the travel time of radio signals significantly.

• Closest Distance: Mars is near opposition, a point where Earth is directly between Mars and the Sun.
• Farthest Distance: Occurs roughly when the two planets are on opposite sides of the Sun in their respective orbits.

Knowing these distances helps in planning missions and communications with spacecraft on Mars.