Problem 6 - An Introduction to Physical Science

Question

A space probe on the surface of Mars sends a radio signal back to the Earth, a distance of \(7.86 \times 10^{7} \mathrm{km} .\) Radio waves travel at the speed of light \(\left(3.00 \times 10^{8} \mathrm{m} / \mathrm{s}\right) .\) How many seconds does it take the signal to reach the Earth?

Step-by-Step Solution

Answer

The signal takes 262 seconds to reach Earth.

Step 1: Convert Distance to Meters

The given distance from Mars to Earth is \(7.86 \times 10^{7}\) km. First, convert this distance to meters because the speed of light is given in meters per second. We know \(1\) km is \(1000\) meters. Therefore, the distance in meters is:\[7.86 \times 10^{7} \text{ km} \times 1000 \text{ m/km} = 7.86 \times 10^{10} \text{ m}\]

Step 2: Apply the Speed-Distance-Time Formula

Use the formula \( \text{Time} = \frac{\text{Distance}}{\text{Speed}} \) to find the time it takes for the radio signal to reach Earth. The distance is \(7.86 \times 10^{10}\) m, and the speed of light is \(3.00 \times 10^{8}\) m/s. Thus, the time it takes is:\[\text{Time} = \frac{7.86 \times 10^{10} \text{ m}}{3.00 \times 10^{8} \text{ m/s}}\]

Step 3: Calculate the Time

Perform the division to find the time:\[\text{Time} = \frac{7.86 \times 10^{10}}{3.00 \times 10^{8}} = 2.62 \times 10^{2}\]This calculation yields that the time it takes for the signal to reach Earth is \(262\) seconds.

Key Concepts

Speed of Light

The speed of light is an essential constant in the field of physics. It represents the fastest speed at which information and matter can travel through the universe. Light travels at an incredible speed of approximately \(3.00 \times 10^{8}\, \text{m/s}\). This applies not only to visible light but also to other types of electromagnetic waves, such as radio waves. It's crucial to understand this speed when studying phenomena like radio wave propagation, satellite signals, and space exploration.

When dealing with problems involving the speed of light, we often use the formula \( \text{Time} = \frac{\text{Distance}}{\text{Speed}} \). This equation helps us calculate how long it will take a signal to travel from one point to another in space. The constant nature of light's speed in a vacuum provides a reliable tool for making these calculations.

Distance Conversion

Distance conversion is a necessary step when different units of measurement are involved in a problem. In this exercise, the given distance from Mars to Earth was initially provided in kilometers \(7.86 \times 10^{7}\,\text{km}\). To use this distance in the time calculation, you must convert it to meters, because the speed of light is expressed in meters per second.

To perform the conversion, remember that \(1\,\text{km} = 1000\,\text{m}\). Therefore, you multiply the distance in kilometers by 1000 to obtain the distance in meters. This gives us the formula:

• Distance in meters = Distance in kilometers \(\times\) 1000

By applying this, Mars' distance to Earth becomes \(7.86 \times 10^{10}\,\text{m}\). This step ensures compatibility with the speed of light measurement, allowing for precise calculations.

Time Calculation

Calculating the time it takes for a signal to travel a given distance at a known speed involves a straightforward application of the speed-distance-time formula: \( \text{Time} = \frac{\text{Distance}}{\text{Speed}} \). In our problem, we need to calculate how long it takes for a radio signal to travel from Mars to Earth.

Given:

• Distance \(= 7.86 \times 10^{10}\, \text{m}\)
• Speed of Light \(= 3.00 \times 10^{8}\, \text{m/s}\)

Plug these values into the formula:

• \( \text{Time} = \frac{7.86 \times 10^{10}}{3.00 \times 10^{8}} \)

Solving this provides \(\text{Time} = 262\,\text{s}\).

This calculation shows that it takes approximately 262 seconds for the radio signal to reach Earth. Understanding this concept is crucial for tasks involving communication over long distances, such as those undertaken in space missions.