Question

A surface probe on Mars transmits radiowaves at a frequency of \(6.0 \times 10^{5} \mathrm{sec}^{-1}\) to an earth station \(8.0 \times 10^{7} \mathrm{~km}\) away. How long does it take radiowaves to traverse this distance?

Short answer

The radio waves take approximately \(266.67 \ \mathrm{s}\) to traverse the given distance of \(8.0 \times 10^7 \ \mathrm{km}\) from Mars to the Earth station.

Step-by-step solution

Identify the Speed of Light

Since radio waves travel at the speed of light, we will use the speed of light \(c\) as their speed. The speed of light \(c\) is approximately \(3.0 \times 10^8 \ \mathrm{m/s}\).

Convert the Speed of Light into km/s

To convert the speed of light into km/s, we can just divide it by 1000 since 1 km = 1000 m. The speed of light in km/s is \(3.0 \times 10^5 \ \mathrm{km/s}\).

Calculate the Time Required for Radiowaves to Travel the Distance

We are given the distance (\(8.0 \times 10^7 \ \mathrm{km}\)) and the speed of light in km/s. Now, we can use the formula for time: time = distance / speed \(t = \frac{8.0 \times 10^7 \ \mathrm{km}}{3.0 \times 10^5 \ \mathrm{km/s}}\)

Simplify the Expression

We can cancel out the units km and divide the numbers: \(t = \frac{8.0 \times 10^7}{3.0 \times 10^5} \ \mathrm{s}\) \(t = \frac{8}{3} \times \frac{10^7}{10^5} \ \mathrm{s}\) \(t = \frac{8}{3} \times 10^2 \ \mathrm{s}\)

Final calculation

The final calculation can be done now: \(t = \frac{8}{3} \times 100 \ \mathrm{s}\) \(t \approx 266.67 \ \mathrm{s}\) The radio waves take approximately 266.67 seconds to traverse the given distance of \(8.0 \times 10^7 \ \mathrm{km}\) from Mars to the Earth station.

Key concepts

Speed of Light

When discussing the fundamentals of physics, the speed of light is a critical constant, not only in science but also for understanding universal workings. This remarkable speed is roughly \(3.0 \times 10^8 \text{ meters per second} (m/s)\). It's the ultimate speed limit of the universe, meaning nothing can move faster according to our current understanding of physics.

Radio waves, which are a form of electromagnetic radiation, travel at this same constant speed in a vacuum like space. This constancy provides a reliable measure for establishing communication times between distances in space, including between planets. For calculations, it is often more practical to convert this speed into kilometers per second, which is \(3.0 \times 10^5 \text{ km/s}\), to match the commonly used distance units in astronomy.

Radio Waves Travel Time

The radio waves travel time can be seen as the interval needed for radio waves to get from one point to another. It is particularly important in astronomy and communications as it affects how we time our transmissions and receive data from space probes on distant planets, like Mars. For instance, knowing the travel time helps in planning when to send signals and when to expect responses. In our specific example, a probe on Mars transmitting data must account for the time radio waves take to reach Earth.

The calculation of this travel time leverages the speed of light. As both light and radio waves travel at the same speed, the travel time is simply the distance divided by the speed of light. Engineers and scientists working on space missions use these calculations to their advantage in order to sync communications with distant spacecraft.

Distance Speed Time Formula

The distance speed time formula is essential in many areas of physics and practical applications like travel and communication over space. It's elegantly simple: time = distance / speed. When you know any two of these variables, you can figure out the third. For instance, if you want to determine the time it takes for a journey, you just divide the distance by the speed at which you're traveling.

For the exercise in question, the distance is \(8.0 \times 10^7 \text{ km}\) and the speed of light (and radio waves) is \(3.0 \times 10^5 \text{ km/s}\). Using the formula, we find that the travel time for radio waves between Mars and Earth approximately equals 266.67 seconds. Such calculations are vital in space exploration as they help in navigation and maintaining communication with distant probes—ensuring the success of missions and safety of any sent equipment.