Question

Space Communications In May 2003 Voyager 1 was 13 billion km from the Sun. Calculate how long it takes for a signal to travel this far assum ing it travels at \(3 \times 10^{8} \mathrm{~m} / \mathrm{s}\).

Short answer

The signal takes approximately 4.33 hours to travel 13 billion kilometers.

Step-by-step solution

Understand the Problem

We want to calculate the time it takes for a signal to travel from the Sun to Voyager 1. We know the distance is 13 billion kilometers and the speed of the signal is \(3 \times 10^8\) meters per second.

Convert Distance to Meters

The distance should be converted from kilometers to meters since the speed is given in meters per second. Since 1 kilometer equals 1000 meters, we have: \[13\text{ billion km} = 13 \times 10^9 \times 10^3 = 13 \times 10^{12} \text{ meters}\]

Apply the Speed Formula

The time \(t\) it takes for the signal to travel is given by the formula \(t = \frac{d}{v}\), where \(d\) is the distance and \(v\) is the velocity of the signal. Substitute the known values into the formula.

Calculate the Time

Using the formula and substituting the values, we have: \[t = \frac{13 \times 10^{12}}{3 \times 10^8} = \frac{13}{3} \times 10^{4} = 4.33 \times 10^{4} \text{ seconds}\]

Convert Time to Minutes

To make the time easier to understand, convert seconds into minutes by dividing by 60: \[t = \frac{4.33 \times 10^{4}}{60} \approx 721.67 \text{ minutes}\]

Key concepts

Speed of Light

The speed of light in a vacuum is one of the fundamental constants in physics. It represents the fastest possible speed at which energy or information can travel in the universe. This speed is commonly denoted by the letter "c" and is approximately equal to \(3 \times 10^8\) meters per second.

Light and other electromagnetic signals, such as radio waves, travel at this remarkable speed. It is so fast that when measured over earthly distances, it seems almost instantaneous. However, over cosmic distances, the finite speed of light becomes more apparent.

• This speed limits how quickly we can receive information from faraway spacecraft or stars.
• Understanding this concept is crucial in fields like astrophysics and telecommunications.

If you imagine the vast distances in space, even the incredibly fast speed of light can result in significant delays when communicating with distant spacecraft, like Voyager 1.

Distance Conversion

When dealing with calculations in physics, having distance in consistent units is important, as seen when working with the speed of light or any velocity. The problem with Voyager 1 highlights the need to convert kilometers into meters.

Conversions are simple once you know the conversion factors. For distance:

• 1 kilometer = 1,000 meters
• Therefore, 1 billion kilometers = \(1 \times 10^9\) kilometers = \(1 \times 10^9 \times 10^3\) meters = \(1 \times 10^{12}\) meters

In this problem, Voyager 1 is 13 billion kilometers away, so conversion to meters gives us \(13 \times 10^{12}\) meters. This step ensures that the velocity and distance units match, allowing correct calculation of time.

Velocity

Velocity is an important concept when calculating time for signals to travel over a distance. For this exercise, velocity refers to the speed at which the signal from the Sun to Voyager 1 travels.

Velocity is a vector quantity, which means it has both magnitude (speed) and direction. Since we are concerned only with speed here, direction is not considered. The speed or velocity of light in this problem is \(3 \times 10^8 \) meters per second, ensuring clarity in calculation.

• Velocity determines how quickly a signal covers a distance over time.
• The greater the velocity, the shorter the time it takes to traverse a given distance.

By knowing the velocity, you can easily calculate how much time it will take to travel a certain distance by using the formula \(t = \frac{d}{v}\), where \(t\) is time, \(d\) is distance, and \(v\) is velocity.

Time Calculation

Calculating time involves determining how long it takes for an object or signal to traverse a given distance at a certain speed. The formula to find time is:\[t = \frac{d}{v}\]where \(t\) is time, \(d\) is distance, and \(v\) is velocity.

In the case of Voyager 1, we identify the following:

• Distance, \(d = 13 \times 10^{12}\) meters (after conversion from kilometers).
• Velocity, \(v = 3 \times 10^8\) meters per second (speed of light).

Plug these values into the formula:

\[t = \frac{13 \times 10^{12}}{3 \times 10^8} = 4.33 \times 10^4 \text{ seconds}\]To make results more comprehensible, converting seconds into minutes can be helpful. Simply divide the total seconds by 60:

• \( t \approx 721.67 \text{ minutes} \)

This shows that a signal from the Sun to Voyager 1 takes about 721.67 minutes, emphasizing the effect of distance on communication time.