Question

A parsec is the distance light travels in 3.26 years. Given the velocity of light, \(3.00 \times 10^{8} \mathrm{~m} / \mathrm{s}\), how many kilometers does light travel in a parsec?

Short answer

Light travels approximately \(3.0871 \times 10^{13}\) kilometers in a parsec.

Step-by-step solution

Understand the Problem

We need to find the distance light travels in a parsec by using the speed of light and the time it takes light to travel that distance, which is 3.26 years.

Convert Years to Seconds

First, convert the time of 3.26 years into seconds. We use the conversion factors: 1 year = 365 days, 1 day = 24 hours, 1 hour = 3600 seconds. Thus, the number of seconds in 3.26 years is:\[3.26 \text{ years} \times 365 \text{ days/year} \times 24 \text{ hours/day} \times 3600 \text{ seconds/hour}\]

Calculate Seconds

Calculate the seconds in 3.26 years:\[3.26 \times 365 \times 24 \times 3600 = 102,902,400 \text{ seconds}\]

Calculate Distance in Meters

Use the formula for distance: Distance = Velocity × Time to find the distance in meters:\[3.00 \times 10^8 \text{ m/s} \times 102,902,400 \text{ s}\]

Calculate Distance Value

Calculate the distance light travels in 3.26 years in meters:\[3.00 \times 10^8 \times 102,902,400 = 3.0871 \times 10^{16} \text{ meters}\]

Convert Meters to Kilometers

Since 1 kilometer = 1000 meters, we convert the distance from meters to kilometers:\[3.0871 \times 10^{16} \text{ meters} \div 1000 = 3.0871 \times 10^{13} \text{ kilometers}\]

Write the Final Result

Thus, the distance light travels in a parsec is approximately \(3.0871 \times 10^{13}\) kilometers.

Key concepts

Velocity of Light

The velocity of light is a fundamental constant of nature. This speed is represented by the letter "c," and it is essential for many calculations in physics. It is the speed at which light travels in a vacuum, which is approximately

• 3.00 x 10⁸ meters per second (m/s)

This constant velocity means that light covers this vast distance every single second. The constant nature of light's speed is foundational for understanding the scale of the universe.
Light speed is not just crucial for tasks like calculating distances in space, but it also plays a role in theories such as Einstein's theory of relativity. Understanding the velocity of light allows us to comprehend how light can travel massive cosmic distances like the ones expressed in parsecs, highlighting how immense distances can be covered in seemingly short cosmic timeframes.

Distance Conversion

Distance conversion is vital in translating measurements into different units. This conversion helps in understanding and comparing the measurements in a format that we find most convenient. In scientific terms, converting between units like meters and kilometers, or years and seconds, is necessary for clear and accurate calculations.
For example, when we talk about the distance light travels in 3.26 years, this time needs to be converted into seconds for calculations using the speed of light. The process involves using necessary conversion factors:

• 1 year = 365 days
• 1 day = 24 hours
• 1 hour = 3600 seconds

By applying these factors, 3.26 years translates into a large number of seconds, specifically 102,902,400 seconds. This number allows us to multiply by the velocity of light to get the distance light covers in those seconds, first in meters and then, after dividing by 1000, in kilometers.
Remembering these conversion steps is crucial when dealing with astronomical units and giant-scale calculations.

Scientific Calculation

Scientific calculation often involves working with very large numbers, especially in astronomy where vast distances need to be expressed succinctly. Calculations such as finding the distance travelled by light in a parsec require precise scientific knowledge and mathematical expression.
Using the formula for distance, which is

• Distance = Velocity × Time

involves high magnitudes of these basic scientific units like time (seconds) and speed (meters per second). For the velocity of light being constant at 3.00 x 10⁸ m/s, timing it by the number of seconds in 3.26 years results in a staggering amount of meters: approximately 3.0871 x 10¹⁶ meters.
For ease of comprehension, converting this massive number of meters into kilometers (since 1 kilometer is 1000 meters) ends up being a more digestible 3.0871 x 10¹³ kilometers. Understanding how to handle and transform these figures is essential for any scientific field dealing with large-scale phenomena.