Question

Explain why the light-year has the dimension of length.

Short answer

Answer: The light-year has the dimension of length because it represents the distance light travels in one year. With a constant speed of light, multiplying it by the time taken (seconds in a year) results in a distance value in meters, confirming that a light-year represents the dimension of length.

Step-by-step solution

Define a light-year

A light-year is a unit of distance used in astronomy that represents the distance light travels in one year. It's calculated by multiplying the speed of light by the time it takes to travel in one year.

Identify the speed of light

The speed of light in a vacuum is approximately \(3.0 × 10^8\, meters/second\) (m/s). This is a constant value and can be represented by the letter \(c\).

Determine the time in seconds for one year

To calculate the distance of one light-year, we need to know the number of seconds in a year. In a non-leap year, there are \(365\, days\), each with \(24\, hours\), \(60\, minutes\), and \(60\, seconds\). Thus, the number of seconds in a year is: \(365\, days \times 24\, hours/day \times 60\, minutes/hour \times 60\, seconds/minute = 31,536,000\, seconds\)

Calculate the distance of one light-year

Now that we know the speed of light and the number of seconds in a year, we can calculate the distance of one light-year. Multiply the speed of light by the number of seconds in a year: \(3.0 × 10^8\, m/s \times 31,536,000\, s = 9.46 × 10^{15}\, meters\)

Demonstrate that a light-year has dimensions of length

A light-year represents a distance – specifically, the distance light travels in one year. As we have shown in our calculation, the value of one light-year is approximately \(9.46 × 10^{15}\, meters\). Since distance or length is typically measured in units such as meters, we can conclude that a light-year indeed has the dimension of length.