A light-year is the distance light travels in one year (at speed = 2.998 × 10⁸ m/s). (a) How many meters are there in 1.00 light-year? (b) An astronomical unit (AU) is the average distance from the sun to the earth, 1.50 × 10⁸km. How many AU are there in 1.00 light-year?

One set of units of a quantity is converted into another by multiplying or dividing the quantity with the conversion factor.

For example, if the length of a piece of cloth is 5.00 inches and you want to measure it in centimeters, you can do this by multiplying the length of the cloth by its conversion factor. The conversion factor to convert inches into centimeters is \(1\,\,{\rm{in}}{\rm{.}} = 2.54\,\,{\rm{cm}}\). Thus, the length of the piece of cloth in centimeters will be \(5.00\,\,{\rm{in}}{\rm{.}} = \left( {5.00\,\,{\rm{in}}{\rm{.}}} \right) \times \left( {2.54\,\,\frac{{{\rm{cm}}}}{{{\rm{in}}{\rm{.}}}}} \right) = 12.7\,\,{\rm{cm}}\)

(a) Given:

Speed of the light =\(2.998 \times {10^8}\,\,{\rm{m/s}}\)

Time taken = 1 year

Here, speed is in m/s, but time is in years. Thus to convert light-years into meters, you need to convert the units of time into seconds.You know that there are 365.25 days in a year, 24 hours in a day, and 3600 seconds in an hour. Thus, you can write one year as follows:

\(\begin{aligned}{c}1\,\,{\rm{y}} = \left( {365.25\,\,{\rm{d}}} \right) \times \left( {24\frac{h}{{\rm{d}}}} \right) \times \left( {3600\frac{{\rm{s}}}{{\rm{h}}}} \right)\\ = 31,557,600\,\,{\rm{s}}\\ = 3.16 \times {10^7}\,\,{\rm{s}}{\rm{.}}\end{aligned}\)

One light-year is defined as the distance traveled by light in one year. Also, speed is the distance traveled per unit of time. Thus, the distance traveled by light can be written as follows:

\(\begin{aligned}{c}{\rm{Distance}} = {\rm{Speed}} \times {\rm{Time}}\\ \Rightarrow 1.00\,\,{\rm{light - year}} = \left( {2.998 \times {{10}^8}\,\,\frac{{\rm{m}}}{{\rm{s}}}} \right) \times \left( {3.16 \times {{10}^7}\,\,{\rm{s}}} \right)\\ = 9.47 \times {10^{15}}\,\,{\rm{m}}\end{aligned}\)

Thus, there are \(9.47 \times {10^{15}}\,\,{\rm{meters}}\)in 1.00 light-year.

(b) Given,\(1\,\,{\rm{AU}} = 1.50 \times {10^8}\,\,{\rm{km}}\)

You know that there are 1000 meters in a km. Thus, 1 AU in meters can be written as follows:

\(\begin{aligned}{c}1\,\,{\rm{AU}} = \left( {1.50 \times {{10}^8}\,\,{\rm{km}}} \right) \times \left( {\frac{{1000\,\,{\rm{m}}}}{{1\,\,{\rm{km}}}}} \right)\\ = 1.50 \times {10^{11}}\,\,{\rm{m}}\end{aligned}\)

So, there are \(1.50 \times {10^{11}}\,\,{\rm{m}}\)in one AU.

Here, you need to convert light-years into AU. Also, you know that there are \(9.47 \times {10^{15}}\,\,{\rm{meters}}\)in 1.00 light-year.Thus, 1 AU can be written as:

\(\begin{aligned}{c}1.00\,\,{\rm{light - year}} = \left( {9.47 \times {{10}^{15}}\,\,{\rm{m}}} \right) \times \left( {\frac{{1\,\,{\rm{AU}}}}{{1.50 \times {{10}^{11}}\,\,{\rm{m}}}}} \right)\\ = 6.31 \times {10^4}\,\,{\rm{AU}}\end{aligned}\)

Thus, there are \(6.31 \times {10^4}\,\,{\rm{AU}}\) in 1.00 light-year.