Question

The speed of light in a vacuum is exactly 299,792,458 m/s. This speed is sometimes used to provide a convenient yardstick for large astronomical distances.(c) A light-year is the distance light travels in one year. The Sun is 28,000 light-years from the center of the Milky Way galaxy. What is this distance in meters?

Answer 1

`\\2.647191386*10^(20)m`

Step-by-step explanation:

To answer this question it is necessary to understand that Light travels in vacuum at a constant speed, which means an important property to estimate distances in Universe. We already know that speed of light in vacuum is exactly 299,792,458 m/s.

To solve this question, we need to answer these two ones:

1. What is the distance that light travels in one year?
2. How many seconds are there in a year?

In the first question, we need to know the distance that light travels in one year, which give us some clue to respond the distance that light travels in 28,000 light-years.

The answer to the second question it is crucial to answer the first question in meters.

How many seconds are there in a year?

`\\ 1year = 365 days * (24h)/(day)*(60min)/(h)*(60s)/(min)=31,536,000s`.

That is, there are 31,536,000 seconds in a year.

What is the distance that light travels in one year?

Because light travels at a constant speed, the distance can be calculated as follows:

`\\ Speed = (distance)/(time)`

or, equivalently,

`\\ distance = Speed * time`

So,

`\\ distance = 299792458(m)/(s) * 31536000s`

But, we can see that these are big numbers, and a better way to deal with this is to use scientific notation or floating-point numbers.

Then,

`\\ 299792458(m)/(s) = 2.99792458 * 10^(8)(m)/(s)`

and,

`\\ 31536000s = 3.1536000*10^(7)s`

So,

`\\ 2.99792458 * 3.1536000 = 9.45425495`, and,

`\\ 10^(8)*10^(7) = 10^(8 + 7) = 10^(15)`

Then, the total distance that light travels in a year is:

`\\ 9.45425495 * 10^(15)m`

But we now that The Sun is 28,000 light-year from the center of the Milky Way galaxy, and that 28000light-year=
`\\2.8 *10^(4)` light-year.

So, the distance in meters of The Sun from the center of the Milky Way galaxy is:

`\\9.45425495 * 10^(15)m * 2.8 *10^(4)light-year =26.47191386*10^(15)*10^(4)=26.47191386*10^(15+4)=26.47191386*10^(19)=2.647191386*10^(20)m`

Then, the answer is:

`\\2.647191386*10^(20)m`.

Likewise, the answer could be found as a matter of proportions, mostly because light travels the same distance at each time:

If light travels
`\\ 9.45425495 * 10^(15)m` in a light-year, how many meters does light travel in 28,000 light-year?

`(9.45425495* 10^(15)m)/(light-year)` =
`(X meters)/(2.8*10^(4)light -year)`. or

`\\ X =(9.45425495* 10^(15)m)/(light-year) * {2.8*10^(4)light -year}=2.647191386*10^(20)m`

That is, the same result.

Notice that we calculate the result using 1 year = 365days. We can add more precision to our answer if we consider 1 year = 365,25days, following the same steps.