A light-year is a unit of distance which is used to express large distances in the universe. It is the distance which light travels in a vacuum in one year. How many metres are there in a light-year, given that the speed of light in a vacuum is 2.98 times 10 to the eight metres per second and a year on Earth is 365.25 days? Give your answer to one decimal place. (A) 1.1 times 10 to the 11 metres. (B) 9.4 times 10 to the 15 metres. (C) 3.9 times 10 to the 14 metres. (D) 1.6 times 10 to the 14 metres.
This question is asking us to work out how many metres there are in a light-year. We’re told that a light-year is a unit of distance and that it’s equal to the distance that light travels in a vacuum in a time of one year. We’re also told the value of the speed of light in a vacuum and the number of days in a year on Earth. So we are given a value of speed, the speed of light, and we’ll label this value 𝑠. So we have that 𝑠 is equal to 2.98 times 10 to the eight metres per second. We are also given a length of time, one year, which we’ll label 𝑡. So we can say 𝑡 is equal to one year.
We know that in a vacuum light travels at this speed 𝑠 and that if it travels for this length of time 𝑡, then the distance that it covers in that time is equal to one light-year. But we want to know how far that distance is in units of metres. We can recall that we have a formula which relates the three quantities speed, distance, and time. For a speed 𝑠, a distance 𝑑, and a time 𝑡, we have that 𝑠 is equal to 𝑑 divided by 𝑡. Since in this question we’re interested in calculating a distance, we should rearrange this formula to make 𝑑 the subject.
If we multiply both sides of the equation by 𝑡, then on the right-hand side, the 𝑡’s in the numerator and denominator cancel each other out. Then, if we swap the left- and right-hand sides of this equation over, we have that distance 𝑑 is equal to speed 𝑠 multiplied by time 𝑡.
There are two steps that we need to take in order to get to our answer. The first step is to make the units of our quantities agree with each other. We have a speed measured in units of metres per second. And we’re looking for a distance measured in metres. But our time is given in units of years. So our first step needs to be converting this time into units of seconds so that the units agree with the other quantities in our formula. Then, our second step is going to be to take our speed 𝑠 in units of metres per second and our time 𝑡 in units of seconds and substitute them into this formula to calculate our value of distance 𝑑 in units of metres.
Let’s make a start on step one. Our value of time 𝑡 is equal to one year, and we want to work out what this is in units of seconds. We are told in the question that there are 365.25 days in one year. So in units of days, we have that 𝑡 is equal to one year multiplied by 365.25 days per year. So we can say that 𝑡 is equal to 365.25 days. Now, there are 24 hours in a day. So if we multiply 365.25 days by 24 hours per day, then we get our time 𝑡 in units of hours. And this value is 8,766 hours.
Then, in each hour, there are 60 minutes. And in each minute, there are 60 seconds. So to get our value of the time 𝑡 in units of seconds, we need to take our time measured in hours and multiply this by 60 minutes per hour and then multiply again by 60 seconds per minute. When we do this multiplication, we get that 𝑡 is equal to 3.16 times 10 to the seven seconds. And this gives us the number of seconds in one year. Notice that since our result is quite a big number, we’ve switched to using scientific notation. We have also written this value of 𝑡 rounded to three significant figures.
It’s worth taking a moment to check out the units throughout this calculation. We began with units of years. Then we multiplied by a quantity with units of days per year so that the years canceled with the per year and we were left with units of days. Then we took this value with units of days and multiplied by a quantity with units of hours per day so that the days canceled with the per day and we were left with a quantity in units of hours. Then in our final step, we took our value of time in units of hours. And we multiplied by a quantity with units of minutes per hour and then by another quantity with units of seconds per minute. Then the hours cancel with the per hour, and the minutes cancel with the per minute. And that’s how we end up with units of seconds.
Now that we have found our value of the time 𝑡 measured in seconds, then the units of all of our quantities agree with each other. And so we’ve achieved our first step. Now it’s time for step two.
We need to take our value for 𝑠 in units of metres per second and our value for 𝑡 in units of seconds and substitute them into this formula to calculate our value of the distance 𝑑 in units of metres. When we do this, we get that 𝑑 is equal to our value of 𝑠, 2.98 times 10 to the eight metres per second, multiplied by our value of 𝑡, 3.16 times 10 to the seven seconds. When we do this multiplication, we get a result of 9.4 times 10 to the 15 metres, where we have rounded this result to one decimal place, like the question asked.
And this value of 𝑑 gives us the distance that light which travels at a speed of 2.98 times 10 to the eight metres per second travels in a time of one year. In other words, this is the number of metres in one light-year, which is what the question was asking us to find. So we have now also achieved the second step in our calculation and substituted in the values to get a result for 𝑑.
If we compare this result to the four possible values presented to us in the question, we see that it agrees with the value given here in option (B). So we have our answer to the question that the number of metres in one light-year is given here in option (B) as 9.4 times 10 to the 15 metres.