Video: Converting between Light-Years and Meters

The speed of light is approximately 3 × 10⁸ m/s. There are approximately 3.16 × 10⁷ seconds in 1 year. Use these values to work out how far a beam of light will travel in 1 year. Give your answer to 2 significant figures.

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Video Transcript

The speed of light is approximately three times 10 to the power of eight meters per second. There are approximately 3.16 times 10 to the power of seven seconds in one year. Use these values to work out how far a beam of light will travel in one year. Give your answer to two significant figures.

Okay, so let’s start by underlining all the important stuff given to us in the question. First of all, we’ve been given the speed of light. That happens to be three times 10 to the power of eight meters per second. Secondly, we’ve also been given the number of seconds in one year.

What we’ve been asked to do is to work out how far — that is, we have to work out a distance — how far the beam of light will travel in one year. We also know that we need to give our answer to two significant figures.

So in this question, we’ve been given two quantities: the speed of light first of all, which is normally labelled 𝑐, and that happens to be three times 10 to the power of eight meters per second, and we’ve also been given the number of seconds in one year. So one year is equal to 3.16 times 10 to the power of seven seconds. Now, we need to work out how far a beam of light will travel in one year. We need to work out that distance. And we can call this distance 𝑑.

Now we’ve got a speed, a time, and a distance. So we need to find a relationship that links the three together. Well, what’s the definition of the speed of light? Well, the speed of light or the speed of anything really is defined as the distance covered divided by the time taken to cover that distance.

And we can write that in symbols as 𝑐 is equal to 𝑑 over 𝑡, where 𝑐 is the speed of light as we’ve already seen, 𝑑 is the distance travelled as we’ve also already seen, and 𝑡 is the time taken 𝑡 for the light to cover that distance.

We’re being asked to calculate the distance, 𝑑, here. So we need to rearrange the equation. What we can do is to multiply both sides of the equation by 𝑡. And what that results in is the cancellation of the 𝑡s on the right- hand side, which leaves us with 𝑡𝑐 is equal to 𝑑.

The time, 𝑡, multiplied by the speed of light, 𝑐, is equal to the distance, 𝑑. Now the time that’s taken is one year, which also happens to be 3.16 times 10 to the power of seven seconds.

The speed of light, which we’ve already been given as well, is three times 10 to the power of eight meters per second. And this is equal to 𝑑. So plugging that into our calculator, we get 9.48 times 10 to the power of 15.

However, that’s not our final answer. There are two things we need to worry about: firstly, the unit; and secondly, the fact that we’ve been asked to give our answer to two significant figures. Let’s start with the units. Those are pretty simple.

The time that we were given was in seconds and the speed was in meters per second. So when we multiply them together, the seconds cancel out and we’re just left with meters. So we can leave a blank for our answer, but we know that the final answer is going to be in the unit of meters.

Now we need to find that value to two significant figures. 9.48 to two significant figures basically turns out to be the following: the first significant figure is nine and the second is four. It’s the third one, the one after the second one, that will tell us whether the second rounds up or stays the same.

In this case, that third value is eight, which is larger than five. Therefore, the second significant figure will round up. And so we get our final answer, which is that the distance travelled by light in one year is 9.5 times 10 to the power of 15 meters, to two significant figures. Coincidentally, this distance that light travels in one year is also known as a light-year. In other words, one light-year is equal to 9.48 times 10 to the power of 15 meters.

But what can be often confusing is the fact that a unit of distance is called a light- year. What-what’s all that about? That’s really confusing. So here’s a slightly simpler way to think about it. It’s not just a “light” year it’s actually a “light-year”. In other words, a better way to think about this is lightspeed year.

This is a much easier way of remembering light-year and why that’s a unit of distance because we’ve already seen that light speed labelled 𝑐 and time 𝑡 when multiplied together give us a unit of distance.

This is what we did to calculate the answer to our question. So if you think of a light-year as a lightspeed year, you’ll remember that light is in fact talking about the speed of light and the year is talking about the time, 𝑡. So multiplying them together gives us a distance. Therefore, a light-year is a unit of distance.

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