A satellite orbits Earth at an orbital radius of 10000 kilometers. Its orbital period is 2.8 hours. How fast is the satellite moving? Give your answer in kilometers per second to two significant figures.
So to answer this question, we can first start by underlining all the important bits of the question. For example, we’ve been given that the orbital radius of the satellite is 10000 kilometers. We also know that its orbital period is 2.8 hours. And we’re asked to find out how fast the satellite is moving. In other words, we have to find its speed.
We’re asked to give our answer in kilometres per second to two significant figures. So guess what? It’s diagram time! And here it is! We’ve got the Earth at the center of the orbit; the Earth is in blue. And we’ve got the orange satellite orbit. It is meant to be a circle by the way, believe me!
But anyway, we can label something on this diagram, we can label the orbital radius of this rather egg-shaped circle, which happens to be 10000 kilometers. Now in this case, we have been given the orbital radius, which happens to be the distance between the center of the orbit and the orbit itself.
The center of the orbit is the center of the Earth. But sometimes we need to be careful. In some questions, we are given the distance between the surface of the Earth and the orbit. In that case, in order to find the radius of the orbit, we need to know the distance between the surface of the Earth and the center of the Earth as well.
However, in this case we’re okay. We’ve been given the orbital radius. We can also write this down on the side of the diagram in order to be able to assign a symbol to the orbital radius. We can say that 𝑟, the radius, is equal to 10000 kilometers. This way we can also label another one of the quantities given to us in the question, the time period 𝑇, which happens to equal 2.8 hours.
Now as we found out earlier, the question wants us to give our answer in kilometers per second. Now the distance that we’re working with, the radius, is already in kilometers. But the time that we have is in hours. We need to convert this to seconds so that we can give our answer in the correct form. In order to do this, we need to think about how many seconds there are in 2.8 hours.
So 2.8 hours, that’s how much we’ve got. And we know that each hour has 60 minutes in it. So in 2.8 hours, there are 2.8 times 60 minutes. But that’s just how many minutes we have. We need to work out how many seconds we have. Well we know that each minute, every single minute, has 60 seconds in it. And so far we’ve got 2.8 times 60 minutes altogether. So the total number of seconds in 2.8 hours is 2.8 times 60 times 60.
And we can plug that into our calculator, which gives us 10080 seconds. So we can replace our time period with 10080 seconds. Now that we’ve sorted out the units of the quantities given to us in the question. We need to think about the speed of the satellite. We can call this 𝑆, and that’s what we’re being asked to find. Now we can use the definition of speed to our advantage here.
Speed is defined as distance, 𝑑, divided by time, 𝑇. In other words, it’s the distance traveled by something divided by the time taken for that distance to be traveled. In this case, the satellite is travelling in a circular orbit. So when it completes one orbit, it travels a certain distance. And that distance happens to be the circumference of that circle.
That’s this distance here, which we’d already drawn in orange. And now we have it in pink. Anyway! So we can also recall how to calculate the circumference of a circle knowing only its radius. The circumference of a circle, 𝐶, is equal to two 𝜋 multiplied by the radius, 𝑟. Now 𝜋 is just a number, it’s just a constant, so two 𝜋 is also just a constant. And the circumference is the distance that the satellite has traveled when it completes one orbit.
So we can safely say that the circumference, 𝐶, is equal to the distance traveled, 𝑑. And so if we want to calculate the speed, we can say that 𝑆 is equal to the circumference, 𝐶, divided by the time period, 𝑇. This is because it takes the time period 𝑇 to travel exactly once around the circle. We also know that the circumference 𝐶 is equal to two 𝜋 multiplied by the radius 𝑟. So we can substitute that in.
Now all that remains is to plug in our numbers. We can say that 𝑆 is equal to two 𝜋 multiplied by 10000, which is the radius 𝑟, divided by the time period 𝑇 which is 10080. Plugging that into our calculator, we get that 𝑆 is equal to 6.233 dot dot dot. But that’s not our final answer. Firstly, we need to worry about units. And secondly, we need to worry about the number of significant figures.
Well we don’t actually need to worry about the units because we’d already converted all the distances to kilometers and the times to seconds. But we do need to think about putting them in. So if we look back at our calculation for 𝑆, we know that we multiplied two 𝜋 by 10000 kilometers and we divided that by 10080 seconds.
So our final answer is going to be in kilometers per second. That’s exactly how we want it. And we can put that in there. Also we need to worry about the number of significant figures that we’re giving our answer to. The question asked us to give the answer to two significant figures. So here’s the first one, and here’s the second one is.
It’s the one after the second one that will tell us whether the second significant figure needs to be rounded up or if it stays the same. In this case, that value is a three. This is less than five. So the second significant figure stays the same. And so our final answer is that the speed of the satellite is 6.2 kilometers per second to two significant figures.