Question Video: Finding the Orbital Velocity from the Radius and Period for Circular Orbits Physics • 9th Grade

Titan is the largest moon of Saturn. It has a roughly circular orbit, and it orbits Saturn at a distance of 1220000 km with a period of 15.9 days. At what speed is Titan moving along its orbit? Give your answer to the nearest kilometer per second.

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Video Transcript

Titan is the largest moon of Saturn. It has a roughly circular orbit, and it orbits Saturn at a distance of 1220000 kilometers with a period of 15.9 days. At what speed is Titan moving along its orbit? Give your answer to the nearest kilometer per second.

Okay, so this question is about Saturn’s largest moon, Titan. And we’re told that it has a roughly circular orbit around the planet. Let’s suppose that this pink circle in the middle is Saturn and this here is Titan. This orange circle shows the circular orbit that Titan moves along. We are told that Titan orbits Saturn at a distance of 1220000 kilometers. This distance is the radius of the orbit, which we’ve marked on our diagram and labeled as 𝑟. We’re also told that the period of the orbit is 15.9 days. We’ll label this period as capital 𝑇. And this defines the time taken to complete one full orbit.

We are asked to work out the speed at which Titan is moving. We can recall that the speed of an object is defined as the distance moved by that object divided by the time taken to move that distance. In this case, we know that the period of the orbit is 15.9 days and that this period defines the time taken to travel the distance corresponding to one full orbit. During one full orbit, Titan goes once around this circle. This means that the distance traveled in a time of one period, capital 𝑇, is equal to the circumference of the circle, which we’ll label as capital 𝐶.

So, if we label the speed that Titan moves along its orbit as 𝑣, then we have 𝑣 is equal to the circumference 𝐶 divided by the period 𝑇. The question hasn’t told us what the circumference is, but it has told us the radius of the orbit. We can recall that the circumference 𝐶 of a circle is equal to two 𝜋 multiplied by the circle’s radius 𝑟. So we can calculate the circumference of Titan’s orbit by subbing in this value for the radius into this equation. Once we’ve subbed in the value of 𝑟, we can then evaluate this expression to calculate the circumference 𝐶 as 7665486.07 kilometers, where the ellipses indicate that there are further decimal places.

We now have values for both the circumference 𝐶 and the period 𝑇 of the orbit. So we could go ahead and sub those into this equation to calculate the speed 𝑣. However, we can first make our lives a little bit easier. Looking back at the question text again, we see that we are asked to give our answer to the nearest kilometer per second. In order to calculate a speed with units of kilometers per second, we need a distance in units of kilometers and a time in units of seconds. The circumference that we’ve calculated has units of kilometers, which is good. However, the period that we’ve got has units of days. So, before we sub these values into the equation for speed, we should convert the period into units of seconds.

To do this, we can note that there are 24 hours in one day, there are 60 minutes in one hour, and there are 60 seconds in one minute. So, to convert a time from units of days into units of seconds, we take the value in units of days and we multiply this by 24 hours per day, then again by 60 minutes per hour, and finally by 60 seconds per minute. Let’s have a look what the units are doing in this expression. The units of days cancel with the per day. The units of hours cancel with the per hour. And the units of minutes cancel with the per minute. This leaves us with units of seconds. When we evaluate the expression, we find that the period 𝑇 is equal to 1373760 seconds.

So we now have a circumference in units of kilometers and a period in units of seconds. This means that we’re ready to sub the values into this equation to calculate Titan’s speed. Doing this gives us this expression here for the speed 𝑣. Evaluating the expression gives a speed of 5.5799 kilometers per second, where the ellipses show that there are further decimal places. The question asks for our answer to the nearest kilometer per second. To this level of precision, the result rounds up. And our answer to the question is that the speed that Titan is moving along its orbit is six kilometers per second.

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