Video Transcript
Phobos is the largest moon of Mars. It orbits Mars at a speed of 2.14 kilometers per second. Assuming that the moon follows a circular orbit, what is the radius of its orbit? Use the value of 6.42 times 10 to the 23 kilograms for the mass of Mars and the value of 6.67 times 10 to the negative 11 meters cubed per kilogram second squared for the universal gravitational constant. Give your answer to the nearest kilometer.
Here, we have Phobos, which is in the special case of circular orbit around Mars. So in this question, it will be appropriate to use the orbital speed formula 𝑉 equals the square root of 𝐺𝑀 over 𝑟, where 𝑉 is the moon’s orbital speed. 𝐺 is the universal gravitational constant. 𝑀 is the mass of the large body at the center of orbit, which is Mars in this case. And 𝑟 is the orbital radius of the moon, which we want to solve for.
Now, let’s get organized and write down all of our known values, starting with 𝑉, which we know is 2.14 kilometers per second. And remember, we’ve been asked to give our final answer in units of kilometers. So we could leave orbital speed written in kilometers per second. But it’s generally a good idea and a good habit to convert all values to base SI units before we calculate. So to convert from kilometers per second to meters per second, recall that one kilometer equals 10 to the three meters.
Let’s make this substitution in the numerator. And we have 𝑉 equals 2.14 times 10 to the three meters per second. Moving on, we’ve also been given the value for the universal gravitational constant, 6.67 times 10 to the negative 11 meters cubed per kilogram second squared. And notice that it’s already expressed in base SI units. So it’s good to go. Next, we also know the mass of Mars, 6.42 times 10 to the 23 kilograms, the base SI unit of mass. And of course, we don’t know the orbital radius yet. So we’ll rearrange the orbital speed formula to solve for 𝑟 and calculate its value.
Now, copying the formula down here, let’s square both sides to undo the radical that 𝑟 appears under. Then, we’ll multiply both sides of the equation by 𝑟 over 𝑉 squared to cancel 𝑟 from the right-hand side and 𝑉 squared from the left-hand side. And now, we’re ready to substitute in the values for 𝐺, 𝑀, and 𝑉. Now calculating, this comes out to about 9350467 meters.
But remember, our answers should be in kilometers. So again, recalling that a kilometer is 10 to the three or 1000 meters, we can convert by moving the decimal place of the meter value one, two, three places to the left. So let’s move the decimal place of our orbital radius value three places to the left. And we have 9350.467 kilometers. And finally, giving our answer to the nearest kilometer, we have found that Phobos orbits Mars at a radius of 9350 kilometers.
