Video Transcript
The table below shows data for four
of the moons of Saturn. Which moon moves fastest along its
orbit? Assume that all four moons have
circular orbits.
In this question, we are shown data
for four moons of Saturn and we are asked to determine which moon moves fastest
along its orbit. To do this, we want to calculate
the orbital speed of each moon. We can assume that all four moons
have circular orbits. So all four moons will orbit Saturn
like in this diagram. Recall that for circular orbits, we
have the equation 𝑆 equals two 𝜋𝑟 over 𝑇, where 𝑆 is the orbital speed, 𝑟 is
the radius of the orbital path, and 𝑇 is the orbital period. This is simply the formula speed
equals distance over time for a circular orbit.
The total distance traveled on a
single revolution of circular orbit is the circumference of the orbit, which is
equal to two 𝜋𝑟, and period 𝑇 is the time taken for one orbit. Note that the orbital speed 𝑆 is
the same at all points around the orbit.
In the table provided, we are given
the values of the orbital radius and the orbital period for each moon. So we can calculate the orbital
speed using this equation and compare their speeds to find out which moon moves the
fastest. However, before we can substitute
these values into the equation for orbital speed, we should take note of the units
being used. The orbital radius of each moon is
given in kilometers, and the orbital period is given in days. This means that if we substitute
these values into the equation, we will get units of kilometers per day as our units
for orbital speed. These are not SI units. And usually, we would convert these
into SI units to make sure all units are consistent with the formula that we are
using.
For this question though, we are
comparing the speeds of each moon and determining which moves the fastest along its
orbit. So it doesn’t really matter which
units we use. Kilometers per day is a perfectly
valid unit of speed. And we can still compare the speeds
as we would if the units were meters per second. So it’s fine to keep the units as
they are, as long as we use the same units for each moon. With this in mind, we can now go
ahead and calculate the orbital speed for each moon.
Let’s begin with Titan. The orbital radius is given as
1220000 kilometers and the orbital period as 15.9 days. Substituting these values into our
equation, we see that the orbital speed of Titan is equal to two 𝜋 times 1220000
kilometers divided by 15.9 days. Completing this calculation, we
find that the orbital speed of Titan is equal to 482000 kilometers per day to three
significant figures.
Now let’s calculate the orbital
speed of Rhea. Reading off the values from the
table and substituting them into our orbital speed equation, we find that the
orbital speed of Rhea is equal to two 𝜋 times 527000 kilometers divided by 4.52
days, which is equal to 733000 kilometers per day to three significant figures.
Now let’s calculate the orbital
speed of Iapetus. Reading off the values from the
table and substituting them into our orbital speed equation, we find that the
orbital speed of Iapetus is equal to two 𝜋 times 3560000 kilometers divided by 79.3
days, which is equal to 282000 kilometers per day to three significant figures.
Now let’s calculate the orbital
speed of Dione. Reading off the values from the
table and substituting them into our orbital speed equation, we find that the
orbital speed of Dione is equal to two 𝜋 times 377000 kilometers divided by 2.74
days, which is equal to 865000 kilometers per day to three significant figures.
We have now calculated the orbital
speeds of each of the moons in kilometers per day, and we can see that Dione has the
fastest orbital speed. Therefore, we have arrived at our
final answer. The moon that moves the fastest
along its orbit is Dione.
