Video Transcript
The table below shows data for four
of the moons of Jupiter. 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 Jupiter. 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
Jupiter 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.
Now then, in the table provided, we
are given the values of the orbital radius and orbital period for each moon. So we can calculate the orbital
speed of each moon using this equation and compare their speeds to find out which
moon moves the fastest. However, before we 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 really doesn’t 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 is 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 Ganymede. The orbital radius is given as
1,070,000 kilometers, and the orbital period is given as 7.15 days. Substituting these values into our
equation, we see that the orbital speed of Ganymede is equal to two 𝜋 times
1,070,000 kilometers divided by 7.15 days. Completing this calculation, we
find that the orbital speed of Ganymede is equal to 940,000 kilometers per day to
three significant figures.
Now, let’s calculate the orbital
speed of Europa. Reading off the values from the
table and substituting them into our orbital speed equation, we find that the
orbital speed of Europa is equal to two 𝜋 times 671,000 kilometers divided by 3.55
days, which is equal to 1,190,000 kilometers per day to three significant
figures.
Now, let’s calculate the orbital
speed of Callisto. Reading off the values from the
table and substituting them into our orbital speed equation, we find that the
orbital speed of Callisto is equal to two 𝜋 time to 1,880,000 kilometers divided by
16.7 days, which is equal to 707,000 kilometers per day to three significant
figures.
Finally, let’s calculate the
orbital speed of Io. Reading off the values from the
table and substituting them into our orbital speed equation, we find that the
orbital speed of Io is equal to two 𝜋 times 422,000 kilometers divided by 1.77
days, which is equal to 1,500,000 kilometers per day to three significant
figures.
We have now calculated the orbital
speeds of each of these moons in kilometers per day. And we see that the moon Io has the
fastest orbital speed. Therefore, we have arrived at our
final answer. The moon that moves the fastest
along its orbit is Io.
