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 the 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 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 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 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 Himalia. The orbital radius is given as
11500000 kilometers, and the orbital period is given as 252 days. Substituting these values into our
equation, we see that the orbital speed of Himalia is equal to two 𝜋 times 11500000
kilometers divided by 252 days. Completing this calculation, we
find that the orbital speed of Himalia is equal to 287000 kilometers per day to
three significant figures.
Now let’s calculate the orbital
speed of Elara. Reading off the values from the
table and substituting them into our orbital speed equation, we find that the
orbital speed of Elara is equal to two 𝜋 times 11700000 kilometers divided by 258
days, which is equal to 285000 kilometers per day to three significant figures.
Now let’s calculate the orbital
speed of Lysithea. Reading off the values from the
table and substituting them into our orbital speed equation, we find that the
orbital speed of Lysithea is equal to two 𝜋 times 11600000 kilometers divided by
256 days, which is equal to 285000 kilometers per day to three significant
figures.
Now let’s calculate the orbital
speed of Leda. Reading off the values from the
table and substituting them into our orbital speed equation, we find that the
orbital speed of Leda is equal to two 𝜋 times 11200000 kilometers divided by 242
days, which is equal to 291000 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 Leda has
the fastest orbital speed. Therefore, we have arrived at our
final answer. The moon that moves the fastest
along its orbit is Leda.
