Video Transcript
Europa is a moon of Jupiter that
has a mass of 4.80 times 10 to the power of 22 kilograms and a radius of 1,560
kilometers. At what speed would an object have
to be launched vertically off the surface of Europa in order to escape its
gravity? Give your answer to three
significant figures.
So, in this question, we’re
thinking about an object on the surface of one of Jupiter’s moons, known as
Europa. And we’re being asked to find a
speed, let’s call this 𝑣, at which we could launch this object vertically off the
surface of Europa that would enable it to escape its gravity. Let’s start by thinking about
exactly what it means for this object to escape Europa’s gravity.
Well, our object on the surface of
Europa, or indeed any object on the surface of any planet, has a certain
gravitational potential energy 𝐸 p, which is given by this equation, where 𝐺 is
the gravitational constant, big 𝑀 is the mass of the planet or moon in question —
in this case, it would be Europa’s mass — little 𝑚 is the mass of the object, and
𝑟 is the distance between the centers of mass of the two masses that we’re
considering. So, in this case, it would be the
distance between the center of mass of Europa and the center of mass of our
object.
Now, we can see that this potential
energy is negative. In physics, we use negative
potential energies to represent when something is bound to something else. For example, in this question, our
object is bound to Europa by a gravitational force. The fact that this is negative
represents the fact that we would need to give the object energy in order to free
it. Effectively, the magnitude of all
this is the amount of kinetic energy we would need to give our object in order for
it to be freed from the gravitational pull of Europa. So, we can say that our object will
escape the gravitational pull of Europa when its kinetic energy, given by a half
𝑚𝑣 squared, is equal to 𝐺 times big 𝑀 times little 𝑚 divided by 𝑟.
Now, if we rearrange this
expression to make 𝑣 the subject, we obtain this expression. This tells us the speed at which an
object must be launched in order to escape the gravity of some other mass,
represented by big 𝑀. The speed is known as the escape
speed. And it’s often denoted as 𝑣 sub
e. We can use this equation directly
to work out the escape speed of our object on Europa. In this case, the value of big 𝑀
is the mass of Europa, which we’re told is 4.80 times 10 to the power of 22
kilograms. And 𝑟 in this equation is the
distance between the center of mass of our object and the center of mass of
Europa.
Now, we haven’t been given this
quantity exactly. However, we have been told the
radius of Europa is 1,560 kilometers. Because planets and moons are
pretty much spherical, this means that center of mass is located in the middle. And as long as our object isn’t too
big, its center of mass will be very close to the surface of Europa. This means that we can use the
radius of Europa as the distance between our two centers of mass. And since this value has been given
in kilometers, we’ll multiply it by 1,000 to give us the value in meters.
Finally, the other value that we
need in our equation is 𝐺, the gravitational constant. This has a value of 6.67 times 10
to the power of negative 11 meters cubed per kilogram per second squared. Substituting these three values
into our equation gives us this expression. And if we enter all of this into
our calculator, we obtain a value of 2,025.99 and so on meters per second, which to
three significant figures is 2,030 meters per second. This is the speed at which an
object with any mass would have to be launched off the surface of Europa in order to
escape its gravity. In other words, it’s the escape
speed.
