This video is about orbiting and escaping a planet.
We'll start by discussing orbiting a planet and then we'll move on to escaping a planet.
Let's begin with orbiting.
Imagine that we have a satellite orbiting a planet in a circular orbit.
This is the planet and this is the satellite.
In our discussion here I will use a planet and a satellite, but the concepts that we'll
cover, apply to any situation when an object orbits another object.
Let's assume that the mass of the satellite is lowercase m, the mass of the planet is
uppercase m, and the distance between the satellite and the center of the planet is r.
In subtopic 6.2 when we learned about Newton's law of gravitation, we derived the formula
for orbital speed.
This formula gives us the speed of a satellite that is orbiting a planet in a circular orbit
at constant speed.
Here are the variables that are present in this formula.
Next, let's explore the energy of an orbiting satellite.
E.T. the total energy of an orbiting satellite has two components.
Gravitational potential energy, E.P. plus kinetic energy, E.K.
Earlier in this subtopic we learned that gravitational potential energy is equal to
negative G capital M lowercase m divided by r.
Recall from subtopic 2.3 that kinetic energy is equal to one-half times m v squared.
v in this case is the orbital speed, so as the next step let's substitute the square
root of gm over r, so the formula for orbital speed, in place of v.
This gives us negative G capital M lowercase m over r plus one-half m multiplied by g
capital M over r under the square root to the power of 2.
As we work further, we still have the expression for the gravitational potential energy and
since in the second term the square root and the square cancel each other out, we get one-half
multiplied by g capital M lowercase m divided by r.
From here we can see that the magnitude of the kinetic energy is half of the magnitude
of the gravitational potential energy and that kinetic energy is positive.
While gravitational potential energy is negative.
Adding the two terms we get negative G capital M lowercase m over 2r.
This is the total energy of a satellite that is orbiting a planet in a circular orbit at
constant speed.
The derivation that you see here for the total energy tends to come up on Ib physics exams
in different contexts, so it might be a good idea to fully understand and be able to carry
out this derivation.
Let's add a few important points.
As we already mentioned, the magnitude of the kinetic energy is half of the magnitude
of the gravitational potential energy and kinetic energy is positive while gravitational
potential energy is negative.
Let's see what happens when a satellite moves from one orbit to another.
Here we'll consider moving from a lower to a higher orbit, which means that r, the orbital
radius increases, but of course these concepts also apply when the satellite moves from a
higher to a lower orbit, except then these concepts are reversed.
Consider the orbital speed formula.
When r, so the denominator of the fraction in the formula increases, the value of the
fraction decreases since G and M do not change.
Therefore orbital speed decreases, hence kinetic energy also decreases.
Next, consider the formula for gravitational potential energy, so negative G capital M
lowercase m over r.
When r, the denominator here increases, the value of the entire fraction decreases, however
since gravitational potential energy is the negative of this fraction, gravitational potential
energy becomes less negative, therefore it increases.
Since the magnitude of the gravitational potential energy is just G uppercase m, lowercase m
over r, the magnitude of the gravitational potential energy decreases.
Using a similar reasoning, we can also conclude that the total energy increases while its
magnitude decreases.
Next, we'll learn how to escape from a planet.
One way to define escape speed is that it is the minimum speed an unpowered object must
have in order to escape a planet from the surface of the planet.
An unpowered object simply means that once the object leaves the surface, it will not
gain further kinetic energy.
A simple example is a bullet fired from a gun.
A bullet does not have a propulsion system to further increase its kinetic energy.
On the other hand, a rocket that is burning fuel is able to gain further kinetic energy
after it has left the surface, so such a rocket would not be an unpowered object.
Logically, in order for an unpowered object to escape the surface of a planet, work must
be done on this object.
This work is simply equal to the kinetic energy of the object.
Theoretically, we are trying to bring the object to infinity, where the gravitational
potential energy is zero.
This means that EP gravitational potential energy plus EK kinetic energy must be equal
to zero.
Gravitational potential energy is negative g uppercase m lowercase m over r and kinetic
energy is one half MV squared.
This sum is equal to zero.
We can cancel lowercase m and rearrange for v squared.
This gives us that v squared is equal to 2g capital M over r.
Taking the square root gives us the formula for the escape speed that can be found in
the IP Physics data booklet.
Here are the variables that are present in the formula.
Let's summarize what we have learned in this video.
We began by drawing a diagram of a satellite orbiting a planet and then refreshed the
orbital speed formula that we learned in subtopic 6.2 along with the variables.
Next, we derived the formula for the total energy of a satellite in orbit and mentioned
that this derivation tends to come up on IB Physics exams.
We finished up our discussion about orbits by listing several important points related
to the energy of a satellite in orbit.
Next we defined escape speed and derived the formula for escape speed defining all the
variables in this formula.
This completes our discussion of orbiting and escaping a planet.
In the next video, we'll learn about charges in orbit.