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