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If you are standing somewhere on Earth... this is the Earth, the mass of the Earth, radius of the Earth, and you're here.

And let's assume for simplicity that there's no atmosphere that could interfere with us, and I want to give you one huge kick, an enormous speed, so that you never, ever come back to Earth, that you escape the gravitational attraction of the Earth.

What should that speed be? Well, when you're standing here and you have that speed, your mechanical energy —

which we often simply call E, the total energy —

is the sum of your kinetic energy —

this is your mass; this is your escape velocity squared —

plus the potential energy, and the potential energy equals minus m Mg divided by the radius of the Earth.

So this is your kinetic energy and this is your potential energy —

always negative, as we discussed before.

Mechanical energy is conserved, because gravity is a conservative force.

So no matter where you are on your way to infinity, if you are at some distance r, that mechanical energy is the same.

And so this should also be one-half m v at a particular location r squared minus m M earth G divided by that little r.

And so at infinity, when you get there —

little r is infinity, this is zero, potential energy at infinity is zero —

and if I get U at infinity with zero kinetic energy, then this term is also zero.

And that's the minimum amount of energy that I would require to get you to infinity and to have you escape the gravitational pull of the Earth.

If I give you a higher speed, well, then, you end up at infinity with a little bit net kinetic energy, so the most efficient way that I can do that is to make this also zero, so you reach infinity at zero speed.

So this is for r goes to infinity.

And so this E equals zero then.

And so this term is the same as this term for your escape velocity.

And so we find that one-half m v escape squared equals m M earth G divided by the radius of the Earth.

I lose my little m, and I find that the escape velocity that I have to give you is the square root of two M earth G divided by the radius of the Earth.

And this is enough, is sufficient to get you all the way to infinity with zero kinetic energy.

If you substitute in here the mass of the Earth and the radius of the Earth, then you will find that this is about 11.2 kilometers per second.

That is the escape velocity that you need.

It's about 25,000 miles per hour.

Again, we assume that there is no air that could interfere with you.

If the total energy when you leave the Earth with that velocity —

if the total energy is larger than zero, you do better than that.

You reach infinity with kinetic energy which is a little larger than zero.

We call this unbound orbit —

larger or equal.

If E is smaller than zero, that the total energy that you have is negative, then you will never escape the gravitational pull of the Earth, and you will be one way or another in what we call a bound orbit.

Let's pursue the idea of circular orbits.

Later in the course we will cover elliptical orbits, but now let's exclusively talk about circular orbits.

Now, this is the mass of the Earth, and in a circular orbit is an object with mass m, a satellite, and m is way, way, way smaller than the mass of the Earth.

And the radius of the orbit is R, and this object has a certain velocity v, tangential speed.

And let's assume for simplicity that there's no atmosphere that could interfere with us, and I want to give you one huge kick, an enormous speed, so that you never, ever come back to Earth, that you escape the gravitational attraction of the Earth.

What should that speed be? Well, when you're standing here and you have that speed, your mechanical energy —

which we often simply call E, the total energy —

is the sum of your kinetic energy —

this is your mass; this is your escape velocity squared —

plus the potential energy, and the potential energy equals minus m Mg divided by the radius of the Earth.

So this is your kinetic energy and this is your potential energy —

always negative, as we discussed before.

Mechanical energy is conserved, because gravity is a conservative force.

So no matter where you are on your way to infinity, if you are at some distance r, that mechanical energy is the same.

And so this should also be one-half m v at a particular location r squared minus m M earth G divided by that little r.

And so at infinity, when you get there —

little r is infinity, this is zero, potential energy at infinity is zero —

and if I get U at infinity with zero kinetic energy, then this term is also zero.

And that's the minimum amount of energy that I would require to get you to infinity and to have you escape the gravitational pull of the Earth.

If I give you a higher speed, well, then, you end up at infinity with a little bit net kinetic energy, so the most efficient way that I can do that is to make this also zero, so you reach infinity at zero speed.

So this is for r goes to infinity.

And so this E equals zero then.

And so this term is the same as this term for your escape velocity.

And so we find that one-half m v escape squared equals m M earth G divided by the radius of the Earth.

I lose my little m, and I find that the escape velocity that I have to give you is the square root of two M earth G divided by the radius of the Earth.

And this is enough, is sufficient to get you all the way to infinity with zero kinetic energy.

If you substitute in here the mass of the Earth and the radius of the Earth, then you will find that this is about 11.2 kilometers per second.

That is the escape velocity that you need.

It's about 25,000 miles per hour.

Again, we assume that there is no air that could interfere with you.

If the total energy when you leave the Earth with that velocity —

if the total energy is larger than zero, you do better than that.

You reach infinity with kinetic energy which is a little larger than zero.

We call this unbound orbit —

larger or equal.

If E is smaller than zero, that the total energy that you have is negative, then you will never escape the gravitational pull of the Earth, and you will be one way or another in what we call a bound orbit.

Let's pursue the idea of circular orbits.

Later in the course we will cover elliptical orbits, but now let's exclusively talk about circular orbits.

Now, this is the mass of the Earth, and in a circular orbit is an object with mass m, a satellite, and m is way, way, way smaller than the mass of the Earth.

And the radius of the orbit is R, and this object has a certain velocity v, tangential speed.

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