The Four Kinds of Trajectories for Celestial Objects

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  • 0:01 Hyperboles and Hyperbolas
  • 0:25 Conic Sections and…
  • 0:53 Circles and Ellipses
  • 2:12 Parabolas and Hyperbolas
  • 3:14 Lesson Summay
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Lesson Transcript
Instructor: Artem Cheprasov
What does the slicing apart of a cone and the resultant shapes have to do with trajectories and escape velocity? This lesson will tell you as it goes over ellipses, circles, hyperbolas, and parabolas.

Hyperboles and Hyperbolas

Although I've told my doctor a million times that I'm so hungry I could eat a horse and I eat a ton as a result, he still tells me I'm as skinny as a pencil. So skinny, that you can knock me over with a feather.

Wait just a minute, this lesson isn't about hyperboles! It's about hyperbolas and other orbital shapes. My boss is going to kill me for this.

Conic Sections and Trajectories

We'll get to defining a hyperbola in just a second, but first we need to define conic sections, which are the four types of curves that are obtained by slicing sections of a cone. These curves represent four kinds of trajectories, or paths taken by objects in space when under the influences of forces such as gravity.

The four conic sections obtained when a cone is sliced with a plane are:

  • A circle
  • An ellipse
  • A parabola
  • The hyperbola

Circles and Ellipses

You've probably heard that the planets circle the sun. A circle is a kind of curve on which any point is equidistant from the center. The circle, looks like a, well, circle, when the cone is cut by a plane as shown on your screen:

cone cut by plane to create circle

Although you may have heard that the planets circle the sun, more technically they actually 'ellipse' the sun. And while ellipse may not be a verb, it is now.

More formally, an ellipse is a noun that's defined as a conic section obtained when a right circular cone is cut by a plane that intersects the cone's axis and surface to produce a closed elongated curve. A picture speaks louder than words in this case, so you can see what I mean by the image on your screen:

cone cut by plane to produce ellipse

So, what does this all have to do with astronomy? It all has to do with escape velocity. Escape velocity is the minimum velocity a body needs to escape the gravitational attraction of a more massive body. If a body in space is moving slower than escape velocity, it follows a circular or elliptical path. Examples of this include the planetary orbits about the sun, which are almost circular, but technically elliptical.

That sort of helps to explain why we don't fly off into space, doesn't it? If Earth moved faster than escape velocity, not only would we be nowhere near the sun right now, we'd also move in a different path.

Parabolas and Hyperbolas

The path Earth would take to escape the solar system would have to be parabolic or hyperbolic.

A parabola is a conic section obtained when a right circular cone is cut by a plane parallel to the cone's side in order to produce an open curve. This means that it doesn't form a closed shape like that of an ellipse or circle. A body following a parabolic path moves at escape velocity. Examples of this include some comets that pass the sun but leave the solar system forever thereafter.

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