Orbits: Elliptical vs. Parabolic

[Math Is Hard]

My Calculus textbook (Calculus , James Stewart, 5th edition, p. 720) says this: "In the 16th century Galileo showed that the path of a projectile that is shot into the air at an angle to the ground is a parabola."
My astronomy teacher says that this is not technically correct, and that the path of the projectile is actually a partial ellipse - a "bound orbit".
Which do you believe is correct? With all due respect to Mr. Stewart, (and Galileo) my money is on my Astro prof.

Thank you in advance for your response.

 

[Labguy]

Originally posted by Math Is Hard
My Calculus textbook (Calculus , James Stewart, 5th edition, p. 720) says this: "In the 16th century Galileo showed that the path of a projectile that is shot into the air at an angle to the ground is a parabola."
My astronomy teacher says that this is not technically correct, and that the path of the projectile is actually a partial ellipse - a "bound orbit".
Which do you believe is correct? With all due respect to Mr. Stewart, (and Galileo) my money is on my Astro prof.

Thank you in advance for your response.

The path of a projectile (bullet, rock, canon shell, etc.) is a partial ellipse. This was re-confirmed/calculated by Wiley Ley and Werner Van Braun during WWII when Germany's long "Paris guns" were missing the target. The trajectory had been calculated using a parabolic path, and the two rocket scientists had them change it to an elliptical path. They hit the target.

For short-range ballistics, like my rifles at the shooting range, or for throwing stones at streetlights, the curves of a parabola and ellipse are nearly indistinguishable (sp?). The difference shows up at larger ranges, bit it is an ellipse.

 

[selfAdjoint]

This was re-confirmed/calculated by Wiley Ley and Werner Van Braun during WWII when Germany's long "Paris guns" were missing the target.

Umm, during WWII Ley was in the USA and von Braun was in Germany. And are you sure the "Paris Gun" was a WWII weapon? I am aware of a gun with that name used in WWI, but both Ley and von Braun would have been little kids then.

Agree with you about the ellipse. The parabola is what you get if the Earth is assumed to be flat (i.e. the other focus of the ellipe recedes to infinity).

 

[Nereid]

And let's not forget air resistance (which is a point being strongly made in another thread (General Physics>Simple question about gravity and force), and the rotation of the Earth, and ...

All of which are, IIRC, neatly packaged into the computers which calculate the 'firing solution' for guns on warships, tanks, etc. How do they establish the range? lasers? GPS?

 

[Labguy]

Originally posted by selfAdjoint
Umm, during WWII Ley was in the USA and von Braun was in Germany. And are you sure the "Paris Gun" was a WWII weapon? I am aware of a gun with that name used in WWI, but both Ley and von Braun would have been little kids then.

Agree with you about the ellipse. The parabola is what you get if the Earth is assumed to be flat (i.e. the other focus of the ellipe recedes to infinity).

Ley and Von Braun co-wrote a book in the late "40's - early "50's which described the "parabola vs. ellipse" problem. I'm not sure where Ley was, but von Braun was in Germany for sure! (During WWII).

The first, and famous "Paris gun" was that 70-80 foot barreled thing in WWI that lobbed shells about 65-70 miles. But, in WWII, Germany had several huge guns on railroad cars, with crews of about 70 troops each, that were also lobbing shells at long distance. The name "Paris gun(s)" were also given to some of there early-on during the battle for France. Just as an interesting note, each shot wore a huge amount out of the barrel-bore, so each shell was numbered with each fired being larger in diameter than each previous shell. After several shots, they had to install a new barrel and start over.

This large concentration of material, manpower and other recources for so little "military gain" was the subject of a History Channel program titled "Great Blunders in History." The History Channel is where I get most of the information I post on PF...

But, I did own and read the book by Ley and von Braun a long time ago.

 

[russ_watters]

I was under the impression that a parabola could be thought of as a special case of an ellipse - with an infinite distance between the foci. A circle is a special case of an ellipse with zero distance between the foci.

 

[Labguy]

Originally posted by russ_watters
I was under the impression that a parabola could be thought of as a special case of an ellipse - with an infinite distance between the foci. A circle is a special case of an ellipse with zero distance between the foci.

Never heard of that. We are just talking the shapes (and math) of conic sections; circle, ellipse, parabola and hyperbola. The last two end up with diverging or parallel "edges". Only the ellipse, of the four, has two focii. (Focuses..?)...

It is true, especially at shorter ranges, that the curves of the parabola, ellipse and hyperbola are so close to each other as to be almost indistinguishable. The circle is the obvious loser.

 

[selfAdjoint]

Originally posted by Labguy
Never heard of that. We are just talking the shapes (and math) of conic sections; circle, ellipse, parabola and hyperbola. The last two end up with diverging or parallel "edges". Only the ellipse, of the four, has two focii. (Focuses..?)...

It is true, especially at shorter ranges, that the curves of the parabola, ellipse and hyperbola are so close to each other as to be almost indistinguishable. The circle is the obvious loser.

This is really sad. They don't teach basic conic sections any more? What are the relations of the simplest ellipse equation x²/a² + y²/b² = 1 to the foci of the ellipse? What if one focus is moved down the x axis? How does the equation change? What is the limit of this as x -> 00?

 

[Labguy]

Originally posted by selfAdjoint
This is really sad. They don't teach basic conic sections any more? What are the relations of the simplest ellipse equation x²/a² + y²/b² = 1 to the foci of the ellipse? What if one focus is moved down the x axis? How does the equation change? What is the limit of this as x -> 00?

Just curious, what's the "sad" part"?? That this is here on PF, or that they don't teach the basics anymore?

Labguy
(Ellipse-Head)

 

[selfAdjoint]

Not teaching the basics. Everyone is in such a hurry to teach calculus, advanced calculus, etc. that they don't stop to smell the "roses" of analytical geometry.

 

[Janus]

Originally posted by Math Is Hard
My Calculus textbook (Calculus , James Stewart, 5th edition, p. 720) says this: "In the 16th century Galileo showed that the path of a projectile that is shot into the air at an angle to the ground is a parabola."
My astronomy teacher says that this is not technically correct, and that the path of the projectile is actually a partial ellipse - a "bound orbit".
Which do you believe is correct? With all due respect to Mr. Stewart, (and Galileo) my money is on my Astro prof.

Thank you in advance for your response.

The crux of this argument lies in how you consider gravity to behave. If you expect it to act in one uniform direction (the direction of force always in parallel), then you will get a parabolic trajectory.

If you consider as acting towards a central point then you will a trajectory that is a section of an ellipse.

Of course gravity acts in the second manner so we get an ellipse.

But over small enough distances over the Earth's surface, the convergence of gravity is so small that for all intents and purposes we can treat it as acting in parallel, and we can treat the path as a parabola, depending on the degree of accuracy we need.

So I'm going to assume that it is the simplified case of parallel acting force which Mr. Stewart is talking about.

Of course, this also assumes that the velocity of the object is below escape velocity.

If it is traveling exactly at escape velocity then it actually will follow a parabolic path. At greater than escape velocity it will follow a hyperbolic path.

 

[Jeebus]

Originally posted by Janus
If it is traveling exactly at escape velocity then it actually will follow a parabolic path. At greater than escape velocity it will follow a hyperbolic path.

That is very true.

 

[russ_watters]

Originally posted by Labguy
Never heard of that. We are just talking the shapes (and math) of conic sections; circle, ellipse, parabola and hyperbola. The last two end up with diverging or parallel "edges". Only the ellipse, of the four, has two focii. (Focuses..?)...

It is true, especially at shorter ranges, that the curves of the parabola, ellipse and hyperbola are so close to each other as to be almost indistinguishable. The circle is the obvious loser.

You can mathematically derive the relationship (its actually not that hard). I'm positive about the relationship between circles and ellipses, just not so sure about ellipses and parabolas.

This is really sad. They don't teach basic conic sections any more?

Ah yes, that's the easiest way to look at the relationship between them. They do teach it (at least they did 15 years ago), I just forgot about it.