# Is a free ballistic path a parabola or a cycloid?

1. May 24, 2005

### Zanket

Feynman says that the path of projectile in a uniform gravitational field is a parabola, but the bottom of http://online.redwoods.cc.ca.us/instruct/darnold/CalcProj/Fall98/NateB/definition.htm [Broken] says it's a cycloid. My calculation shows a parabola. Which is correct?

If parabola, why does a hypothetical Big Bang to Big Crunch scenario (a closed Friedmann model) plot, for the distance over time for a pair of galaxies, a cycloid? Is that because in GR in this scenario the galaxies are in free fall in signficiant expanding space (the expanding space paradigm) whereas for the projectile the expanding space is negligible?

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2. May 24, 2005

### dextercioby

In Newtonian physics is neither of them.It's an arch of en ellipse,with the center of the earth in the closest focus.

Daniel.

3. May 24, 2005

### Zanket

An arc of an ellipse is a parabola, no?

4. May 24, 2005

### dextercioby

Nope,an arch of an ellipse is just an arch of an ellipse (pardon the tautology).

Daniel.

5. May 24, 2005

### Zanket

You mean "arc"? What is the difference between arc and arch as you're using it? An arc of an ellipse looks like a parabola to me. How is it different? Feynman didn't know what he was talking about?

6. May 24, 2005

### dextercioby

I dunno what he was talking about.Yes,"arc",if you prefer.

Give me a reference on Feynman to check him out.

Daniel.

7. May 24, 2005

### EL

Yes.

By uniform gravitational field he means a field with the same amplitude and direction at all points.

The Earth's gravitational field is directed towards its centre (i.e. different directions at different points), and is getting weaker further away.

8. May 24, 2005

### Garth

That link states "The "altitude" vs time of an object in gravitational free-fall is a cycloid"; that is not the same as the plot of altitude vs distance, which is a parabola on a flat Earth. As we are not on a flat Earth, although the real Earth can be approximated as such for small distances, the trajectory is part of an ellipse as stated by Daniel.

Garth

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9. May 24, 2005

### Zanket

That makes sense, thanks. Except, isn't part of an ellipse a parabola?

Last edited: May 24, 2005
10. May 24, 2005

### Zanket

From Six Not-So-Easy Pieces, pg. 140: “...a parabola—the same curve followed by something that moves on a free ballistic path in the gravitational field”

11. May 24, 2005

### EL

Nope. An ellipse closes itself. For a parabola, say y=x^2, y goes to infinity when x does.

12. May 24, 2005

### Zanket

Ah but wait--I had thought before about the altitude/time vs. altitude/distance difference. I thought they were synonymous plots because the horizontal velocity is constant. What am I missing?

13. May 24, 2005

### Zanket

OK. Is the path of the projectile an arc of a parabola on a (hypothetical) flat Earth and an arc of an ellipse on a round Earth? Edit: Never mind, I see that is just what Garth said. Maybe I'm getting this now.

14. May 24, 2005

### Zanket

Looks like Feynman is correct, because he was talking about a uniform gravitational field. You are correct about a nonuniform field.

15. May 24, 2005

### EL

You sure are!

16. May 24, 2005

### Garth

The round Earth?

Garth

17. May 24, 2005

### dextercioby

Incidentally,"ballistic" trajectories should keep account on many subtleties,on of them being the Earth's curvature,hence elliptical shape of orbit,neglecting rotatation & air motion & viscosity.

You know,ballistical missiles (so caled "strategical weapons") have ranges in thousands of kilometers...To assume earth's flat is insanity.

Daniel.

18. May 24, 2005

### Zanket

Can you elaborate? Given a flat Earth, I'm thinking that if the horizontal velocity is 1 km/s, say, then a curve of altitude/time will be the same shape as altitude/distance. Given a paraboloidal curve for altitude/distance, I don't see how altitude/time will be cycloidal.

19. May 24, 2005

Staff Emeritus
Garth's point is that absent atmosphere, on a spherical earth the arc would be a section of an ellipse with a focus at the earth's center. Geometrically if you allow that focus to move away "to infinity", i,.e. so far way the error is below your maxiumum accuracy, then the arc of the ellipse is indistinguishable from an arc of a parabola.

20. May 24, 2005

### Zanket

Thanks, that agrees with what I learned above. What I'm asking now is, for a hypothetical uniform gravitational field, why is the path paraboloidal for an altitude/distance plot but cycloidal for an altitude/time plot (as told by the link in the first post in this thread)? Given that the horizontal velocity is constant, I would think these plots would have the same shape.