## Water Ellipses?

A funny thing happened on the way to the graphing utility.

I thought I’d use **Geogebra** to estimate the equation of the water parabola I saw at the Detroit Airport.

So I pasted the photo into **Geogebra**, dropped five points on the arc, and then used “Construct Conic Through Five Points”. The results are on the right.

Now the weird part: the equation is not a parabola, but an **ellipse**. I thought that perhaps I had done a poor job of selecting points, but no matter how I did it, it came up as an ellipse. Note the presence of both an x² **and** a y² in the equation below.

Is this a limitation of Geogebra?

Is this an anomaly caused by rendering the digital picture? Or is my assumption that the path of the water is parabolic faulty?

Or is it the case that the paths of projectiles really are ellipses, but are just locally approximated by parabolas on the surface of the earth, as one colleague suggested?

*Click here to see more on Application.*

Water does travel differently in air than solid objects, but not that much different.

So what are the differences?

Try plotting three points in geogebra, then solve for the three coefficients of the corresponding quadratic manually. Plot the resulting equation on the same image and see where it differs from the photograph of the water stream.

Perhaps some combination of air resistance and the surface tension of the water stream conspire to produce a shape that is more elliptical than parabolic.

Fascinating! I’ve been using videos and pictures of that fountain to motivate parabolas without even stopping to question it.

I get the feeling that any half-ellipse can be decently approximated by a well-chosen parabola. So many this isn’t really that surprising.

Could the velocity at which the water is being projected throw off the eccentricity to that of an ellipse?

If it was a ball being projected, we’d expect a parabola, regardless of the velocity. I guess that’s where I’m starting in my informal analysis of the situation. If something is indeed changing the parabolic path to an elliptical one, I’d be more inclined to look at some characteristic of the water (like surface tension, as Whit Ford suggested).

Of course, the water can’t complete a full ellipse, as it will crash into the ground. So it’s really only the top of an ellipse. And maybe tops of ellipses aren’t any different from parabolas?

……that is very true. I was thinking that the short span travelled by the water may not give gravity enough time to reflect its true properties on a fast moving stream.