What is the proper geometrical name for a a rectangle with a semi-circle at each end? I'm talking about the shape made up of a rectangle with a semi-circle at each end.  Does it have a particular name? Does it begin with e?
 A: I'm pretty sure they actually use four clothoid arcs joined together in practice, e.g. this. This has a lot to do with the fact that the clothoid is the curve whose curvature is directly proportional to its arclength; an abrupt variation in curvature would equate to an abrupt variation in centripetal force, which can be bad for the racehorses (or even racecars, for that matter).

Here's a simulated clothoid track drawn in Mathematica:

Just to show that the bends are honest-to-goodness clothoids, I drew the clothoid corresponding to the lower right portion of the track in full (the dashed gray one).
The parametrization used is
$$(x\qquad y)=\left(\sqrt{\frac{\pi}{2}}C\left(\sqrt{\frac{2}{\pi}}s\right)\qquad \sqrt{\frac{\pi}{2}}S\left(\sqrt{\frac{2}{\pi}}s\right)\right)$$
where $C(x)$ and $S(x)$ are the Fresnel integrals; I leave you to verify using those expressions that the curvature of the clothoid is indeed directly proportional to the arclength.
A: It's called a stadium. See http://en.wikipedia.org/wiki/Glossary_of_shapes_with_metaphorical_names or http://mathworld.wolfram.com/Stadium.html
A: Obround, apparently.  I don't know Wiktionary's source.  This definition of obround does not appear in OED, for example.  Googling indicates that this definition is commonly used for machine parts having this shape.
A: An oval.
Racetracks, arenas, stadiums and round pens have one things in common - the shape, and the animals who use them at different speeds. The spaces are created based upon the strides of an animal creating velocity and balance as they move through their gaits of walk, trot, lope/canter to gallop. These forces can be seen in action during any training such as round pen training or barrel racing where the rider uses the shape, spacing and pattern angles to achieve speed and then cuts back, creates a circle and curves in speed to the next barrel. 
I believe it is called an oval, but then again I am just a cowboy. :)
A: Another shape called " Bermuda Bottle" has meridian curvature proportional to $x$. Also ratio of shell curvatures is $2$ when rotated about minor axis, however minor axis measures only $\approx 0.6 $ times major axis.
EDIT1:
It is is same shape as filled parachutes.
The differential equation of Cornu Spiral by J.M. is not a mathematician is ( $s$ is arc ):
$$ \frac{d\phi}{ds} = s/a $$
and the Bermuda Bottle (not sketched) is
$$\frac{d\phi}{ds} = x/b $$
They are similar in appearance. Clothoid curve has a third order discontinuity at sharpest corners.
