“How fast can a watermelon roll down a hill?” That was my brothers favorite question to my father when he was a kid. We always joked at him for asking such a question. Now, he’s older and he says there’s an actual answer for it. Anybody have the slightest clue on how you could even find that out?
Is it a laden or unladen watermelon?
A 7 lb. wheel of cheese a foot in diameter will roll slowly enough downhill that a running person can keep up with it.
If he doesn’t fall down first.
http://www.cnn.com/WORLD/9705/27/fringe/cheese.rolling/
http://www.ellisclinic.com/bodyinjury.htm
So, call it 20 mph, more or less.
The quick answer is as fast as the size of the hill will allow.
But practically, this actually sounds like a nifty problem. The greatest speed will be attained when the axis of rotation is the long axis of the watermelon. Let’s idealize it as a prolate spheroid. The long axis has the smallest moment of inertia. But, I think, that mode of rotation is unstable. So if the watermelon gathers enough speed before degenerating into chaotic spin (read tumbling) it will disintegrate. Cool.
::MonkeyMensch runs off to grocery store::
Nope, you have made an assumption not in evidence. The cheese chasers don’t have to keep up with the cheese to win, merely cross the line at the bottom of the hill first (after the cheese). Which is a good thing, as the cheese can reach speeds of 40 miles per hour.
This is the only place in the whole wide world where there are people discussing, in perfect seriousness, how fast a watermelon would roll down a hill. 
Okay, but the cheese would roll faster than a watermelon, wouldn’t it? Being a disc shape rather than a round tubby shape…The watermelon would give up more friction to the ground, and so would roll slower.
Well, I guess it depends on how steep a hill you can have until it is not considered rolling. Suppose you have an almost vertical hill, and it is basically falling except for those few times it clips the ‘wall’. Is that rolling?
Perhaps MonkeyMensch is on the right track, and we need to calculate how fast you can spin a watermelon before it sprays chunks all over the place.
Duck Duck Goose, at my best I could get about 25mph going slightly uphill, but that was only for about 75 yards. I think that chasing a cheese downhill would increase my speed, but I would probably lose a kneecap somewhere along the way…
For a sufficiently steep, smooth and high hill, and constant contact between melon and said hill, I would guess the spinning melon would continue to accelerate until the centripetal force caused by its own spinning ripped it apart. Find that force, calculate moment, get surface speed of melon at that point, and that is your terminal velocity. Or half that is your terminal, I forget.
Or not.
My own WAG is that the Watermelon would hit a terminal velocity of air resistance before it ripped it self apart centripetally. I just think that a watermelon out husk would allow it to reach pretty damn high rpms. IF only I had a high speed, high horsepower lathe to try it on ;). Anybody work in a well equiped shop. If so we could find the terminal rotaitional velocity of a watermelon, calculate the speed it would be going using its circumfrance and compare to the wind resitance of its shape at sea level. Of course in a vacuum it probabbly would detonate eventually.
If anyone is missing a copy of the word ‘out’ it somehow jumped into my post before ‘husk’.
Would freezing the watermelon first help keep it together up to unsafe (for watermelon) speeds?
Guess we’ll have to wait for the International Commission on Watermelon Rolling to codify the rules for alpine watermelon rolling to see how close we can flirt with banned substances, like R134-a. 
Would a frozen watermelon roll faster than a non-frozen watermelon? There’s never a physics geek around when you need one…
Well, someone’s just going to have to go out and get a watermelon and gather some empirical data for us.
Well, don’t look at me–I’m a Flatlander out here on the Wide Prair-ee. The closest thing we’ve got to a hill is the Macon County Landfill.
What? You can spin a watermelon in a lathe and see what angular velocity causes it to fly apart. Not having hills is no excuse.
Florida has the lowest highest point of any state, and that is way up north. Otherwise I’d be all over this one. One for the Master?
But the OP wasn’t asking, “How fast would you have to spin a watermelon to get it to fly apart?”, the OP was asking, “How fast can a watermelon roll downhill?”. And in order to gather empirical data for that, you would need a hill.
Well, lets see.
Say the angle between the horizontal and the slope is è. The force due to gravity is g (9.8 m/s^2). The watermelon’s mass is m. Its acceleration down the hill is a. The normal force from the hill is Fn. The frictional force opposing motion is Ffr.
Now we rotate our co-ordinate axis so that +X is in the +a direction (ie, so +X points down the hill).
Summing Y forces we see that Fn = mg cosè
Summing X forces,
mg sinè - Ffr = ma
And since Ffr = uFn, where u is the coefficient of kinetic friction
mg sinè - mg cosè = ma
g sinè - g cosè = a (cancelling the m from both sides)
…from this we conveniently eliminate the mass of the watermelon! How exciting!
Now, if you happen to know how long the watermellon’s been rolling for, we can find its speed, v, with v = a*t (we’ll assume it started at rest).
In this example I am assuming a perfect watermelon that does not “wobble” from side to side; thus a perfect circular cross-section through the middle will always touch tangent to the hill.
Does that answer your question?
Get a grant.
I think the only way to predict it would be to test it. Why you ask? Well, way back in the days of Physics, we used to do these types of problems wherer you would change the potential energy of the height of the hill to kinetic energy. This kenetic energy would be reperesented in regular momentum (the movement down) and angular momentum (the energy of the spinning melon). To be able to tell which amount of energy went to which part, spinning or moving, you would need a specific factor for each type of solid. For instance, a hollow wheel had a different time than a solid wheel (it was faster, I think).
In some sense you can think of it like this: The movement going down the hill is inhibited by the rotational inertia of the object. I would assume that a watermelon is basically consistent in density throughout, so it would probably have the rotational intertia of a solid wheel.
As for trigonal planar’s explanation, I don’t really understand why friction is necessary. It isn’t sliding down the hill. If the watermellion is rolling, then it never actually breaks the threshold. The only way I could see friction becoming an issue is if the rotational intertia was greater than the friction threshold, so it wouldn’t actually roll. I don’t really feel like brushing up on high-school physics, now, though ; )