Conference rusure::math

It's about 21.8 miles to work.  This assumes the cup is cylindrical, the
surface of the Earth is a plane, the cup is positioned vertically throughout
the trip, and the coffee manages not to spill when you change from
acceleration to deceleration.


				-- edp

If the car's horizontal acceleration is equal and perpendicular to the
acceleration of gravity, the coffee will be subjected to an acceleration
at 45� degrees to the perpendicular, so it just barely avoids spilling.

Through half the trip, the car accelerates at 32m/s�, and through the
other half it decelerates at 32m/s�.  Under constant acceleration a,
the distance d covered in time t (starting from a dead stop) is given by:

	d = at�/2.

So, the overall distance of the trip is double the distance covered in
half the time, namely:

	2 (32m/s�)(60s)�/2 = 115.2km

Or:
	115.2km x 1mi/1.609km = 71.6mi

Note that at peak velocity, the car is travelling at 6912 km/h, or 4296 mph.
That explains why you weren't stopped by a local law enforcement official.
It also explains why Officer O'Grady thought he was reliving a chase scene
from Star Wars.

By the way, the above assumes there were no turns.  So,... given a turning
radius of 10m, how fast can you take the turns?

					- Gilbert

I think Peter has confused his constants. The coffee will sustain a 1g
horizontal acceleration, which is 9.8 m/s^2 or 32 f/sec^2, not 32m/s^2.
The result in .1 sounds right. - Lynn 

Sorry to bring reality into the picture, but

1. You must compensate for Corolis forces.

2. You must compensate for Sun/Moon tides as your position changes.

3. At the instants when accelleration changes, you must compensate
   for the kind of coffee and amount of sugar (or for the viscosity
   of the fluid, if you want to be picky).

Have at it.

"Sorry to bring reality into the picture, but"

You all forgot to *drink* some of the coffee during the trip, which would
of course allow more accelaration.

No, only kidding, but it's interesting that when I asked my girlfriend this
puzzle, she brought that aspect up immediately, and I might mention she's not
very mathematically inclined.

When I formulated the problem, I forgot to consider deccelarating at all.
With the simple accelerate all the way case, I calculated an answer of
72 km.  Do others agree ?

	My formula is d = 1/2 * a * t**2.  a = 10, t = 60*2, so
	    d = 1/2 * 10 * 120**2 = 5 * 14400 = 72000 m = 72 km.

By the way, I don't believe the cup need by cylindrical.  For instance,
a cubical cup ought yield same answer, right ?  However, perhaps only
certain orientations of the cubical cup are optimal.  For instance,
intuitively, I'd say by symmetry that if face or edge is forward, we're
o.k., but what about intermediate rotations ?  (obviously we need to hold
all coffee cups level though)

/Eric

re .5 - for two minutes it's 120 seconds, so 1/2 g t^2 is 4 times larger
than that, or 16*14400 ft.

A cube IS a cylinder. Any cylinder with central or front-back symmetry will 
do.

"coreolis".

re .3:	Thanks for correcting my gravitational constant.

re .4,.5:
	The shape of the cylinder is important.  For example,
	consider an oval-shaped cylinder, with the long axis
	parallel to the direction of acceleration.  Here's a
	side view:

			|			|
			|			|
			|			|
			+-----------------------+

	Note that an acceleration of 1g will cause the surface of the
	liquid to be at a 45� angle, and you'll get a lap-full of java.

	But if you put the long axis perpendicular to the direction of
	acceleration, you'd be able to acccelerate much faster than 1g
	without spillage.  Here's another side view:

				|   |
				|   |
				|   |
				+---+

	Again, assuming a cylindrical coffee cup, how fast *can* you take
	a curve?

No, "Coriolis".

>re .5 - for two minutes it's 120 seconds, so 1/2 g t^2 is 4 times larger
>than that, or 16*14400 ft.
>
>A cube IS a cylinder. Any cylinder with central or front-back symmetry will 
>do.

First of all, I think you're working in feet, and I'm working in metric.
Are both of our answers correct ?

Secondly, since when is a cube a cylinder ??  I thought a cylinder's definition
is a *circle* projected through space.  Or is a cylinder really considered
by definition to be *anything* projected through space ?

/Eric

My dictionary has a definition for "cylinder", doesn't yours?

no

Well, here's what mine says:

cyl-in-der n. 1. a. a surface generated by a straight line moving parallel
to a fixed straight line and intersecting a plane curve. b. a solid bounded
by two parallel planes and such a surface having a closed curve, esp. a circle.
2. Something shaped like a cylinder. 3. A chamber in which a piston moves.
4. The rotating chamber of a revolver that holds the cartidges.

Re .12:

I think you need a new dictionary.  The JOYOFLEX file contains some comments
about various dictionaries.


				-- edp

Please see note 412 for a related problem.

/Eric

    RE: .1
    
    I assume edp's answer was correct for HIS ride to work
    
    31.2 miles is correct for MY ride to work
    
    I fell off my dinosaur when I first heard a problem similar to this.
    
    Reread the original question!!!!