The Scale of the universe

[Brumby]

... algebra that describes this:

Now, a harder one: the amount of additional rope needed is always 6 meters, independent of the radius of the globe, even for a globe with an infinitely large radius.

[mtdoc]

Just use junior high school algebra.

Or junior high geometry. Is that what this forum has come to?   

[dannyf]

Easy: as the diameter tends to infinity, so will the length of the rope. Thus we'll get a length of infinity + 2.PI, which is still infinity.

The point is that two approaches, one with a flat surface, and one with a circle approaching a flat surface, provide two different answers for the same measurement.

[Brumby]

The point is -

One IS a flat surface (well, technically, a straight line) - which absolutely excludes any closed shape such as a circle.  Mathematically it is defined in a way that is completely distinct from that of a circle and shares nothing in common - no derivatives, integrals or any other formulae.

The other being a circle will never be - and can never be - a flat surface straight line, BY DEFINITION.  It will certainly be asymptotic, but it can never achieve straightness.  The formulae to describe it is completely distinct.  Certainly you can apply the limiting condition to sufficiently small arc segment, but you cannot apply this logic to the full circle.  No matter what the limiting condition is, the complete entity is still a circle.

You are, therefore not making the same measurement.

[zapta]

However, an infinitely large global is essentially a flat surface.

That's an incorrect statement. Just like saying that a as a radius of a circle goes to infinity, the shape of the circle approaches a straight line. It doesn't.

[ruffy91]

A quarter sector of a circle will always have a 90 degree curve, even when the radius approaches infinity.

[zapta]

It's amazing that math can deal with this kind of magnitudes so easily and represent them in such a compact form. Good job!

[bookaboo]

A quarter sector of a circle will always have a 90 degree curve, even when the radius approaches infinity.

True no matter how large the circle is, but how do we "approach" infinity 

[Brumby]

How 'easy' it is to 'approach infinity' depends on the function under consideration.

In the case of an inverse function like  y = 1 / x then when x 'approaches infinity', y 'approaches zero'.  To determine the function value at the limiting condition is simple and easily comprehended.

In the case of a circle, it's not really straightforward other than saying it's bigger than anything you can measure.  It's kinda hard to wrangle infinite stuff outside of maths.

[dannyf]

It's kinda hard to wrangle infinite stuff outside of maths.

Two cars, 1 mile apart, are driving towards each other, at speed 1 and speed 2 respectively. A fly flies, at speed 3 (speed 3 > speed 2 and speed 1) from the first car towards the second car. When it hits the 2nd car, it turns around and flies back towards the first car and so on and so forth. The fly is assumed to be infinitesimal dimension wise.

Q1: how long (time wise) does the fly fly before it is smashed by the two cars?
Q2: how many turns does the fly make before its demise?
Q3: how did the fly do that in Q2?

[Brumby]

... and that's supposed to mean what, exactly?

[miguelvp]

Assuming you mean, speed 1 as in 1 mph and the same with 2 and 3.

Q1: 20 minutes
Q2: infinite number of turns
Q3: 3/5 of a mile on the 1st trip to reach the 2nd car, 1st car only traveled 1/3 of that distance.
Then use what is left and do the same the other way until infinity.

But instead of that you can just ignore the fly and check when the cars are going to collide because the fly does not matter.

[calexanian]

There you go for you young kids who have never seen this.

[zapta]

True no matter how large the circle is, but how do we "approach" infinity 

The mathematical model of approaching infinity is well defined. Realty is something else.

[dannyf]

Q1: 20 minutes
Q2: infinite number of turns

How did the fly make infinite number of turns within a finite period of time?

[zapta]

Q1: 20 minutes
Q2: infinite number of turns

How did the fly make infinite number of turns within a finite period of time?

It's a series of flys at increasing speeds

[Armxnian]

Q1: 20 minutes
Q2: infinite number of turns

How did the fly make infinite number of turns within a finite period of time?

It's a series of flys at increasing speeds

Na, it's a magical fly that has infinite velocity.

[miguelvp]

Q1: 20 minutes
Q2: infinite number of turns

How did the fly make infinite number of turns within a finite period of time?

It's a series of flys at increasing speeds

Na, it's a magical fly that has infinite velocity.

No, the fly is always at the same speed, but the problem mentioned the fly is infinitesimal in size, so it never ends up bouncing.
You can compute the rate of the remaining distance and that rate will be constant ad infinitum

Edit: note I said speed not velocity because that changes per trip with an instant acceleration when bouncing, also the rate to compute is the round trip or the combined rate of each direction.

[Armxnian]

Q1: 20 minutes
Q2: infinite number of turns

How did the fly make infinite number of turns within a finite period of time?

It's a series of flys at increasing speeds

Na, it's a magical fly that has infinite velocity.

No, the fly is always at the same speed, but the problem mentioned the fly is infinitesimal in size, so it never ends up bouncing.
You can compute the rate of the remaining distance and that rate will be constant ad infinitum

Edit: note I said speed not velocity because that changes per trip with an instant acceleration when bouncing, also the rate to compute is the round trip or the combined rate of each direction.

Not sure what you mean. What does the size of the fly have to do with the number of trips it makes? And you only have 20 minutes, so it will run out.

My example is better. With infinite velocity the fly can cover the distance in zero time. Plus infinite velocity implies infinite acceleration, because without infinite acceleration you would never reach infinite velocity. So the fly could fly towards one car, stop instantly, turn around instantly, and fly towards the other car instantly, and repeat, an infinite amount of times as you complete everything in 0 time. 

[miguelvp]

So the fly is going at a speed of 3 and the car coming towards goes at a speed of 2.

The fly will reach the car at 3*t miles where t is the point in time for the 1st trip.
the car2 will reach the fly at 2*t  miles.
The distance of both added will be one mile.

So we have.

dFly+dCar2=1 mile
dFly = 3*t miles
dCar2 = 2*t miles

so 2*t+3*t = 1 or 5*t = 1 so t is 1/5 of an hour.

The fly would have traveled 3/5 miles.
Car1 travels at 1 mph, so when the fly changes direction Car1 has traveled 1/5 of a mile so the fly needs to now travel 3/5-1/5 or 2/5 of a mile to reach car 1.

So dFly+dCar1=2/5 miles
so 1*t+3*t = 2/5 or 4*t = 2/5 so it's t = 1/10 of an hour
in 1/10th of an hour the fly would have traveled 3/10 miles before heading back to car2 once more.

The rate from fly to car2 is the same as in the 1st part just the distance has gotten shorter.
Going back the rate from fly to car1 is the same as in the 2nd part, again the distance has gotten shorter.

This will keep on happening with the same combined rate, but since the fly is infinitesimal, there is always a distance to travel and that distance will approach 0 but never reach it. Also the time will get shorter per trip and will approach 0 but it will never reach that value either.

[miguelvp]

To probably make it more clear, the rate of the fly to car2 is 1/5th of the remaining distance and the rate from the fly to car1 is 1/4th of the remaining distance. Those rates remain constant as the distance and the time per trip approaches 0 but never reaching 0.

[dannyf]

So the fly is going at a speed of 3 and the car coming towards goes at a speed of 2.

Much simpler than that. The time it takes for the two cars to reach each other - they will for sure crash into each other - is 1 mile / (car 1's speed + car 2's speed), an definitive number.

Over that period of time, the fly is flying at speed 3 so the total distance the fly has traveled before it is crashed by the bumpers is the time it takes the two cars to crash (calculated above) * the fly's speed.

that distance will approach 0 but never reach it.

That distance sure will reach zero. If it didn't, we wouldn't have head on collisions.

The interesting thing here is that an infinite number of things (making turns in this case) is being done in a finite period of time -> different for the brain to handle, at least intuitively.

It is, conceptually, the same as the hotel with infinite rooms, even when full, the hotel with infinite rooms can always take on new guests. aka the "infinite" hotel is bigger than infinite.

[Armxnian]

but since the fly is infinitesimal, there is always a distance to travel and that distance will approach 0 but never reach it.

Oh okay, so you're saying that since the fly is infinitesimally small, it can fit in between the atoms of the front bumpers of each car and will never be smashed, so it can fly around all it wants.

[Brumby]

The question does not contain the necessary information to produce a solution:

When it hits the 2nd car, it turns around and flies back towards the first car and so on and so forth. The fly is assumed to be infinitesimal dimension wise.

The phrase 'it turns around' defines a specific action, but no parameter is supplied to describe the time taken for this action to be performed.

You can certainly assert that this took no time at all, where the above efforts make sense - but if you really want to make it more interesting in a practical setting, give that parameter a finite value.

[Brumby]

but since the fly is infinitesimal, there is always a distance to travel and that distance will approach 0 but never reach it.

Oh okay, so you're saying that since the fly is infinitesimally small, it can fit in between the atoms of the front bumpers of each car and will never be smashed, so it can fly around all it wants.

Nice observation.