Wednesday, April 22, 2009

Wednesday Math, Vol. 68: Nigh unto INFINITE POWER!

Last week, I was talking about the ideas of string theory, which have as one of their consequences that the big bang did not come from a single infintesimal point, pressurized to infinitely high tempertures, but from an incredibly small but still measurable region of space at nearly unfathomable temperatures, but still finite and measurable. While math formulas can produce singularities where a value tends to infinity, the universe just doesn't have room enough for infinite power or time enough for infinite speed. In the real world, there are always limiting factors.

Except that I, Maty Boy, have stumbled on an experiment. It's childishly simple, actually, but it will give me access to virtually INFINITE POWER!


Sorry, mad scientist laugh. Hard habit to break.

Let's say we have a ten foot tall ladder. It has wheels on both ends and a track along the floor and a track up the wall, so that as we slide it, it stays in contact with both with the ground and the wall. Let's also stipulate that it is connected to a machine sliding the top of the ladder up the wall at a very leisurely one foot per second, which is less than one mile per hour. This means the bottom of the ladder is sliding as well, along the floor from right to left in this picture.

If the ladder started flat on the floor, at nine seconds the top is nine feet high along the wall. Where is the bottom of the ladder? We just need to use the Pythagorean Theorem, a^2 + b^2 = c^2. The ladder length of 10 is c and the height of 9 is a. 10^2 = 100 and 9^2 = 81, so the bottom of the ladder is sqrt(19) feet from the wall, about 4 feet, 4 inches away from the wall. In the last second, the bottom of the ladder, which traveled less than six feet in the first nine seconds, will have to be moving at least 4 feet per second.

Some of the keener students will here notice that 4 feet per second is not infinite power.

Wait, it gets better.

The top of the ladder is moving at a fixed rate, but the bottom of the ladder is getting faster and faster. In the last tenth of a second, the ladder will have to average 14 feet per second. In the last hundredth of a second, it will have to be moving at nearly 45 feet per second. The math says the closer we get to 10 seconds and the top of the ladder getting to ten feet high at a nice constant rate, the faster the speed of the bottom of the ladder with no upper bound.

Searching around the internets, sports physics websites say the amount of time a golf ball stays in contact with the club is measured at about a half a millisecond, or 0.0005 of a second. If there was a hole in the wall at the bottom and I teed up a golf ball with the edge of the ball about 1.2 inches away from the wall on the inside, and a golf club head was attached to the bottom of the ladder, the club would be moving at 200 feet per second and accelerating when it made contact with the ball. In comparison, when Tiger Woods swings his driver, the club head is moving about 183 feet per second.

Nice shot, Alice. Does your husband play golf, too?


Of course, if the ladder was being lifted faster, the bottom of the ladder would be moving faster. If the ladder was moving two feet per second going up, the ball should be teed up at 1.7 inches inside the wall, and the club head speed would be about 280 feet per second and accelerating. The increase is by a factor of the square root of 2.

If the ladder and the wheel tracks are made of strong enough stuff with low enough friction coefficients, we could try this experiment and probably get some impressive whacks on a golf ball. The problem is the whole infinite power thing. Stresses to the connections to the wall and the floor would be great. If we pull a ladder away from a wall it isn't connected to, it loses contact with the wall part of the way down and the rest of its fall is due to gravity and not the Pythagorean Theorem. In the last tiniest fractions of a second, heat and torque will conspire to slow the moving ladder down so that it can't be going at infinite speeds.


So what will we be doing next Wednesday, Matty Boy?

The same thing we do every Wednesday, hypothetical question asker. We will try to take over the world!

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