Archive for September, 2007

100-Meter Sprint/Dash — a Source for Interesting Calculations

Monday, September 10th, 2007

Asafa Powell Bonita Jamaica Fastest

Originally uploaded by Bonita Jamaica

On Sept. 9, 2007, Asafa Powell broke his own 100-meter sprint/dash world record, his new record is now 9.74 sec. This race and his record can be a source for interesting calculations:

1. how many steps it takes the runners to cover 100 m?

2. what is the average step size?

3. What is their speed in terms of mph?

4. How much time during the race they spend in the air (as oppose to touching the ground)?

This can be evaluated/estimated by viewing the video frame by frame.

I believe it is more than half the time.

Look at it another way, with respect to distance, not time:
Most of the 100-meter distance he covers while he is airborne. If so, in a sense, with respect to distance, he is flying. But this is a misrepresentation because he must touch the ground every step in order to propel his airborne self for the next segment of his flight.

5. Assuming that Asafa Powell’s weight is still 88 kg (per Wikipedia’s older article), then:

5.1. What is his acceleration?

5.2. How much force he spent?

5.3. How much energy?

Friendly Gifts/Favors and Mathematical Reasoning

Monday, September 10th, 2007

It is common among friends and relatives of professionals to expect a favorable treatment, that is, a discount or a freebie, when they need the service or product of their professional friend. And it is almost just as common practice for the professional to oblige with such an expectation. For example, if you have a friend who is a plumber, I dare presume that, when your toilet is plugged and you urgently need a plumber, you might call your plumber friend, ask his assistance and expect him to give you a discount or perhaps even a freebie. You may then reciprocate buying him a dinner or a bottle of wine but the value of this thank-you gift is much lower than the value of the service.

Or, say, your friend is an author who just published a new book. You probably expect her to give you a free copy of the book, perhaps even an autographed one with a personal dedication.

My question is this:

In mutual relationship, why it is the pro who has to favor you? Why dont you favor the pro?

Consider the example of your author friend. Why should she give you a free copy of her book? Why shouldnt you buy her book and pay double its price?

OK, I know, the bookseller cant take a payment larger than what they sell it for but you get the idea. Beside, we can figure out a workaround this formal limitation. For if you truly like your friend and want to (a) encourage her writing and/or (b) encourage her publisher to publish more of her books and/or publish more books of this kind, then you can send the extra payment with an explanation to either the publisher or your friend the author. Or at least, buy the hardbound book at a full-price retailer, not a paperback at discounter, and then, when the paperback comes out, buy it too.

Where is the math here?

Think of positive and negative numbers and especially think of the duality between the positives and negatives. In this case, why the positive should be a discount for you and the financial negative to your friend and not the other way around?

This is a clear example how positive and negative numbers are often set by the relevant context. For example, if I owe you money, then, as far as I am concerned, my debt to you has a negative value while, from your perspective the debt has a positive value. Similarly, for pilots going up is a positive experience and ascending is indicated by positive numbers and descending by negative numbers. On the other hand, for a scuba diver going into the ocean depth is a most positive experience, so for her descending is measured by positive numbers, which also indicate the increase in pressure, while ascending is measured by negative numbers.

When I talk to students, teachers and others about negative and positive numbers, I like to say:

There is nothing negative about the negative numbers.