Simple Nature by Crowell B.

By Crowell B.

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The cgs system, which is now obsolete, was based on the centimeter, the gram, and the second. In the cgs system, the unit of energy is not the joule but the erg, 1 erg=1 g · cm2 /s2 . How many ergs are in one joule? The simplest approach is to treat the units as if they were algebra symbols. kg · m2 s2 1000 g kg · m2 × × =1 s2 1 kg 1J=1 100 cm 1m 2 g · cm2 s2 7 = 10 erg = 107 If you have trouble understanding this example, you should study ch. 0 and 1 of my book Newtonian Physics. Cabin air in a jet airplane example 3 A jet airplane typically cruises at a velocity of 270 m/s.

45175395145262565 Now we’re ready to attack a problem that challenged the best minds of Europe back in the days when there were no computers. In 1696, the mathematician Johann Bernoulli posed the following famous question. Starting from rest, an object slides frictionlessly over a curve joining the point (a, b) to the point (0, 0). Of all the possible shapes that such a curve could have, which one gets the object to its destination in the least possible time, and how much time does it take? ” The solution to the brachistochrone 54 Chapter 2 Conservation of Energy problem evaded Bernoulli himself, as well as Leibniz, who had been one of the inventors of calculus.

Sqrt(2*g*(b-y)))/2. sqrt(dx**2+dy**2) # Pythagorean thm. 2 Numerical Techniques 55 As a first guess, we could try a straight diagonal line, y = x, which corresponds to setting c1 = 1, and all the other coefficients to zero. The result is a fairly long time: >>> b=1. >>> n=10000 >>> c1=1. >>> c2=0. 63887656499994161 What we really need is a curve that’s very steep on the right, and flatter on the left, so it would actually make more sense to try y = x3 : >>> c1=0. >>> c2=0. 59458339947087069 This is a significant improvement, and turns out to be only a hundredth of a second off of the shortest possible time!

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