alittlebitofcalculus-pdf-january2011-111112001007-phpapp01

Proving convergence becomes very very difficult a

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Unformatted text preview: efinitely over a FINITE time interval. Correspondingly, y ’(t) will have COUNTABLY many points of discontinuity over a FINITE time interval [0, T]. (ii) The ball may bounce indefinitely over an INFINITE time interval. Correspondingly, y ’(t) will have COUNTABLY many points of discontinuity over the INFINITE time interval [0, +∞). So eventually the ball will run out of ener g y a nd come to rest. From a practical point of view energ e ner we may say that the ball disappears into TIME. energ We know that ener g y E i s a function of displacement. In simple terms e ner E = m.a.s, or more precisely E = m.a.ds, where m = mass, a = acceleration, m.a.ds and ds = a n element of displacement. In the case of the bouncing ball ds dy(t) = y ’ (t) d t . ds = d 172 b Depending on how the ball comes to rest, in the integral ∫ to compute the energy a E, b = some finite instant T or b = +∞. Now comes the question: b Does the integral ∫ a CONVERGE ? If it does CONVERGE then the energy function E is INTEGRABLE. When there is such a loss of energy (damping) on each bounce we can see that: energy E = ∫ m.a.ds must CONVERGE. m.a.ds From the Physics point of view this CONVERGENCE means that the energy E Physics energy required for the ball to bounce indefinitely is finite. What if the ball bounces without loss or gain of energy ? The energy will change energy form between kinetic and potential. But the total energy will be CONSTANT. From the engineering point of view we will have a perpetual motion machine. We have a similar situation in the hydrogen atom. The electron revolves around the proton in an elliptical orbit known as the path or curve of CONSTANT energy. Now let us look at the situation where the ball gains energy on each bounce. Eventually the ball will disappear from sight. From a practical point of view we may say that the ball disappears into SPACE. This is similar to saying a swing or pendulum increases in amplitude with each oscillation. The displacement of such a pendulum is described in the diagram below. a m p l i t u d e 0 T 2T 173 3T t In this case i...
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