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By K. O. Friedrichs
The most thread operating via this slightly unorthodox method of the designated concept of relativity is the Pythagorean theorem. it sounds as if in its most simple geometric shape within the very starting of this monograph. Then it reappears in algebraic apparel, is extra changed and at last reinterpreted to play the position of 1 of the most characters within the unique conception of relativity.
The first 4 chapters are simply obtainable to highschool sophmores or juniors. the rest a part of the publication could be a little tricky for college kids who by no means studied physics, even if the writer really employs purely the proposal of influence and presupposes no heritage in physics. using the vector geometry brought prior, he leads the reader from the effect dialog legislation to the well-known formulation e=mc2.
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Extra info for From Pythagoras to Einstein
Sample text
First of all we note that, corresponding to every vector c‘, there is a “unit vector” 5, that is, a vector of length 1 ii I = 1, such that z= Ic‘I5. In case I c‘ I # 0, we need only set if I c‘ I = 0 we may take any unit vector for 5. We shall refer to ii as the unit vector (positively) in the direction of c‘. Next we observe that through the notion of vector one is quite naturally led to the interpretation of the real numbers as directed numbers in the manner described at the end of Chapter 2.
2) . $2)- Using identities (*) we arrive at the fundamental formula -- ii b = C& + CU&. Observe that this formula expresses the inner product of two vectors as the sum of the products of their components. ~n important special case of this formula arises in case 6 = ii. ~n this case we have ii ii = I ii l2 and our formula for the inner product yields COMPONENTS AND COORDINATES' 35 Clearly, the first formula here may be regarded as an expression of the Pythagorean thewem; for, the vector a' runs along the hypotenuse of the right triangle formed by the vector a 1 W and the transplanted vector a2G(2), while the absolute values 1 a1 I , I a2 I of the components al,a2 are evidently the lengths of these legs.
With 51 being the velocity of a platform as above, I 5 12 = I 51 12 + 251 - 5 2 + I 52 12. In particular, if platform and particle move in perpendicular directions, we have 15 12 = I51 1’ + I52 12, a formula related to the Pythagorean theorem. If a particle is under the influence of forces its velocity changes in the course of time. To describe the motion of such a particle, it is necessary to know how these forces act and how they affect its velocity. We shall not discuss these matters. We shall confine ourselves to discussing a particular type of motion-the fundamental process called “impact”-that can be determined in essential features without reference to these forces.