By Gerard G. Emch
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Extra info for Algebraic Methods in Statistical Mechanics and Quantum Field Theory (Physics & Astronomical Monograph)
These problems are called relative velocity problems. In the example, the velocity of the plane relative to the air in which it travels was used to find its true velocity for a given wind speed. Other sources for relative velocity problems are: swimmers in fast flowing rivers; the direction of steam from a moving train; wind assistance in athletics events; the direction assumed by the sail of a yacht. 8 1 A swimmer who can swim at a speed of 5 ms-1 in the still water of a swimming pool needs to cross a river whose width is 20 m.
2 3 4 5 Fig. 5 Solution Each ball has equal mass, m. When ball 1 is pulled aside and released, it swings down and hits ball 2, which is of course stationary. Ball 1 stops, but what has happened to its momentum? Usually, the balls are made of metal, which can be assumed to be perfectly elastic. As ball 1 swings down, its speed increases, and so does the magnitude of its momentum as a result. ) Ball 1 collides with ball 2. The impact produces an impulse which must be equal to the momentum lost by ball 1 on impact, from the integral of Newton's second law, equation (1).
Most classical work on systems of forces was done using trigonometric and geometric methods on the resulting triangle of forces, and we too have followed this path. This method is very useful for such problems. However, when more than three forces are involved in a system, it can be very tedious combining forces using the triangular law of addition. For instance, to find the resultant of five forces would require as many as four applications of the addition law. One refinement of the triangle law which is achieved by its repeated application is the law of polygon of forces.