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25th_IPhO_1994 - THE EXAMINATION XXV INTERNATIONAL PHYSICS...

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THE EXAMINATION XXV INTERNATIONAL PHYSICS OLYMPIAD BEIJING, PERPLE’S REPUBLIC CHINA THEORETICAL COMPETITION July 13, 1994 Time available: 5 hours READ THIS FIRST! INSTRUCTIONS: 1. Use only the ball pen provided. 2. Your graphs should be drawn on the answer sheets attached to the problem. 3. Your solutions should be written on the sheets of paper attached to the problems. 4. Write at the top of the first page of each problem: ● The total number of pages in your solution to the problem ● Your name and code number 1
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Theoretical Problem 1 RELATIVISTIC PARTICLE In the theory of special relativity the relation between energy E and momentum P or a free particle with rest mass m 0 is 2 4 2 0 2 2 mc c m c p E = + = When such a particle is subject to a conservative force, the total energy of the particle, which is the sum of 4 2 0 2 2 c m c p + and the potential energy, is conserved. If the energy of the particle is very high, the rest energy of the particle can be ignored (such a particle is called an ultra relativistic particle). 1) consider the one dimensional motion of a very high energy particle (in which rest energy can be neglected) subject to an attractive central force of constant magnitude f . Suppose the particle is located at the centre of force with initial momentum p 0 at time t =0. Describe the motion of the particle by separately plotting, for at least one period of the motion: x against time t , and momentum p against space coordinate x . Specify the coordinates of the “turning points” in terms of given parameters p 0 and f . Indicate, with arrows, the direction of the progress of the mothon in the ( p , x ) diagram. There may be short intervals of time during which the particle is not ultrarelativistic. However, these should be neglected. Use Answer Sheet 1. 2) A meson is a particle made up of two quarks. The rest mass M of the meson is equal to the total energy of the two-quark system divided by c 2 . Consider a one--dimensional model for a meson at rest, in which the two quarks are assumed to move along the x -axis and attract each other with a force of constant magnitude f It is assumed they can pass through each other freely. For analysis of the high energy motion of the quarks the rest mass of the quarks can be neglected. At time t=0 the two quarks are both at x =0. Show separately the motion of the two quarks graphically by a ( x , t ) diagram and a ( p , x ) diagram, specify the coordinates of the “turning points” in terms of M and f , indicate the direction of the process in your ( p , x ) diagram, and determine the maximum distance between the two quarks. Use Answer Sheet 2. 3) The reference frame used in part 2 will be referred to as frame S , the Lab frame, referred to as S , moves in the negative x -direction with a constant velocity v =0.6 c . the coordinates in the two reference frames are so chosen that the point 2
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x =0 in S coincides with the point 0 = x in S at time 0 = = t t . Plot the motion of the two quarks graphically in a ( x , t ) diagram. Specify the coordinates of the turning points in terms of M , f and c
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