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A motorcycle has to move with a constant speed on an overbridged which is in the form of a circular arc of radius R and has a total length L. Suppose the motorcycle starts from the highest point,
(a) What can its maximum velocity be for which the contact with the road is not broken at the highest point ?
(b) If the motorcycle goes at speed 1/ 2 times the maximum found in part (a), where will it lose the contact with the road?
(c) What maximum uniform speed can it maintain on the bridge if it does not lose contact anywhere on the bridge?
2. Two protons, each having a speed of 0.935c in the laboratory, are moving toward each other. Determine
(a) The momentum of each proton in the laboratory.
(b) The total momentum of the two protons in the laboratory,
(c) The momentum of one proton as seen by the other proton.
3. An antiproton p0 has the same rest energy as a proton. It is created in the reaction p + p ? p + p + p + p . In an experiment, protons at rest in the laboratory are bombarded with protons of kinetic energy KL, which must be great enough so that kinetic energy equal to 2m0c2 can be converted into the rest energy of the two particles. In the frame of the laboratory, the total kinetic energy cannot be converted into rest energy because of conservation of momentum. However, in the zero-momentum reference frame in which the two initial protons are moving toward each other with equal speed u, the total kinetic energy can be converted into rest energy.
(a) Find the speed of each proton u such that the total kinetic energy in the zero-momentum frame is 2m0c2.
(b) Transform to the laboratory’s frame in which one proton is at rest, and find the speed u’ of the other proton.
(c) Show that the kinetic energy of the moving proton in the laboratory’s frame is KL = 6m0c2.
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4. Creating a Particle two protons (each with rest mass M = 1.67 x 10-7 kg) are initially moving with equal speeds in opposite directions. The protons continue to exist after a collision that also produces a 71° particle. The rest mass of the 71° is m = 9.75 x 10-28 kg.
(a) If the two protons and the 71° are all at rest after the collision, find the initial speed of the protons, expressed as a fraction of the speed of light.
(b) What is the kinetic energy of each proton? Express your answer in MeV.
(c) What is the rest energy of the 71°, expressed in MeV?
(d) Discuss the relationship between the answers to parts (b) and (c).
5. Two protons approach each other head on at 0.5c relative to reference frame S’.
(a) Calculate the total kinetic energy of the two protons as seen in frame S’.
(b) Calculate the total kinetic energy of the protons as seen in reference frame S, which is moving with speed 0.5c relative to S’ such that one of the protons is at rest.
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