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Chapter 36: Q. 34 (page 1060)
At what speed, as a fraction of , is a particle’s momentum twice its Newtonian value?
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One of the important ways in which the Higgs boson was detected at the Large Hadron Collider was by observing a type of decay in which the Higgs—which decays too quickly to be observed directly—is immediately transformed into two photons emitted back to back. Two photons, with momenta , were detected. What is the mass of the Higgs boson? Give your answer as a multiple of the proton mass.
Event A occurs at space-time coordinates .
a. Event B occurs at space time coordinates . Could A possibly be the cause of B? Explain.
b. Event C occurs at space time coordinates . Could A possibly be the cause of C? Explain
The star Delta goes supernova. One year later and away, as measured by astronomers in the galaxy, star Epsilon explodes. Let the explosion of Delta be at role="math" localid="1649750409129" and. The explosions are observed by three spaceships cruising through the galaxy in the direction from Delta to Epsilon at velocities, , and . All three spaceships, each at the origin of its reference frame, happen to pass Delta as it explodes.
a. What are the times of the two explosions as measured by scientists on each of the three spaceships?
b. Does one spaceship find that the explosions are simultaneous? If so, which one?
c. Does one spaceship find that Epsilon explodes before Delta? If so, which one?
d. Do your answers to parts b and c violate the idea of causality? Explain.
A ball of mass m traveling at a speed of 0.80c has a perfectly inelastic collision with an identical ball at rest. If Newtonian physics were correct for these speeds, momentum conservation would tell us that a ball of mass 2m departs the collision with a speed of 0.40c. Let’s do a relativistic collision analysis to determine the mass and speed of the ball after the collision.
a. What is gp, written as a fraction like a/b?
b. What is the initial total momentum? Give your answer as a fraction times mc. c. What is the initial total energy? Give your answer as a fraction times mc2 . Don’t forget that there are two balls.
d. Because energy can be transformed into mass, and vice versa, you cannot assume that the final mass is 2m. Instead, let the final state of the system be an unknown mass M traveling at the unknown speed uf. You have two conservation laws. Find M and uf.
Let’s examine whether or not the law of conservation of momentum is true in all reference frames if we use the Newtonian definition of momentum: . Consider an object A of mass 3m at rest in reference frame S. Object A explodes into two pieces: object B, of mass m, that is shot to the left at a speed of c/2 and object C, of mass 2m, that, to conserve momentum, is shot to the right at a speed of c/4. Suppose this explosion is observed in reference frame S′ that is moving to the right at half the speed of light.
a. Use the Lorentz velocity transformation to find the velocity and the Newtonian momentum of A in S′.
b. Use the Lorentz velocity transformation to find the velocities and the Newtonian momenta of B and C in S′.
c. What is the total final momentum in S′?
d. Newtonian momentum was conserved in frame S. Is it conserved in frame S′?
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