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Chapter 36: Q.35 (page 1060)
What is the speed, as a fraction of c, of a particle whose momentum is mc?
Speed of particle is 0.707c.
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This chapter has assumed that lengths perpendicular to the direction of motion are not affected by the motion. That is, motion in the -direction does not cause length contraction along the or axes. To find out if this is really true, consider two spray-paint nozzles attached to rods perpendicular to the axis. It has been confirmed that, when both rods are at rest, both nozzles are exactly 1 m above the base of the rod. One rod is placed in the reference frame with its base on the axis; the other is placed in the reference frame with its base on the axis. The rods then swoop past each other and, as FIGURE P36.60 shows, each paints a stripe across the other rod.
We will use proof by contradiction. Assume that objects perpendicular to the motion are contracted. An experimenter in frame finds that the nozzle, as it goes past, is less than above the axis. The principle of relativity says that an experiment carried out in two different inertial reference frames will have the same outcome in both.
a. Pursue this line of reasoning and show that you end up with a logical contradiction, two mutually incompatible situations.
b. What can you conclude from this contradiction?
At what speed, as a fraction of does a moving clock tick at half the rate of an identical clock at rest?
At , a firecracker explodes at in reference frame S. Four seconds later, a second firecracker explodes at . Reference frame S′ moves in the x-direction at a speed of . What are the positions and times of these two events in frame S′?
The star Alpha goes supernova. Ten years later and away, as measured by astronomers in the galaxy, star Beta explodes.
a. Is it possible that the explosion of Alpha is in any way responsible for the explosion of Beta? Explain.
b. An alien spacecraft passing through the galaxy finds that the distance between the two explosions is. According to the aliens, what is the time between the explosions?
A proton is accelerated to 0.999c.
a. What is the proton’s momentum?
b. By what factor does the proton’s momentum exceed its Newtonian momentum?
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