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Chapter 2: Q41E (page 64)
A muon has a mean lifetime of 2.2in its rest frame. Suppose muons are travelling at 0.92c relative to Earth. What is the mean distance a muon will travel as measured by an observer on Earth?
Themean distance travelled by muon on Earth is 1549 m.
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In the collision shown, energy is conserved because both objects have the same speed and mass after as before the collision. Since the collision merely reserves the velocities, the final (total) momentum is opposite the initial. Thus. momentum can be conserved only if it is zero.
(a) Using the relativistically correct expression for momentum. Show that the total momentum is zero-that momentum is conserved. (Masses are in arbitrary units).
(b) Using the relativistic velocity transformation. find the four velocities in a frame moving to the right at 0.6c.
(c) Verify that momentum is conserved in the new frame.
Verify that the special case leads to equation (2-6) when inserted in linear transformations (2-4) and that special case. in turn leads to (2-8).
Bob is on Earth. Anna is on a spacecraft moving away from Earth at 0.6c . At some point in Anna's outward travel, Bob fires a projectile loaded with supplies out to Anna's ship. Relative to Bob, the projectile moves at 0.8c . (a) How fast does the projectile move relative to Anna? (b) Bob also sends a light signal, " Greetings from Earth:' out to Anna's ship. How fast does the light signal move relative to Anna?
The boron-nucleus (mass: 14.02266 u) "beta decays," spontaneously becoming an electron (mass: 0.00055 u) and a carbon- nucleus (mass: 13.99995 u). What will be the speeds and kinetic energies of the carbon-nucleus and the electron? (Note: A neutrino is also produced. We consider the case in which its momentum and energy are negligible. Also, because the carbon- nucleus is much more massive than the electron it recoils ''slowly''; .)
Question: Here we verify the conditions under which in equation (2-33) will be negative. (a) Show that is equivalent to the following:
(b) By construction v, cannot exceed , for if it did, the information could not catch up with Amy at event 2. Use this to argue that if , then must be positive for whatever value is allowed to have. (c) Using the fact that , show that the right side of the expression in part (a) never exceeds . This confirms that when v need not exceed to produce a negative .
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