6.2 Problems 323
where the Lorentz factorγis defined as usual
(ii) Hence determine the lifetime of muons at rest, knowing that when trav-
elling at a speedc√
8 /3 through the apparatus described above (withL=
200 m) N 1 and N 2 were measured to be 10,000 and 8,983, respectively.
[adapted from University of London, Royal Holloway and Bedford New
College]6.38 A particleXat rest is a sphere of rest-massmand radiusrand has a proper
lifetimeτ. If the particle is moving with speed√
3
2 cwith respect to the lab
frame (cis the speed of light):
(a) Determine the total energy of the particle in the lab frame
(b) The average distance the particle travels in the lab before decaying
(c) Sketch the shape and dimensions of the particle when viewed perpendicular
to its motion in the lab frame, include an arrow to indicate its direction of
motion on your sketch.
[adapted from the University of London Royal Holloway and Bedford New
College 2006]6.39 The Lorentz velocity transformation isν′= 1 −νu−ν/uc 2 , whereν′andνare the
velocities of an object parallel touas measured in two inertial frames with
relative velocityu. Show that a photon moving atc, the speed of light will
have the same speed in all frames of reference.
6.2.3 Mass,Momentum,Energy ..........................
6.40 The mean life-time of muons at rest is 2. 2 × 10 −^6 s. The observed mean life-
time of muons as measured in the laboratory is 6. 6 × 10 −^6 s. Find
(a) The effective mass of a muon at this speed when its rest mass is 207me
(b) its kinetic energy (c) its momentum
6.41 Calculate the energy that can be obtained from complete annihilation of 1 g of
mass.
6.42 What is the speed of a proton whose kinetic energy equals its rest energy?
Does the result depend on the mass of proton?
6.43 What is the speed of a particle when accelerated to 1.0 GeV when the
particle is (a) proton (b) electron
6.44(a) Calculate the energy needed to break up the^12 C nucleus into its con-
stituents. The rest masses in amu are:(^12) C12.000000; p 1.007825; n 1.008665;α 4. 002603
(b) If^12 C nucleus is to break up into 3 alphas. Calculate the energy that is
released.