Modern Physics
Problems on relativistic collisions
        
1)  A particle of rest mass mo  and kinetic energy 2 moc2  strikes and sticks to a  stationary   particle of 
     rest mass 2 mo .        Find the rest mass Mo   of the final particle (in terms of mo  ). 
        
2) Two particles, each of rest mass mo   (each moving at   4/5 c ) collide head on and stick together.   
     What is the rest mass and velocity of the resulting particle?

3) A particle of rest mass Mo is at rest in the laboratory when it decays into three identical particles, each 
    of rest mass mo. .  Two of the particles (labeled 1 and 2) have velocities in the directions shown.
    a) Find the direction of motion for the  third particle.                  4/5 c                          
                                                                                                                                                                                           
    b) Find the speed of the third particle.                     
                         
    c) Find the ratio Mo /mo.                                                              3/5 c
        
4)  A particle of rest mass mo  moving at a speed of 0.8 c collides with an identical type of 
     particle (i.e., same rest mass) that is initially at rest.  Only one particle remains after the  
     collision.    Find the rest mass and the speed of the composite particle.
 
5)  In introductory physics we used the phrase “completely inelastic” to describe the type of 
     collision in which particles stuck together?  What did we say happened to the kinetic 
     energy in these collisions studied in 355/305 (at least 3 examples)?  
     What conclusions can you make about what happens to mass and/or kinetic energy in a completely 
     inelastic relativistic collision?  On what basis?  Hint: Use your results to problems above.  
     Is there a particularly good frame in which you should analyze the collision in order to generalize the 
     your conclusions to all completely inelastic collisions?

    What happens to mass and kinetic energy in explosions?  (Same hints and questions as in the 
    previous part.)
       
6)  A neutral pi meson 0 (also called a pion) decays into two gamma rays (and nothing else).  
     Suppose that a pion (rest mass energy = 135 Mev) moving with a kinetic energy equal to a 
     value of N times the rest-mass energy decays.  (N is some spcific number.) 
        
    a) Determine the pion's speed before decaying in terms of N
    b) What are the energies of the two gamma rays (in terms of N) if the decay process causes   
        them to be emitted in opposite directions along the pion's original line of motion? 
    c) Evaluate the answers to (a) and (b) for the specific case of N = 2.
    d) How do the answers to (a) and (b) depend on N as N approaches infinity?  
    e) half-page graphs of (a) and (b) as a function of N? (integrating the answers to d)

7)  Show that if a photon of wavelength    collides with a particle at rest, the photon  cannot 
     be completely absorbed (assuming that the particle keeps its identity,  i.e., has the same  
     rest mass).    
     (Assume that it is possible:  write down the appropriate conservation laws, solve, and show 
     that a contradiction results.)

8)  Show that a single photon of wavelength    cannot  perform pair production.  See hint 
     given in the previous problem.  (This problem is not anywhere near as easy.)

9)            e-    +    e+    --->    -   +   + 
        
     An electron and a positron, each moving at 0.99999 c, make a head-on collision.  
     In the collision the electron/positron pair disappears and is replaced by a muon pair
     (rest energy = 105.7 Mev) which move off in opposite directions.
        
     What is the kinetic energy of each muon?  how fast are they moving compared to 
     the original electron/positron? 
        
10)             0    --->          +    
        
     A neutral pi meson (rest energy = 135 Mev) moving through the laboratory at a speed of 
     0.98 c decays into two gamma rays that move in directions that make equal angles with 
     the direction of motion of the initial pi meson. 
        
     Find the angle    and the energies of the gamma rays.   
     What happens to  if we increase the speed of the original  0  ?  Don’t forget to do an asymptotic 
     limit as you did in problem 6 above.
        
11) Show that the energy expression for particles can be rewritten as
        
             E  =  moc2 [ 1 +  ( p / mo  c )2  ]1/2
        
      and use the binomial expansion to show that if v << c, then
        
             E       moc2   +  p2/2mo         moc2    +   1/2  mov2 
        
       What if we kept the third term of the binomial expansion?  What would it be?
       What is the size of the third term relative to the second term for

	a) a fast plane?
	b) the earth’s speed around the sun?
	c) something moving one-tenth the speed of light? 
        
12) Suppose a pion at rest decays into a muon and an antineutrino (of the muon kind):      
      
+  --->   +  +    
              
      Find the speed (v/c) of the muon. Look up the rest masses of muon and pion.

13) A moving electron collides with a stationary electron and electron-positron pair comes into being as a result (a positron is a positively charged electron).  When all four particles have the same velocity after the collision, the kinetic energy required for this process turns out to be a minimum.  Show that Kmin = 6 moc2, where mo is the rest mass of the electron.

14) A K0S particle with a speed 0.80c relative to the laboratory decays into 2 pions, + and - .
a) Calculate the speed of each pion in the rest frame of the K0S .  The mass of the K0S  is 498 Mev/c2 and masses of the pions are 140 Mev/c2 .

b) Assume that the pions move along a line parallel to the original direction of motion of the K0S .  Calculate the speed of the pions in the laboratory.