# S18-TOPIC- The energy-momentum vector ## The energy-momentum four-vector

### PROBLEM SET

PROBLEM 1. Show that the expression

p t

=

c

m

0 has units of momentum. PROBLEM 2. Show that the expression

p i

= 

m

0 

V

has units of momentum. PROBLEM 3. Show that the expression

E

0 =

m

0

c

2 has units of energy. PROBLEM 4. Show that in the expression

E

= (

m

0 2

c

4 +

c

2

p

2 ) 1/2 has units of energy. has units of momentum. PROBLEM 6. Show that 1

c E

term in the expression

p

 (

E

, 

p

)  2 = 1 + + ... = Then find the next term.

c

2 2 + + ... . PROBLEM 7. Find the energy of an electron traveling at

V

= 4

c

/5 using the equation

E

= (

m

0 2

c

4 +

c

2

p

2 ) 1/2 . What percent of the energy is due to the rest mass of the electron? PROBLEM 8. Suppose 0.5 kg of mass is completely converted into energy. How many joules will this be? Express your answer in the units of megatons of TNT, where 1 megaton of TNT releases 4.2

 10 15 J of energy. PROBLEM 9. The Hiroshima atomic bomb was equivalent to 13 kilotons of TNT. How much mass was converted to energy to supply this energy? PROBLEM 10. An unstable and short-lived particle called a neutral pion, with the symbol  0 , can be created during collisions in a particle accelerator. It is observed in a very short time to decay into two gamma photons represented with the symbol symbol for the Lorentz factor  (not to be confused with the  = [1 -

V

2 /

c

2 ] -1/2 ) according to the process  0   +  . Given that the rest mass of the neutral pion is 135 MeV, find the energy (in joules) of each photon. Hint: 1 eV = 1.602

 10 -19 J. PROBLEM 11. The sun's core contains the three isotopes of hydrogen given in the table. One of the reactions in the sun's core is the so-called d-d reaction: d + d  t + p. SYMBOLS NAME MASS p 1 H proton d 2 H deuteron t 3 H triton

m p

= 1.00783 u

m p

= 2.01410 u

m p

= 3.01605 u where 1

atomic mass unit

= 1 u = 1.661

 10 -27 kg. Determine whether the d-d reaction requires energy, or releases it. Express the quantity absorbed or released in millions of electron volts (MeV). What percent of the reactant's original energy is this?