﻿Fundamental 
  Laws 
  of 
  Matter 
  and 
  Energy. 
  711 
  

  

  Its 
  momentum 
  and 
  kinetic 
  energy 
  will 
  change 
  according 
  to 
  

   (11) 
  and 
  (12) 
  by 
  the 
  amounts 
  

  

  dM=zfdt, 
  

  

  Hence 
  dV 
  =fdl=fvdt. 
  

  

  dE' 
  = 
  vdM 
  (13) 
  

  

  So 
  far 
  the 
  equations 
  are 
  those 
  of 
  Newtonian 
  mechanics, 
  but 
  

   now 
  in 
  substituting 
  for 
  M 
  from 
  equation 
  (10) 
  we 
  must 
  regard 
  

   m 
  as 
  a 
  variable 
  and 
  write 
  

  

  d$j' 
  = 
  mvdv 
  + 
  v 
  2 
  dm 
  (14) 
  

  

  This 
  will 
  be 
  our 
  fundamental 
  equation 
  connecting 
  the 
  kinetic 
  

   energy 
  of 
  a 
  body 
  with 
  its 
  mass 
  and 
  velocity. 
  

  

  Introducing 
  now 
  the 
  relation 
  of 
  mass 
  to 
  energy 
  given 
  in 
  

   equation 
  (7) 
  we 
  may 
  write, 
  

  

  dW 
  = 
  Y 
  2 
  dm, 
  

  

  and 
  combining 
  this 
  equation 
  with 
  (14) 
  gives 
  

  

  Y 
  2 
  dm 
  = 
  mvdv 
  + 
  v 
  2 
  dm. 
  

  

  This 
  equation, 
  containing 
  only 
  two 
  variables, 
  m 
  and 
  v 
  and 
  

   the 
  constant 
  V, 
  may 
  readily 
  be 
  integrated 
  as 
  follows. 
  By 
  a 
  

   simple 
  transformation 
  

  

  ( 
  1 
  ""V 
  2 
  )^ 
  m 
  = 
  

  

  mvdv 
  

  

  "V2- 
  

  

  Writing 
  /3 
  = 
  r/V, 
  and 
  noting 
  that 
  

  

  we 
  see 
  that 
  

  

  dm_ 
  1 
  <1 
  -/3 
  2 
  ) 
  

   m 
  ~ 
  2 
  (1-/3 
  2 
  )' 
  

   Hence 
  

  

  log 
  m 
  =—± 
  log 
  ( 
  1 
  - 
  /3 
  2 
  ) 
  + 
  log 
  m 
  , 
  

  

  where 
  log 
  m 
  is 
  the 
  integration 
  constant. 
  Therefore 
  

   log— 
  =log(l-/3*)-* 
  

  

  m 
  

   m 
  

  

  or 
  m 
  1 
  

  

  m 
  ( 
  15 
  > 
  

  

  This 
  is 
  the 
  general 
  expression 
  for 
  the 
  mass 
  of 
  a 
  moving 
  body 
  

   in 
  terms 
  of 
  /3, 
  the 
  ratio 
  of 
  its 
  velocity 
  to 
  the 
  velocity 
  of 
  

  

  