\magnification=\magstep2

$$
{\rm E_A} \quad = \quad 5 {\rm\ GeV}  \quad = \quad 8 \times 10^{10} {\rm\ J}
$$

$$
{\rm E_A} \quad = \quad  \gamma_A {m_e} c^2 
$$

$$
\gamma_A = 9782
$$

$$
\eqalign{
v_A \quad &= \quad c {\sqrt{1 - {{1}\over{\gamma_A^2}}} }  \quad
          = \quad c {\left(1 - \gamma_A^{-2} \right)^{1/2}} \quad
          \simeq \quad c{\left(1 - {{1}\over{2}} \gamma_A^{-2} \right) } \cr
          &\phantom{=} \cr
          &\simeq \quad c {\left( 1 - 5 \times 10^{-9} \right)} \cr
}
$$


$$
\eqalign{
  t_A \quad &= \quad { {1.5 \times 10^{11} {\rm\ m}} 
                  \over {c  {\left( 1 - 5 \times 10^{-9} \right)} } }
            \quad = \quad 
                     { {1.5 \times 10^{11} {\rm\ m}} \over {c} } 
                       {\left( 1 - 5 \times 10^{-9} \right)^{-1}}   \cr
          &\phantom{=} \cr
            & \simeq \quad 
                     { {1.5 \times 10^{11} {\rm\ m}} \over {c} } 
                       {\left( 1 + 5 \times 10^{-9} \right)}   \cr
          &\phantom{=} \cr
           & \simeq \quad 500.4 {\rm \ s \ }
                       {\left( 1 + 5 \times 10^{-9} \right)}  
           \quad \simeq \quad 500.4 + {2.6 \times 10^{-6} {\rm \ s } }  \cr
}
$$


$$
{\rm E_B} \quad = \quad 10 {\rm\ GeV}  \quad = \quad 16 \times 10^{10} {\rm\ J}
$$

$$
{\rm E_B} \quad = \quad  \gamma_B {m_e} c^2 
$$

$$
{\gamma}_B = 19563
$$

$$
\eqalign{
v_B \quad &= \quad c {\sqrt{1 - {{1}\over{\gamma_B^2}}} }  \quad
          = \quad c {\left(1 - \gamma_B^{-2} \right)^{1/2}} \quad
          \simeq \quad c{\left(1 - {{1}\over{2}} \gamma_B^{-2} \right) } \cr
          &\phantom{=} \cr
          &\simeq \quad c {\left( 1 - 1.3 \times 10^{-9} \right)} \cr
}
$$


$$
\eqalign{
  t_B \quad &= \quad { {1.5 \times 10^{11} {\rm\ m}} 
                  \over {c  {\left( 1 - 1.3 \times 10^{-9} \right)} } }
            \quad = \quad 
                     { {1.5 \times 10^{11} {\rm\ m}} \over {c} } 
                       {\left( 1 - 1.3 \times 10^{-9} \right)^{-1}}   \cr
          &\phantom{=} \cr
            & \simeq \quad 
                     { {1.5 \times 10^{11} {\rm\ m}} \over {c} } 
                       {\left( 1 + 1.3 \times 10^{-9} \right)}   \cr
          &\phantom{=} \cr
           & \simeq \quad 500.4 {\rm \ s \ }
                       {\left( 1 + 1.3 \times 10^{-9} \right)}  
           \quad \simeq \quad 500.4 + {0.7 \times 10^{-6} {\rm \ s } }  \cr
}
$$




$$
t_A - t_B \quad = \quad {\left( 500.4 + 2.6 \times 10^{-6} \right) }
                    - 
                        {\left( 500.4 + 0.7 \times 10^{-6} \right) }
                 \quad \simeq \quad
                        2 \times 10^{-6} {\rm \ s } 
$$

\bye

