弾性衝突
v 1 ′ {\displaystyle v_{1}'} ≈ m 2 ( u 2 − u 1 ) m 1 + m 2 {\displaystyle {\frac {m_{2}(u_{2}-u_{1})}{m_{1}+m_{2}}}}
v 2 ′ {\displaystyle v_{2}'} ≈ m 1 ( u 1 − u 2 ) m 1 + m 2 {\displaystyle {\frac {m_{1}(u_{1}-u_{2})}{m_{1}+m_{2}}}}
v 1 {\displaystyle v_{1}} ≈ v 1 ′ + v c {\displaystyle v_{1}'+v_{c}} ≈ m 2 u 2 − m 2 u 1 + m 1 u 1 + m 2 u 2 m 1 + m 2 = u 1 ( m 1 − m 2 ) + 2 m 2 u 2 m 1 + m 2 {\displaystyle {\frac {m_{2}u_{2}-m_{2}u_{1}+m_{1}u_{1}+m_{2}u_{2}}{m_{1}+m_{2}}}={\frac {u_{1}(m_{1}-m_{2})+2m_{2}u_{2}}{m_{1}+m_{2}}}}
v 2 {\displaystyle v_{2}} ≈ u 2 ( m 2 − m 1 ) + 2 m 1 u 1 m 1 + m 2 {\displaystyle {\frac {u_{2}(m_{2}-m_{1})+2m_{1}u_{1}}{m_{1}+m_{2}}}}
それゆえ、衝突する両方の物体の速度が光速(秒速30万km)よりもはるかに遅い場合、古典的な計算が成り立つ。
s 1 − s 2 = − s 3 + s 4 {\displaystyle s_{1}-s_{2}=-s_{3}+s_{4}}
非自明な解を出す最後の式から、 s 2 {\displaystyle s_{2}} を解き、従属式に代入すると e s 1 {\displaystyle e^{s_{1}}} とさらに e s 2 {\displaystyle e^{s_{2}}} を得る。
e s 1 = e s 4 m 1 e s 3 + m 2 e s 4 m 1 e s 4 + m 2 e s 3 {\displaystyle e^{s_{1}}=e^{s_{4}}{\frac {m_{1}e^{s_{3}}+m_{2}e^{s_{4}}}{m_{1}e^{s_{4}}+m_{2}e^{s_{3}}}}}
e s 2 = e s 3 m 1 e s 3 + m 2 e s 4 m 1 e s 4 + m 2 e s 3 {\displaystyle e^{s_{2}}=e^{s_{3}}{\frac {m_{1}e^{s_{3}}+m_{2}e^{s_{4}}}{m_{1}e^{s_{4}}+m_{2}e^{s_{3}}}}}
これは問題の解であるが、速度のパラメータで表現されている。速度の解を得るために戻って代入すると以下の式となる。
v 1 / c = tanh  ( s 1 ) = e s 1 − e − s 1 e s 1 + e − s 1 {\displaystyle v_{1}/c=\tanh(s_{1})={\frac {e^{s_{1}}-e^{-s_{1}}}{e^{s_{1}}+e^{-s_{1}}}}}
p T {\displaystyle p_{T}} ≈ m 1 u 1 + m 2 u 2 {\displaystyle m_{1}u_{1}+m_{2}u_{2}}
v c {\displaystyle v_{c}} ≈ m 1 u 1 + m 2 u 2 m 1 + m 2 {\displaystyle {\frac {m_{1}u_{1}+m_{2}u_{2}}{m_{1}+m_{2}}}}
u 1 ′ {\displaystyle u_{1}'} ≈ u 1 − v c {\displaystyle u_{1}-v_{c}} ≈ m 1 u 1 + m 2 u 1 − m 1 u 1 − m 2 u 2 m 1 + m 2 = m 2 ( u 1 − u 2 ) m 1 + m 2 {\displaystyle {\frac {m_{1}u_{1}+m_{2}u_{1}-m_{1}u_{1}-m_{2}u_{2}}{m_{1}+m_{2}}}={\frac {m_{2}(u_{1}-u_{2})}{m_{1}+m_{2}}}}
u 2 ′ {\displaystyle u_{2}'} ≈ m 1 ( u 2 − u 1 ) m 1 + m 2 {\displaystyle {\frac {m_{1}(u_{2}-u_{1})}{m_{1}+m_{2}}}}
運動量中心系における衝突前の速度 u 1 ′ {\displaystyle u_{1}'} と u 2 ′ {\displaystyle u_{2}'} は:
u 1 ′ = u 1 − v c 1 − u 1 v c c 2 {\displaystyle u_{1}'={\frac {u_{1}-v_{c}}{1-{\frac {u_{1}v_{c}}{c^{2}}}}}}
u 2 ′ = u 2 − v c 1 − u 2 v c c 2 {\displaystyle u_{2}'={\frac {u_{2}-v_{c}}{1-{\frac {u_{2}v_{c}}{c^{2}}}}}}
v 1 ′ = − u 1 ′ {\displaystyle v_{1}'=-u_{1}'}
v 2 ′ = − u 2 ′ {\displaystyle v_{2}'=-u_{2}'}
v 1 = v 1 ′ + v c 1 + v 1 ′ v c c 2 {\displaystyle v_{1}={\frac {v_{1}'+v_{c}}{1+{\frac {v_{1}'v_{c}}{c^{2}}}}}}
v 2 = v 2 ′ + v c 1 + v 2 ′ v c c 2 {\displaystyle v_{2}={\frac {v_{2}'+v_{c}}{1+{\frac {v_{2}'v_{c}}{c^{2}}}}}}
u 1 ≪ c {\displaystyle u_{1}\ll c} かつ u 2 ≪ c {\displaystyle u_{2}\ll c} のとき、