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動量守恆律者,物理要論也。其用系算碰撞也。
其義為:使一閉系統毋外力加之,則其動量常數也。若觀乎二物之相撞,則 m 1 v 1 + m 2 v 2 = m 1 v 1 ′ + m 2 v 2 ′ {\displaystyle m_{1}v_{1}+m_{2}v_{2}=m_{1}v_{1}'+m_{2}v_{2}'} 。 其中 m 1 {\displaystyle m_{1}} 為物件一之質, m 2 {\displaystyle m_{2}} 乃物件二之質, v 1 {\displaystyle v_{1}} 乃物件一之始速, v 2 {\displaystyle v_{2}} 乃物件二之始速, v 1 ′ {\displaystyle v_{1}'} 乃物件一之終速, v 2 ′ {\displaystyle v_{2}'} 乃物件二之終速。
證:
∫ F 1 d t = m 1 v 1 ′ − m 1 v 1 . . . ( 1 ) {\displaystyle \int F_{1}\mathrm {d} t=m_{1}v_{1}'-m_{1}v_{1}...(1)}
∫ F 2 d t = m 2 v 2 ′ − m 2 v 2 . . . ( 2 ) {\displaystyle \int F_{2}\mathrm {d} t=m_{2}v_{2}'-m_{2}v_{2}...(2)}
以 F 1 = F 2 {\displaystyle F_{1}=F_{2}} 之故(見牛頓第三定律),得:
F 1 = F 2 {\displaystyle F_{1}=F_{2}}
∫ F 1 d t = ∫ F 2 d t {\displaystyle \int F_{1}\mathrm {d} t=\int F_{2}\mathrm {d} t}
( 1 ) = ( 2 ) {\displaystyle (1)=(2)}
m 1 v 1 ′ − m 1 v 1 = m 2 v 2 ′ − m 2 v 2 {\displaystyle m_{1}v_{1}'-m_{1}v_{1}=m_{2}v_{2}'-m_{2}v_{2}}
m 1 v 1 + m 2 v 2 = m 1 v 1 ′ + m 2 v 2 ′ {\displaystyle m_{1}v_{1}+m_{2}v_{2}=m_{1}v_{1}'+m_{2}v_{2}'}