Title: 碰撞 (Collide (Skillet album))
Section: 弹性碰撞

两个物体互相碰撞，能量不转换为内能（如热或变形），碰撞前动能和与碰撞后动能和相等。在动量守恒定律中碰撞前的动量（向量）和同样等于碰撞后的动量和。理想弹性碰撞在宏观上是一个物理模型。由于摩擦和其他因素的存在，系统总会损失动能。相关的模型如台球和橡胶球。在原子和基本粒子的碰撞中，依据量子力学存在一个最小能，这个最小能给原子或其他粒子以推动力，或在量子物理学中创造和和转换粒子提供必要条件。这个能量仍然不足以发生理想弹性碰撞。對於理想彈性碰撞，碰撞前后的动能和必须相等：                                                                                                                                                            m                                                  1                                                                    ⋅                                              |                                                                    v                                                  1                                                                                                                      |                                                                          2                                                                                      2                                                  +                                                                                                    m                                                  2                                                                    ⋅                                              |                                                                    v                                                  2                                                                                                                      |                                                                          2                                                                                      2                                                                                              =                                                                                                    m                                                  1                                                                    ⋅                                              |                                                                    v                                                  1                                                ′                                                                                              |                                                                          2                                                                                      2                                                  +                                                                                                    m                                                  2                                                                    ⋅                                              |                                                                    v                                                  2                                                ′                                                                                              |                                                                          2                                                                                      2                                                                                                                                                                                          m                                                  1                                                                    ⋅                                              v                                                  1                                                                          2                                                                                      2                                                  −                                                                                                    m                                                  1                                                                    ⋅                                              v                                                  1                                                                          ′                                                      2                                                                                                                2                                                                                              =                                                                                                    m                                                  2                                                                    ⋅                                              v                                                  2                                                                          ′                                                      2                                                                                                                2                                                  −                                                                                                    m                                                  2                                                                    ⋅                                              v                                                  2                                                                          2                                                                                      2                                                                                                                                                                  m                                              1                                                              2                                                  ⋅                (                                  v                                      1                                                  −                                  v                                      1                                    ′                                )                (                                  v                                      1                                                  +                                  v                                      1                                    ′                                )                                                            =                                                                            m                                              2                                                              2                                                  ⋅                (                                  v                                      2                                    ′                                −                                  v                                      2                                                  )                (                                  v                                      2                                    ′                                +                                  v                                      2                                                  )                                                          {\displaystyle {\begin{aligned}{\frac {m_{1}\cdot |v_{1}|^{2}}{2}}+{\frac {m_{2}\cdot |v_{2}|^{2}}{2}}&={\frac {m_{1}\cdot |v_{1}'|^{2}}{2}}+{\frac {m_{2}\cdot |v_{2}'|^{2}}{2}}\\{\frac {m_{1}\cdot v_{1}^{2}}{2}}-{\frac {m_{1}\cdot v_{1}'^{2}}{2}}&={\frac {m_{2}\cdot v_{2}'^{2}}{2}}-{\frac {m_{2}\cdot v_{2}^{2}}{2}}\\{\frac {m_{1}}{2}}\cdot (v_{1}-v_{1}')(v_{1}+v_{1}')&={\frac {m_{2}}{2}}\cdot (v_{2}'-v_{2})(v_{2}'+v_{2})\\\end{aligned}}}  按照动量守恒定律，速度向量为：                                                                        (                                  m                                      1                                                  ⋅                                  v                                      1                                                  )                +                (                                  m                                      2                                                  ⋅                                  v                                      2                                                  )                                                            =                (                                  m                                      1                                                  ⋅                                  v                                      1                                    ′                                )                +                (                                  m                                      2                                                  ⋅                                  v                                      2                                    ′                                )                                                                    (                                  m                                      1                                                  ⋅                                                                                                    v                                                  1                                                                    →                                                                      )                −                (                                  m                                      1                                                  ⋅                                                                                                    v                                                  1                                                ′                                            →                                                                      )                                                            =                (                                  m                                      2                                                  ⋅                                                                                                    v                                                  2                                                ′                                            →                                                                      )                −                (                                  m                                      2                                                  ⋅                                                                                                    v                                                  2                                                                    →                                                                      )                                                                                      m                                      1                                                  ⋅                (                                                                                                    v                                                  1                                                                    →                                                                      −                                                                                                    v                                                  1                                                ′                                            →                                                                      )                                                            =                                  m                                      2                                                  ⋅                (                                                                                                    v                                                  2                                                ′                                            →                                                                      −                                                                                                    v                                                  2                                                                    →                                                                      )                                                          {\displaystyle {\begin{aligned}(m_{1}\cdot v_{1})+(m_{2}\cdot v_{2})&=(m_{1}\cdot v_{1}')+(m_{2}\cdot v_{2}')\\(m_{1}\cdot {\vec {v_{1}}})-(m_{1}\cdot {\vec {v_{1}'}})&=(m_{2}\cdot {\vec {v_{2}'}})-(m_{2}\cdot {\vec {v_{2}}})\\m_{1}\cdot ({\vec {v_{1}}}-{\vec {v_{1}'}})&=m_{2}\cdot ({\vec {v_{2}'}}-{\vec {v_{2}}})\\\end{aligned}}}  动量的方向不可忽略，因为向量和在n维空间（n>1）中是一个大数值。向量平方在能量守恒定律中视作标量。因此请注意，以下算式中速度与碰撞方向相同（相切），而不是相交。 在一维空间中两个物体的速度满足两个未知量                               v                      1                    ′                      {\displaystyle v_{1}'}   、                              v                      2                    ′                      {\displaystyle v_{2}'}   :                              v                      1                    ′                =                                            (                              m                                  1                                            −                              m                                  2                                            )              ⋅                              v                                  1                                            +              2                                             m                                  2                                            ⋅                              v                                  2                                                                                    m                                  1                                            +                              m                                  2                                                                          {\displaystyle v_{1}'={\frac {(m_{1}-m_{2})\cdot v_{1}+2\ m_{2}\cdot v_{2}}{m_{1}+m_{2}}}}                                v                      2                    ′                =                                            (                              m                                  2                                            −                              m                                  1                                            )              ⋅                              v                                  2                                            +              2                                             m                                  1                                            ⋅                              v                                  1                                                                                    m                                  1                                            +                              m                                  2                                                                          {\displaystyle v_{2}'={\frac {(m_{2}-m_{1})\cdot v_{2}+2\ m_{1}\cdot v_{1}}{m_{1}+m_{2}}}}                                m                      1                          =                  m                      2                                {\displaystyle m_{1}=m_{2}}   是以下方程的解:                              v                      1                    ′                =                  v                      2                                {\displaystyle v_{1}'=v_{2}}                                v                      2                    ′                =                  v                      1                                {\displaystyle v_{2}'=v_{1}}  在二维或多维空间中必须将碰撞依据碰撞角拆开分析。