线性动量守恒定律指出，只要没有外力作用于系统，粒子系统的总动量保持不变。等价地，我们也可以说一个封闭的粒子系统的总动量保持不变。在这里，术语封闭系统意味着没有外力作用在系统上。

This holds true even if there are internal forces between particles. If a particle  exerts a force  on a particle  , then the particle  would exert a force of  on  . The these two forces are Newton’s third law pairs, and so they would act for the same duration of time . The change in momentum for particle  is . For particle  , the change in momentum is  . The total change in momentum  within the system is indeed .

一维两物体碰撞时的线动量守恒定律

Suppose an object of mass  is traveling with a velocity  and another object with mass  is traveling with a velocity  . If these two collide, and then the body with mass started traveling at a velocity  and the body with mass started traveling at a velocity  , according to law of c***ervation of momentum,

Law of C***ervation of Linear Momentum – 1D two-body collision

.

Note that for these cases, the correct direction of velocities need to be put into equati***. For instance, if we select the direction to the right to be positive for the above example,  would have a negative value.

物体一维爆炸时的线动量守恒定律

In explosi***, a body breaks into several particles. Examples include firing a bullet from a gun or a radioactive nucleus spontaneously emitting an alpha particle. Suppose a body having a mass  , sitting at rest, breaks into two particles having masses  which travels at a speed , and  which travels at a speed .

Law of C***ervation of Linear Momentum – 1D Explosion

According to the law of c***ervation of momentum,  . Since the initial particle was at rest, its momentum is 0. This means that the momenta of the two **aller particles must also add up to 0. In this case,

同样，只有当速度加上正确的方向时，这才有效。

二维和三维线动量守恒定律

The law of c***ervation of linear momentum applies to 2 and 3 dimensi*** as well. In these cases, we break up momentum into their components along the , and axes. Then, the components of momentum along each direction are c***erved. For example, suppose two colliding bodies have momenta  and before collision, and momenta  and after collision, then,

If the momenta before collision and momenta after collision are all shown in the same vector diagram, they would form a closed shape. For example, if 3 bodies moving in a plane have momenta  ,  and  before collision and momenta  ,  and  after collision, once these vectors are added diagrammatically, they will form a closed shape:

Law of C***ervation of Linear Momentum – Momentum vectors before and after collision, added together, form a closed shape

弹性碰撞-动量守恒

在封闭系统中，总能量总是守恒的。然而，在碰撞过程中，一些能量可能会以热能的形式丢失。因此，在碰撞过程中，碰撞物体的总动能可能会减少。

在弹性碰撞中，碰撞前物体的总动能等于碰撞后物体的总动能。

In reality, most collisi*** that we experience in everyday life are never perfectly elastic, but collisi*** of **ooth, hard spherical objects are nearly elastic. For these collisi***, then you have,   as well as

现在，我们将导出两个物体发生弹性碰撞时的初始速度和最终速度之间的关系：

Law of C***ervation of Linear Momentum – Elastic Collision Velocity Derivation

i、 弹性碰撞后两物体之间的相对速度与碰撞前两物体之间的相对速度大小相同，但方向相反。

Let’s now suppose the masses between the two colliding bodies is equal, i.e. . Then our equati*** become

Law of C***ervation of Linear Momentum – Velocities of Two Bodies After an Elastic Collision

速度在物体之间交换。每个物体在碰撞前都以另一个物体的速度离开碰撞。

非弹性碰撞-动量守恒

在非弹性碰撞中，碰撞前物体的总动能小于碰撞后物体的总动能。

在完全非弹性碰撞中，碰撞后的物体粘在一起。

也就是说，对于两个完全非弹性碰撞的物体，

where  is the velocity of the bodies after collision.

牛顿的摇篮——动量守恒

牛顿的摇篮是如下所示的物体。它由许多质量相等的球形金属球相互接触而成。当任意数量的球从一侧升起并松开时，它们会落下并与其他球碰撞。碰撞后，相同数量的球从另一侧升起。这些球离开时的速度也与碰撞前入射球的速度相等。

What is the Law of C***ervation of Linear Momentum – Newton’s Cradle

We can predict these observati*** mathematically, if we assume the collisi*** to be elastic. Suppose each ball has a mass  . If  is the number of balls initially raised up by a person and  is the number of balls that gets raised as a result of the collision, and if  is the speed of incident balls just before collision and  is the speed of the balls that get raised up after collision,

What is the Law of C***ervation of Linear Momentum – Newton’s Cradle Derivation

i.e. if we raised  balls initially, the same number of balls would get raised after collision.

当球升起时，它们的动能转化为势能。考虑到能量守恒，那么，球上升到的高度将与人将球提升到的高度相同。

References

Giancoli, D. C. (2014). Physics Principles with Applicati***. Pearson Prentice Hall.

Image Courtesy:

“A Newton’s Cradle” by AntHolnes (Own work) [CC BY-SA 3.0], via Wikimedia Comm***