線性動量守恆定律指出，只要沒有外力作用於系統，粒子系統的總動量保持不變。等價地，我們也可以說一個封閉的粒子系統的總動量保持不變。在這裡，術語封閉系統意味著沒有外力作用在系統上。

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 .

一維兩物體碰撞時的線動量守恆定律

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.

物體一維爆炸時的線動量守恆定律

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,

同樣，只有當速度加上正確的方向時，這才有效。

二維和三維線動量守恆定律

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

彈性碰撞-動量守恆

在封閉系統中，總能量總是守恆的。然而，在碰撞過程中，一些能量可能會以熱能的形式丟失。因此，在碰撞過程中，碰撞物體的總動能可能會減少。

在彈性碰撞中，碰撞前物體的總動能等於碰撞後物體的總動能。

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

現在，我們將匯出兩個物體發生彈性碰撞時的初始速度和最終速度之間的關係：

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

i、 彈性碰撞後兩物體之間的相對速度與碰撞前兩物體之間的相對速度大小相同，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

速度在物體之間交換。每個物體在碰撞前都以另一個物體的速度離開碰撞。

非彈性碰撞-動量守恆

在非彈性碰撞中，碰撞前物體的總動能小於碰撞後物體的總動能。

在完全非彈性碰撞中，碰撞後的物體粘在一起。

也就是說，對於兩個完全非彈性碰撞的物體，

where  is the velocity of the bodies after collision.

牛頓的搖籃——動量守恆

牛頓的搖籃是如下所示的物體。它由許多質量相等的球形金屬球相互接觸而成。當任意數量的球從一側升起並鬆開時，它們會落下並與其他球碰撞。碰撞後，相同數量的球從另一側升起。這些球離開時的速度也與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.

當球升起時，它們的動能轉化為勢能。考慮到能量守恆，那麼，球上升到的高度將與人將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***