在这里，我们将看看如何解决动量问题，在一维和二维使用守恒定律的线性动量。根据这个定律，只要没有外力作用，粒子系统的总动量就保持不变。因此，解决动量问题需要计算系统在相互作用前后的总动量，并将两者相等。

如何解决动量问题

一维动量问题

例1

一个质量为0.75 kg、速度为5.8 m s-1的球与另一个质量为0.90 kg、同样以2.5 m s-1的速度在同一距离内移动的球相撞。碰撞后，较轻的球以3.0 m s-1的速度在同一方向上移动。求大球的速度。

How to Solve Momentum Problems – Example 1

According to the law of c***ervation of momentum,  .

Taking the direction to the right on this digram to be positive, 

Then, 

例2

一个质量为0.32kg的物体以5m s-1的速度与一个质量为0.90kg的静止物体相撞。碰撞后，两个粒子粘在一起运动。找出它们的速度。

According to the law of c***ervation of momentum,   .

Then, 

例3

一颗质量为0.015公斤的子弹从一把2公斤重的枪上射出。射击后，子弹立即以300米s-1的速度移动。假设枪在射击前是静止的，求出枪的后坐力速度。

Let the recoil speed of the gun be . We will assume the bullet travels in the “positive” direction. The total momentum before firing the bullet is 0. Then,

.

我们认为子弹的方向是肯定的。所以，负号表示枪在回答中行驶，表示枪在相反方向行驶。

例4：弹道摆

The speed of a bullet from a gun can be found by firing a bullet at a suspended wooden block. The height () that the block rises by can be measured. If the mass of the bullet () and the mass of the wooden block () are known, find an expression to calculate the speed of the bullet.

根据动量守恒，我们得到：

(where is the speed of the bullet+block immediately after collision)

根据能量守恒定律，我们得出：

.

Substituting this expression for in the first equation, we have

二维动量问题

As mentioned in the article on the law of c***ervation of linear momentum, to solve momentum problems in 2 dimensi***, one needs to c***ider momenta in   and   directi***. Momentum will be c***erved along each direction separately.

例5

A ball of mass 0.40 kg, traveling at a speed of  2.40 m s-1 along the  axis collides with another ball of mass 0.22 kg traveling at a speed of  mass 0.18, which is at rest. After the collision, the heavier ball travels with a speed of 1.50 m s-1 with an angle 20o to the  axis, as shown below. Calculate the speed and direction of the other ball.

How to Solve Momentum Problems – Example 5

例6

表明当一个物体与另一个静止质量相同的物体发生弹性碰撞时，两个物体之间会以90度的角度移动。

Suppose the initial momentum of the moving body is  . Take the momenta of the two bodies after the collision to be  and  . Since the momentum is c***erved, we can draw up a vector triangle:

How to Solve Momentum Problems – Example 6

since   , we can represent the same vector triangle with vectors  , and . Since  is a common factor to each side of the triangle, we can produce a similar triangle with just the velocities:

How to Solve Momentum Problems – Example 6 Velocity vector Triangle

我们知道碰撞是有弹性的。那么，

.

排除共同因素，我们得到：

According to Pythagors’ theorem, then,  . Since , so then  . The angle between the two bodies’ velocities is indeed 90o. This type of collision is common when playing billiards.