Linear Momentum
AP Physics C: Mechanics Topic 4 7:00 English narration · English + 中文 subtitles burned in
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Transcript
A car stops in a crash.
汽车在碰撞中停下。
With an airbag, the driver walks away. Without one, they do not.
有安全气囊，司机可以走下车；没有安全气囊，就不行。
In both cases the momentum change is exactly the same — from full speed, down to zero.
在这两种情况下，动量的变化完全相同——从全速降到零。
Only the time is different.
不同的只有时间。
And that single difference decides whether the force is survivable.
而正是这一个差别，决定了受到的力是否还能承受。
This is Unit Four: linear momentum.
这是第四单元：线动量。
The quantity that survives every collision, however violent.
无论碰撞多么剧烈，它都能守恒下来。
We will meet impulse, conservation, and the difference between elastic and inelastic.
我们会学到冲量、守恒，以及弹性碰撞与非弹性碰撞的区别。
Let's begin.
让我们开始吧。
Momentum is mass times velocity.
动量等于质量乘以速度。
It is a vector, pointing exactly along the velocity — so a momentum to the left is a negative momentum, and the sign does real work in every problem.
它是矢量，方向与速度完全一致—— 所以向左的动量就是负动量，这个符号在每道题里都真正起作用。
A heavy lorry moving slowly can carry the same momentum as a light motorbike moving fast.
一辆缓慢行驶的重卡，可以和一辆高速行驶的轻摩托拥有相同的动量。
For a group of objects, add the momenta as vectors.
对一组物体，把各自的动量按矢量相加。
There is a neat shortcut: the total momentum equals the total mass, times the velocity of the centre of mass.
有一个漂亮的捷径： 总动量等于总质量乘以质心的速度。
Any number of parts becomes one object's worth of bookkeeping.
不论有多少个部分，都简化成一个物体的账。
Newton's second law is really a statement about momentum: the net force is the rate of change of momentum.
牛顿第二定律其实是关于动量的陈述：合力等于动量的变化率。
When the mass is constant that becomes mass times acceleration, but the momentum form also handles a changing mass.
当质量不变时，它变成质量乘以加速度；而动量形式还能处理质量变化的情形。
The impulse of a force is its integral over time, and the impulse-momentum theorem says that impulse equals the change in momentum.
力的冲量是它对时间的积分，冲量—动量定理说：冲量等于动量的变化。
Read it off graphs both ways. On a force-time graph, the impulse is the area under the curve. On a momentum-time graph, the net force is the slope.
从图像上可以两头读它：在力—时间图上，冲量是曲线下的面积； 在动量—时间图上，合力是斜率。
A force of ten t newtons acts on a two kilogram object for two seconds.
一个大小为十 t 牛顿的力作用在两千克的物体上，持续两秒。
Find the change in its speed.
求它速度的变化。
Integrate the force over the time. The integral of ten t is five t squared, so between zero and two the impulse is twenty newton seconds.
对力关于时间积分：十 t 的积分是五 t 平方，所以从零到二，冲量是二十牛顿秒。
Or read it as the triangle under the line — the same twenty.
也可以读作直线下的那个三角形面积——同样是二十。
Divide by the two kilogram mass, and the speed changes by ten metres per second.
除以两千克的质量， 速度的变化就是十米每秒。
Now the payoff.
现在来看回报。
Two force-time curves can have exactly the same area — the same impulse, the same momentum change — and look completely different.
两条力—时间曲线可以有完全相同的面积—— 相同的冲量、相同的动量变化——但形状却截然不同。
The one that lasts longer has a much lower peak force.
持续时间更长的那条，峰值力要小得多。
That is the whole idea behind an airbag, a crumple zone, a landing mat, and bending your knees when you jump down.
这正是安全气囊、溃缩区、落地垫，以及跳下时屈膝的全部道理。
You cannot change the momentum you must lose. You can choose how long you take to lose it.
你无法改变必须失去的动量，但你可以选择用多长时间去失去它。
Inside any system, forces come in third-law pairs.
在任何系统内部，力都成对出现，构成第三定律的作用与反作用对。
They can shuffle momentum from one part to another, but they can never change the total.
它们可以把动量从一部分转移到另一部分，却永远不能改变总量。
Momentum only enters or leaves through a net external force.
动量只能通过外力的合力进出系统。
So the condition is simple: if the net external force is zero, the total momentum before equals the total momentum after.
所以条件很简单： 如果外力的合力为零，碰撞前的总动量就等于碰撞后的总动量。
Choose the system large enough that the collision forces are internal, and momentum is conserved in every collision, however violent.
把系统取得足够大，使碰撞力都成为内力，那么无论碰撞多么剧烈，动量都守恒。
And apply it one axis at a time.
而且要一个轴一个轴地用。
A six kilogram shell sits at rest, then explodes into two pieces.
一个六千克的炮弹静止，然后炸成两块。
A two kilogram piece flies east at nine metres per second.
两千克的一块以九米每秒向东飞出。 另一块会怎样？
What does the other piece do?
爆炸前总动量为零。
Before the explosion the total momentum is zero. The explosion is internal, so the total must still be zero afterwards — so the four kilogram piece must carry an equal momentum the other way.
爆炸是内部作用，所以之后总动量仍必须为零—— 因此四千克的那一块必须带着大小相等、方向相反的动量。
Two times nine is eighteen, divided by four gives four point five metres per second, west.
二乘九等于十八，再除以四，得到四点五米每秒，方向向西。
Notice the kinetic energy went up, from nothing to something. It came from stored chemical energy.
注意动能从零增加了，它来自储存的化学能。
Momentum conservation never promises energy conservation.
动量守恒从不保证能量守恒。
Watch a collision.
看一次碰撞。
Whatever happens in the middle — however messy, however violent — the total momentum before and after is the same.
不管中间发生了什么——不管多么混乱、多么剧烈—— 碰撞前后的总动量都相同。
That is what makes momentum so useful: you never need to know the details of the contact. You only need the states before and after.
这正是动量如此好用的原因： 你完全不需要知道接触过程的细节，只需要知道碰撞前后的状态。
All collisions conserve momentum. They differ in what happens to the kinetic energy.
所有碰撞都守恒动量，区别在于动能发生了什么。
In an elastic collision the total kinetic energy is also conserved, although the shares change.
在弹性碰撞中，总动能也守恒，只是各自的份额发生变化。
In an inelastic collision some kinetic energy becomes heat, sound and deformation, so the total falls.
在非弹性碰撞中，一部分动能变成热、声和形变，所以总动能下降。
In a perfectly inelastic collision the objects stick together, move off with one shared velocity, and lose the most kinetic energy possible.
在完全非弹性碰撞中，物体粘在一起，以同一个速度离开，动能损失最大。
So here is the strategy: always write momentum conservation first. Add the energy equation only when the question actually says elastic.
所以策略是：永远先写动量守恒；只有当题目明确说「弹性」时，才加上能量方程。
Two useful elastic facts: equal masses in one dimension simply exchange velocities, and in the centre-of-mass frame each object just reverses its velocity.
两个有用的弹性事实：一维等质量会直接交换速度；在质心系里，每个物体只是把速度反向。
A one thousand kilogram car travelling at twenty metres per second hits a stationary fifteen hundred kilogram car, and they lock together. Pause here and try it.
一辆一千千克的汽车以二十米每秒撞上静止的一千五百千克汽车，两车锁在一起。
Momentum first: twenty thousand kilogram metres per second, shared by two and a half thousand kilograms, gives eight metres per second.
先暂停，自己试一试。 先看动量：两万千克米每秒，由两千五百千克共同分担， 得到八米每秒。
Now audit the energy.
再来核对能量。
Before: two hundred kilojoules. After: one half times two and a half thousand times eight squared, which is eighty kilojoules.
碰撞前是二十万焦耳；碰撞后是二分之一乘两千五百再乘八的平方， 等于八万焦耳。
Sixty percent of the kinetic energy has gone — and yet the momentum is exactly conserved.
百分之六十的动能消失了——而动量却完全守恒。
In two dimensions nothing new is needed.
在二维情况下不需要任何新东西。
Split the collision into two independent equations, one per axis.
把碰撞拆成两个独立的方程，每个轴一个。
A puck moving east at four metres per second strikes an identical puck at rest.
一个冰球以四米每秒向东运动，撞上一个静止的同样的冰球。
Afterwards one puck moves at two metres per second, sixty degrees north of east.
碰后其中一个以两米每秒、沿北偏东三十度方向（即东偏北六十度）运动。
Along the east axis: four equals two times the cosine of sixty, plus the unknown — so the unknown is three.
沿东西轴：四等于二乘六十度的余弦，加上未知量——所以未知量是三。
Along the north axis: zero equals two times the sine of sixty, minus the unknown — so the unknown is one point seven.
沿南北轴：零等于二乘六十度的正弦，减去未知量——所以未知量是一点七。
Combine the two components: three point five metres per second, about thirty degrees south of east.
把两个分量合成：三点五米每秒，大约在东偏南三十度方向。
Three habits that save marks.
三个能保住分数的习惯。
First, state the condition, not the slogan: say that the net external force on the system is zero during the collision, so its total momentum is constant.
第一，写出条件，而不是口号： 要说明碰撞过程中系统受到的外力合力为零，所以总动量不变。
Second, never assume a collision is elastic — compute the kinetic energy before and after and compare them.
第二，绝不要假定碰撞是弹性的——把碰撞前后的动能算出来比较。
Third, in two dimensions write one conservation equation per axis, and solve them separately.
第三，在二维情况下，每个轴写一条守恒方程，分别求解。
Get those three, and this unit is yours.
做到这三点，这个单元就是你的了。