Linear Momentum
AP Physics 1 Topic 4 7:49 English narration · English + 中文 subtitles burned in
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Five steel balls hang in a row, just touching.
五个钢球排成一行，彼此刚好接触。
Lift one, let it go, and one ball flies off the far end.
抬起一个，放开，远端就飞出一个球。
Only one.
只有一个。
Lift two, and exactly two fly off.
抬起两个，就正好飞出两个。
Something passes along that line of steel, and it is counted exactly — never one and a half.
有某种东西沿着这排钢球传递，而且被精确地清点着——绝不会是一个半。
That something is momentum.
那就是动量。
Unit four, in four steps.
第四单元，分四步。
First, what momentum is, and why it is a vector.
第一，什么是动量，以及它为什么是矢量。
Then impulse, and the theorem that links a force to a change in momentum.
然后是冲量，以及把力和动量变化联系起来的定理。
Then the big law: momentum is conserved.
接着是那条重要的定律：动量守恒。
And last, the two kinds of collision, elastic and inelastic.
最后是两种碰撞，弹性碰撞和非弹性碰撞。
Linear momentum is mass in motion.
动量就是运动中的质量。
It is the mass multiplied by the velocity: that is the whole definition.
它等于质量乘以速度，这就是它的全部定义。
It is a vector: it points the way the object is moving, so a total momentum is a vector sum, with signs.
它是矢量：方向正好是物体运动的方向，所以总动量要按矢量相加，带上正负号。
And it grows with both mass and speed.
它随质量和速度一起变大。
On the right: two kilograms at three metres per second, and six kilograms at one.
看右边：二千克、每秒三米，和六千克、每秒一米。
Different objects, the same momentum.
不同的物体，同样的动量。
A net force acting for a time changes momentum.
一个合力持续作用一段时间，就会改变动量。
Force times the time it acts is the impulse.
力乘以它作用的时间，这个乘积叫做冲量。
The impulse-momentum theorem says the impulse equals the change in momentum.
冲量动量定理说：冲量等于动量的变化。
On a graph of force against time, it is the area under the line.
在力对时间的图上，它就是线下面的面积。
Now read the bottom box.
现在看下面那个方框。
For a fixed change in momentum, a longer contact time means a smaller force.
对于固定的动量变化，接触时间越长，力就越小。
That one sentence is the airbag.
这一句话就是安全气囊的原理。
In a real collision the force is not steady.
在真实的碰撞中，力并不稳定。
It grows to a peak and dies away, like this.
它升到一个峰值，然后减小，就像这样。
The impulse is still the area under the curve.
冲量仍然是曲线下面的面积。
We often replace it with one steady average force covering the same area.
我们常常把它换成一个稳定的平均力，覆盖同样的面积。
That is the dashed line.
那就是虚线。
Same area, same change in momentum.
面积相同，动量的变化就相同。
So a tall short spike and a gentle long push can deliver the same impulse.
所以又高又短的冲击，和温和而漫长的推力，可以给出相同的冲量。
A zero point one five kilogram ball hits a wall at twenty metres per second and bounces straight back at fifteen.
一个零点一五千克的球以每秒二十米撞墙，然后以每秒十五米原路弹回。
The contact lasts zero point zero two zero seconds.
接触时间是零点零二零秒。
Find the average force.
求平均力。
First, choose a positive direction.
首先，选定一个正方向。
Take the rebound as positive, so the ball arrives at minus twenty and leaves at plus fifteen.
取弹回的方向为正，于是球以负二十进来，以正十五离开。
The change in velocity is thirty-five, not five.
速度的变化是三十五，不是五。
Multiply by the mass, and the change in momentum is five point two five.
乘以质量，动量的变化就是五点二五。
Divide that by the contact time: two hundred and sixty newtons.
再除以接触时间：二百六十牛顿。
A bounce changes momentum far more than a stop.
会弹回的球，动量变化远大于只是停下来的球。
Now the law that makes momentum so useful.
现在来看让动量如此有用的那条定律。
If the net external force on a system is zero, that system is isolated, and its total momentum stays constant.
如果一个系统受到的外力合力为零， 这个系统就是孤立的，它的总动量保持不变。
Why?
为什么？
Inside, the objects push on each other in third-law pairs, so the internal forces cancel.
在内部，物体之间按牛顿第三定律成对地互相推，所以内力正好抵消。
Only an outside force can change the total.
只有外力才能改变总动量。
And it holds for collisions and explosions alike.
而且它对碰撞和爆炸同样成立。
Conservation of linear momentum — the whole method is in one picture.
整套方法就在这一张图上。
Before: A moves right, B moves left.
碰撞前：A 向右运动，B 向左运动。
Add their momenta as vectors, with signs.
把它们的动量按矢量相加，带上正负号。
After: A has bounced back to the left, B moves right.
碰撞后：A 被弹回向左，B 向右运动。
Add those too.
也把它们加起来。
The two totals are equal, always.
两个总和总是相等。
You never need to know the force, or how long the contact lasted.
你完全不需要知道力有多大，也不需要知道接触持续了多久。
Watch a real case.
来看一个真实的例子。
A two kilogram trolley rolls at three metres per second into a stationary one kilogram trolley, and they lock together.
一辆二千克的小车以每秒三米撞上一辆静止的一千克小车， 两车连在一起。
Before the crash the total is six units, because only the moving trolley contributes.
碰撞前总动量是六个单位，因为只有运动的那辆车有贡献。
Afterwards three kilograms move together, so they must travel at two metres per second.
碰撞后三千克一起运动，所以它们必须以每秒二米前进。
Draw that same crash as momentum against time.
把同一次碰撞画成动量对时间的图。
Trolley A loses momentum, so its line falls.
小车 A 失去动量，所以它的线下降。
Trolley B gains it, so its line rises.
小车 B 得到动量，所以它的线上升。
The total is flat, unchanged straight through the collision.
总动量是一条水平线，在整个碰撞过程中都没有变。
And the deeper point: the slope of a momentum line is the force on that object.
更深一层的意思是：动量线的斜率就是作用在那个物体上的力。
The two slopes are equal and opposite, exactly as Newton's third law demands.
两个斜率大小相等、方向相反，这正是牛顿第三定律要求的。
Explosions work the same way, backwards.
爆炸的道理一样，只是反过来。
A sixty kilogram skater stands still on frictionless ice and throws a two kilogram ball forward at eight metres per second.
一位六十千克的滑冰者静止站在无摩擦的冰面上， 把一个二千克的球以每秒八米向前扔出。
How fast does she slide back?
她会以多快的速度向后滑？
Nothing moves at the start, so the total momentum is zero.
开始时什么都没有动，所以总动量为零。
It must still be zero afterwards.
之后它必须仍然为零。
The ball carries sixteen units forward, so the skater must carry sixteen units backward.
球带走向前的十六个单位，所以滑冰者必须带走向后的十六个单位。
Divide by her sixty kilograms: zero point two seven metres per second.
除以她的六十千克：每秒零点二七米。
That is how a rifle kicks, and how a rocket flies.
步枪的后坐力和火箭的飞行，正是这个道理。
One more idea the exam likes.
还有一个考试喜欢的概念。
A group of objects can be described by a single point, the centre of mass, moving with one velocity.
一组物体可以用一个点来描述，那就是质心，它以一个速度运动。
That velocity is the total momentum divided by the total mass.
这个速度等于总动量除以总质量。
Two carts sit at rest on ice, with a spring squeezed between them.
两辆小车静止在冰面上，中间夹着一个被压紧的弹簧。
Release it.
放开它。
They fly apart, the light one three times faster.
它们向两边飞开，轻的那辆快三倍。
But the centre of mass has not moved at all.
但是质心完全没有移动。
An internal push cannot move that point.
内部的推力不能移动这个点。
Momentum is conserved in every collision.
动量在每一次碰撞中都守恒。
Kinetic energy is not, and that splits collisions into two kinds.
动能却不一定，这正好把碰撞分成两类。
On top, an elastic collision: the objects bounce apart and the total kinetic energy is unchanged.
上面一行是弹性碰撞：物体弹开，总动能保持不变。
Underneath, an inelastic collision: some kinetic energy has become heat and sound. If they also stick together, it is perfectly inelastic.
下面一行是非弹性碰撞：它们粘在一起，一部分动能变成了热和声音。
Momentum is conserved in both rows.
这两行的动量都守恒。
Now we can answer the hook.
现在我们可以回答开头的问题了。
In an elastic collision between two equal masses, the moving ball stops dead and the still one leaves at the incoming speed.
在两个质量相等的物体之间发生弹性碰撞时， 运动的球会立刻停住，静止的球以原来的速度离开。
They simply swap velocities.
它们只是交换了速度。
A Newton's cradle is that swap, repeated down the line to the last ball.
牛顿摆就是这个交换沿着这排球一路重复，直到最后一个球。
One ball in, one ball out.
进去一个球，出来一个球。
Two balls out would carry the right momentum but too little kinetic energy.
飞出两个球虽然动量对，动能却太小。
The full exam question.
来看完整的考题。
A one thousand kilogram car at twenty metres per second hits a stationary fifteen hundred kilogram car, and they lock together.
一辆一千千克的汽车以每秒二十米撞上一辆静止的一千五百千克汽车， 两车锁在一起。
Find their common speed, and the kinetic energy lost.
求它们共同的速度，以及损失的动能。
Momentum first.
先看动量。
Before, only the moving car counts: twenty thousand units.
碰撞前只有运动的那辆车有贡献：二万个单位。
Two and a half tonnes then move as one, so the common speed is eight metres per second.
之后二点五吨一起运动， 所以共同速度是每秒八米。
Now the energy.
再看能量。
Before, two hundred thousand joules; after, eighty thousand.
碰撞前是二十万焦耳；碰撞后是八万焦耳。
So about sixty per cent went into crumpling and heat.
所以大约百分之六十变成了车身变形和热。
Momentum survived.
动量保住了。
Kinetic energy did not.
动能没有。
Real collisions are rarely in a straight line.
真实的碰撞很少沿着一条直线。
Here one ball arrives along the horizontal axis and two leave at angles.
这里一个球沿水平轴射入，两个球以一定角度离开。
The rule does not change; you apply it twice.
规则没有改变，你只要用两次。
Across the page, the momenta before and after must match.
沿横向，碰撞前后的动量必须相等。
Up and down the page they must match too, and since nothing moved that way at the start, those two parts cancel.
沿纵向也必须相等，而由于开始时纵向没有任何运动，这两部分必须互相抵消。
Four marks students throw away.
四个学生常丢的分。
One: momentum is a vector.
第一：动量是矢量。
Write down a positive direction before anything else, and remember that a rebound reverses the sign.
在写别的之前，先写下一个正方向， 并且记住弹回会让符号反过来。
Two: momentum is conserved in every collision, but kinetic energy only when the question says elastic.
第二：动量在每一次碰撞中都守恒， 但动能只有在题目说明是弹性碰撞时才守恒。
Three: impulse is force times time, and it is the area under a force-time graph.
第三：冲量等于力乘以时间，在力对时间的图上就是面积。
Four: for recoil and explosions, the total before is usually zero.
第四：反冲和爆炸，碰撞前的总动量通常是零。