Work, Energy, and Power
AP Physics 1 Topic 3 8:07 English narration · English + 中文 subtitles burned in
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Look up at the first hill of a roller coaster.
抬头看过山车的第一座坡。
A chain drags the train to the top, then lets go.
链条把列车拖到顶端，然后松开。
There is no engine on the train — nothing pushes it again for the whole ride.
列车上没有发动机——接下来的整段行程，再也没有任何东西推它。
Every drop and every loop after this was paid for on the way up.
之后的每一次俯冲、每一个环圈，都是在上坡时就付过的账。
Unit Three follows that payment, in five steps.
第三单元分五步追踪这笔账。
Kinetic energy.
动能。
Work — how a force puts energy in or takes it out.
功——力把能量注入或取出的方式。
Potential energy, stored by position.
势能，仅由位置储存。
Conservation of energy.
能量守恒。
And power: not how much, but how fast.
以及功率：不是有多少，而是有多快。
Anything moving carries kinetic energy: one half the mass, times the speed squared.
任何运动的物体都带有动能：二分之一的质量，乘以速率的平方。
It is a scalar: no direction, and never negative.
它是标量：没有方向，永不为负。
And the speed is squared, so doubling the speed does not double the energy. It makes it four times bigger.
而且速率是平方的， 所以速率加倍并不会让能量加倍，而是变成四倍。
One subtlety AP loves: kinetic energy depends on who is watching.
AP 很爱考的一个细节： 动能取决于谁在观察。
On a moving train your bag has none for you, plenty for the platform.
在行驶的列车上，你的包对你来说动能为零，对站台却很大。
How does energy get in?
能量是怎么进来的？
Through work: energy carried in or out by a force acting over a distance.
通过功：力沿着一段距离把能量带进来或带出去。
But only part of it counts.
但只有一部分算数。
This rope pulls at an angle.
这里绳子以一个角度拉动木块。
The part along the motion, the force times the cosine of the angle, is the only part that does work.
沿着运动方向的那一部分， 也就是力乘以角度的余弦，是唯一做功的部分。
The part at right angles does none: it can steer the block, not speed it up.
与运动垂直的那一部分完全不做功： 它能改变方向，却不能让它加速。
Swing the rope flatter, and the useful part grows.
把绳子拉得更平，有用的那一部分就会变大。
Work is that parallel part, times the distance.
功就是这个平行分量，乘以移动的距离。
The cosine also fixes the sign.
余弦同时决定了正负号。
Push along the motion: the angle is zero and the work is positive.
沿运动方向推：角度为零，功为正。
Push against it — friction, or a brake — the angle is a hundred and eighty degrees, so the work is negative.
逆着运动方向——摩擦力，或刹车——角度是一百八十度，功为负。
At right angles the cosine is zero: no work at all.
垂直方向余弦为零：完全不做功。
Carry a bag across a flat room, and none on the bag.
端着袋子横穿平地，对袋子不做功。
If the force changes as you go, work is the area under the force-position graph.
如果力在过程中变化，功就是力—位移图线下的面积。
Now the theorem tying them together.
现在是把两者联系起来的定理。
The net work on an object equals the change in its kinetic energy.
物体所受的合功等于它动能的变化。
Not one force: the total from every force.
不是某一个力，而是所有力做功的总和。
Positive net work and it speeds up; negative and it slows down; zero and nothing changes.
合功为正，物体加速；为负，物体减速； 为零，速率不变。
See the example: two kilograms, three to five metres per second.
看这个例子：两千克，从每秒三米到每秒五米。
Its kinetic energy rises from nine joules to twenty-five, so the net work was sixteen.
它的动能从九焦耳升到二十五焦耳，所以合功是十六焦耳。
A two-kilogram block slides at three metres per second on a smooth floor.
一个两千克的物块在光滑地面上以每秒三米滑行。
A five-newton force pushes it four metres forward.
一个五牛顿的力沿运动方向推它四米。
How fast is it then?
那时它有多快？
Pause here and try it.
先暂停，自己试一试。
First the work: five newtons times four metres is twenty joules.
先算功：五牛顿乘以四米，是二十焦耳。
That twenty joules is the change in kinetic energy.
这二十焦耳就是动能的变化。
It began with nine joules, so it ends with twenty-nine.
它原来有九焦耳，所以最后是二十九焦耳。
Back to a speed: five point four metres per second.
再换算回速率：每秒五点四米。
No forces, no times — only energy.
不用力，也不用时间——只用能量。
Lift a book straight up, or carry it up a winding staircase to the same shelf: gravity does the same work.
把一本书竖直举上去，或者沿旋转楼梯搬到同一层书架：重力做的功完全相同。
Only the start and the end matter, never the path.
只有起点和终点重要，路径从不重要。
Gravity is a conservative force, and so is a spring.
重力是保守力，弹簧也是。
Bring it down again and gravity returns every joule — around a closed loop its work is zero.
把它放回原处，重力会把每一焦耳还回来——沿闭合路径一周，它做的功为零。
Friction is the opposite: take the long way and it takes more, and gives none back.
摩擦力正相反：绕远路走，它消耗得更多，而且从不归还。
Only conservative forces store potential energy: energy an object has because of where it is, not because it moves.
只有保守力才储存势能：势能是物体因为所处的位置、而不是因为在运动所具有的能量。
Raise a mass and you store gravitational potential energy — the mass, times the field strength, times the height gained.
把一个质量抬高，你就储存了重力势能——质量，乘以重力场强度，乘以升高的高度。
Let it fall and every joule comes back as kinetic energy.
让它落下，每一焦耳都会变回动能。
And you choose where height counts as zero, because only the change ever reaches the answer.
而且零高度的位置由你自己选， 因为只有变化量才会进入答案。
Springs store elastic potential energy.
弹簧也储存能量。
Hang equal masses on a spring and each adds the same extension, so the force-extension graph is a straight line.
在弹簧上挂等重的砝码，每一个都增加同样的伸长量， 所以力—伸长图是一条直线。
The energy stored is the area under that line: a triangle, one half the spring constant times the extension squared.
储存的能量就是这条直线下的面积：一个三角形， 等于二分之一的劲度系数乘以伸长量的平方。
Squared again: stretch it twice as far and it stores four times the energy.
又是平方：把它拉长一倍， 储存的能量是四倍。
That is why a fully drawn bow shoots so far.
这就是弓拉得越满、射得越远的原因。
A warning about that height formula: it only works close to a surface, where the gravitational field is nearly constant.
关于那个高度公式，有一点要小心：它只在靠近表面、重力场几乎恒定的地方成立。
Further out you need the general form — minus the gravitational constant, times the two masses, divided by the distance.
更远处你需要一般形式——负的万有引力常量，乘以两个质量，除以它们之间的距离。
It is negative on purpose: we set the energy to zero at infinite separation.
它是负的，而且是故意的：我们把相距无限远时的能量定为零。
So as a satellite climbs away, its potential energy rises toward zero.
所以当卫星飞离时，它的势能朝零上升。
Here is the payoff.
关键在这里。
Kinetic plus potential is the mechanical energy, and when only conservative forces do work, that total never changes.
动能加势能就是机械能；当只有保守力做功时，这个总和永远不变。
Watch the pendulum.
看这个单摆。
At the ends it stops for an instant: all potential.
在两端它瞬间静止：全是势能。
At the bottom it is fastest: all kinetic.
在最低点它最快：全是动能。
The bars trade back and forth, but their sum stays level.
两根能量条来回交换，但它们的和始终齐平。
So kinetic plus potential at the start equals kinetic plus potential at the end.
所以开始时的动能加势能，等于结束时的动能加势能。
Two more ideas the exam loves.
还有两个考试常考的要点。
First, friction: rub a surface and mechanical energy becomes thermal energy.
第一，摩擦：摩擦会把机械能变成热能。
Nothing is destroyed — total energy is always conserved — but mechanical energy is not, so subtract what friction took.
什么都没有被消灭——总能量始终守恒——但机械能并不守恒， 所以要减去摩擦拿走的那部分。
Second: the answer depends on the system you choose.
第二：答案取决于你选择的系统。
Take the ball alone: gravity acts from outside, so its energy changes.
只取小球：重力来自系统外部，它的能量会变化。
Take the ball and the Earth together, and it becomes potential energy inside.
把小球和地球一起取，它就变成系统内部的势能。
A ball starts from rest at the top of a smooth ramp, five metres high.
一个球从五米高的光滑斜坡顶端由静止释放。
How fast is it at the bottom?
它在底端有多快？
Pause and try it.
先暂停，自己试一试。
At the top the energy is all potential: mass, times gravity, times height.
在顶端，能量全是势能：质量，乘以重力，乘以高度。
At the bottom, all kinetic: one half mass, times speed squared.
在底端全是动能：二分之一质量，乘以速率的平方。
Set them equal: the mass cancels, so the answer never depends on it.
令两者相等：质量被约掉了， 所以答案从不取决于它。
That leaves twice gravity times height — about ninety-eight — so nine point nine metres per second.
剩下二乘以重力再乘以高度——约等于九十八—— 所以是每秒九点九米。
The shape of the ramp never mattered.
斜坡的形状从来无关紧要。
Last idea: power — not how much energy, but how fast.
最后一个概念：功率。 功率不是能量有多少，而是有多快。
Divide the energy transferred by the time taken and you get the average power, in watts — one joule a second.
用转移的能量除以所用的时间，就得到平均功率，单位是瓦特——每秒一焦耳。
On a graph of work against time, power is the slope.
在功对时间的图上，功率就是斜率。
Both machines here do the same work, but the steep line finishes sooner — high power — while the shallow one takes longer, so its power is lower.
图中两台机器做的功完全相同， 但陡的那条线更早完成——功率高——而平缓的那条用时更久，功率更低。
At a steady speed there is a shortcut: power equals force times velocity.
匀速运动时还有一个捷径：功率等于力乘以速度。
A motor lifts a fifty-kilogram load at a steady two metres per second.
一台马达以每秒两米的恒定速度提升五十千克的重物。
What power does it deliver, and what is its efficiency?
它输出多大功率？
Because the speed is steady, nothing accelerates, so the lifting force balances the weight — fifty times nine point eight, four hundred and ninety newtons.
由于速度恒定，没有加速度，所以提升力平衡重力——五十乘以九点八，四百九十牛顿。
Power is force times velocity: four hundred and ninety times two, nine hundred and eighty watts.
功率等于力乘以速度：四百九十乘以二，九百八十瓦特。
No real motor is perfect: this one draws fourteen hundred watts, so it is seventy per cent efficient.
没有哪台真实马达是完美的：这台要消耗一千四百瓦，所以效率是百分之七十。
Four habits before you go.
结束之前，四个习惯。
First, check the angle: work is zero when the force is perpendicular, and negative when it opposes the motion.
第一，先看角度：力与运动垂直时功为零，与运动相反时功为负。
Second, when the path is complicated, reach for energy instead of forces: the work-energy theorem beats Newton's laws almost every time.
第二，当路径复杂时，用能量而不是用力：动能定理几乎每次都胜过牛顿定律。
Third, when friction acts, mechanical energy is not conserved — subtract the heat first.
第三，有摩擦时机械能不守恒——先减去变成热的那部分。
Fourth, kinetic energy goes as the speed squared: double the speed and you quadruple the energy, and the stopping distance.
第四，动能正比于速率的平方：速率加倍，能量和刹车距离都变成四倍。