Work & Energy
A-Level Physics Topic 5 17:09 English narration · English + 中文 subtitles burned in
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A roller coaster has no engine on its track.
过山车的轨道上没有发动机。
A chain hauls it up the very first hill — and after that, it is on its own.
一条链条把它拉上第一座山坡——之后，它就全靠自己了。
So how does it race through every loop and dive that follows?
那么它怎么冲过后面的每一个环圈和俯冲？
Energy.
靠能量。
At the top, it is full of stored energy of height.
在最高点，它满是储存的高度的能量。
As it drops, that height becomes speed.
往下俯冲时，高度就变成速度。
Climb again, and speed becomes height.
再次爬升，速度又变回高度。
The total never changes — it just trades back and forth.
总量从不改变——只是来回转换。
This is the story of energy: work, power, and the great law that energy is never lost.
这是关于能量的故事：功、功率，以及那条伟大的定律——能量永不消失。
Today, how forces do work, how energy changes form, and how fast we can move it.
今天，我们看力如何做功、能量如何改变形式，以及我们能多快地传递它。
Let's begin.
让我们开始吧。
Everywhere around us, energy is changing form.
我们身边，能量处处在改变形式。
These wind turbines take the kinetic energy of moving air and turn it into electrical energy for the grid.
这些风力发电机把流动空气的动能，变成送入电网的电能。
A car engine burns chemical energy into thermal energy and motion.
汽车发动机把化学能烧成热能和运动。
Your body turns food into the work of walking upstairs.
你的身体把食物变成上楼行走的功。
The same story every time: one form becomes another, and nothing is created from nothing.
故事总是一样：一种形式变成另一种，没有任何东西凭空产生。
Work is done when a force moves something along its own direction.
当一个力使物体沿力自身的方向移动时，就做了功。
Work equals the force, times the distance moved in the direction of the force — times the cosine of the angle between them.
功等于力，乘以距离， 再乘以两者夹角的余弦。
Only the part of the force along the motion counts.
只有沿运动方向的那部分力才算数。
Pull a sledge with twenty newtons, at sixty degrees, over five metres, and the work is fifty joules.
用二十牛顿、成六十度角、 拉雪橇走五米，做的功是五十焦耳。
If the force points along the motion, the work is positive.
如果力沿着运动方向，功为正。
At right angles, it does no work at all.
成直角时，完全不做功。
And against the motion — like friction — the work is negative.
而与运动方向相反时——比如摩擦力——功为负。
Say the definition the way the mark scheme does: work done is the product of the force and the distance moved in the direction of the force.
定义要照评分标准的说法讲： 所做的功，等于力与沿力的方向移动的距离的乘积。
The last clause is the whole definition — a force at right angles to the motion does no work at all, however large it is.
最后那半句才是定义的关键—— 与运动方向垂直的力，不管多大，做的功都是零。
Look carefully at this diagram.
仔细看这张图。
The force F is pulled at an angle theta to the displacement s.
力 F 与位移 s 成一个角度西塔。
Split the force into two parts: F cos theta along the motion, and F sin theta straight up, at right angles to the path.
把力拆成两部分：沿运动方向的 F 余弦西塔， 以及垂直于路径的 F 正弦西塔。
Only the along-the-path part does work.
只有沿路径的那部分做功。
The perpendicular part never moves its point of contact along its own line, so it contributes nothing.
垂直部分从未沿自身方向移动作用点， 所以它贡献为零。
That is why the formula always multiplies by the cosine.
这就是公式里总要乘以余弦的原因。
Positive and negative work tell you the direction of energy transfer.
正功和负功告诉你能量传递的方向。
When the force points the same way as the motion, work is positive, and energy is given to the object.
力与运动同向时，功为正，能量交给物体。
When the force points opposite — friction is the classic case — work is negative, and energy is taken from the object.
力与运动反向时——摩擦是典型例子——功为负，能量从物体取走。
The normal contact force on a car rolling on a flat road is at ninety degrees to the motion, so it does no work at all.
平路上行驶的汽车所受的支持力与运动成九十度，所以完全不做功。
And remember: work is a scalar.
还要记住：功是标量。
Its unit is the joule, which is one newton times one metre.
它的单位是焦耳，等于一牛顿乘以一米。
Now put the object on a slope.
现在把物体放在斜坡上。
Climbing a height h against gravity always costs the same work: mass times g times h — whatever path you take.
克服重力爬升高度 h，所做的功总是一样：质量乘以 g 再乘以 h—— 与路径无关。
But if you use a horizontal push over the slope length L, you use the cosine of the slope angle alpha.
但如果你用水平力沿斜坡长度 L 去推，就要用到坡角阿尔法的余弦。
Two different questions, two different formulas.
两个不同的问题，两套不同的公式。
Never confuse the height with the distance along the ramp.
绝不要把高度和沿坡距离搞混。
Energy is never created and never destroyed.
能量既不能被创造，也不能被消灭。
It only changes form, or moves from one place to another.
它只会改变形式，或从一处转移到另一处。
Watch this pendulum.
看这个单摆。
At the top of its swing, all its energy is stored as height — potential energy.
在摆动的最高点，它的能量全部以高度的形式储存——势能。
At the bottom, it moves fastest, and the energy is all kinetic.
在最低点，它运动最快， 能量全是动能。
Up and down it trades, back and forth, but the total stays exactly the same.
它上下往复地转换，来来回回，但总量始终完全不变。
This is the conservation of energy.
这就是能量守恒。
And the principle itself, in the examiner's words: energy cannot be created or destroyed; it can only be transferred from one form to another.
而这条原理本身，用考官的原话是： 能量不能被创造，也不能被消灭；它只能从一种形式转移到另一种形式。
That is the sentence to write; a paraphrase about energy being "used up" contradicts it.
要写的就是这一句； 说能量被「用掉了」的那种改写，恰恰和它相反。
When you write an energy equation, list every form the energy starts as and ends as.
写能量方程时，列出能量起始和结束的每一种形式。
Common forms in this syllabus are kinetic, gravitational potential, then also elastic, electrical, thermal, and chemical energy — and sound, too.
本大纲常见的形式有：动能、重力势能、弹性势能、电能、热能、声能和化学能。
In a closed system, the total of all these stays constant — energy only changes costume.
在封闭系统里，所有这些的总和保持不变——能量只是换装，并不消失。
Here is the same idea as stacked bars.
同样的想法，用堆叠的能量条表示。
At the top of a frictionless ramp, all the energy is gravitational potential.
在无摩擦斜坡顶端，能量全是重力势能。
Halfway down, half has become kinetic.
下到一半时，一半已变成动能。
At the bottom, it is all kinetic.
到了底部，全是动能。
The total height of the bars never changes.
能量条的总高度从不改变。
With friction, some of that energy would become thermal energy of the ramp and the air — the mechanical total would fall, but the true total still balances once you count the heat.
若有摩擦，一部分会变成斜坡和空气的热能——机械能总量会下降， 但一旦把热也算进去，真正的总量仍然守恒。
A bungee jumper is the standard question on this.
蹦极者是这一块的标准题。
From the platform to the point where the cord first goes taut, gravitational potential energy becomes kinetic energy.
从跳台到绳子刚绷直的那一点，重力势能变成动能。
From there the cord stretches and takes energy as elastic potential energy, so the kinetic energy reaches its maximum at the moment the cord's pull first equals the weight — not at the lowest point — and then falls to zero at the bottom, where the gravitational potential energy lost equals the elastic energy stored.
从那里开始绳子被拉伸，把能量取走存成弹性势能， 所以动能是在绳子的拉力第一次等于体重的那一刻达到最大——不是在最低点—— 然后到最低点降为零，那里失去的重力势能等于储存的弹性势能。
Let's put it to work.
我们来用一用。
A ball is released from rest at the top of a smooth ramp, one point two metres high.
一个球在一条光滑斜坡的顶端从静止释放，斜坡高一点二米。
How fast is it going at the bottom?
它到达底部时速度有多快？
All the potential energy of height becomes kinetic energy of motion.
高度的势能全部变成运动的动能。
Set them equal, and the mass cancels out — the speed is the square root of two, times gravity, times the height.
让它们相等，质量就消掉了—— 速度等于二乘以重力加速度再乘以高度，然后开平方。
That gives about four point nine metres per second.
结果约为每秒四点九米。
Where does that potential energy come from?
那个势能从哪里来？
Lift a mass slowly through a height.
把一个质量缓缓抬升一段高度。
You must push up with a force equal to its weight, and you move it that height — so the work you do is the weight, times the height.
你必须用一个等于它重力的力向上推， 并把它移动那段高度——所以你做的功就是重力乘以高度。
That work is stored as gravitational potential energy: in a uniform field, close to a planet's surface, it is the mass, times gravity, times the change in height.
这份功被储存为重力势能： 质量乘以重力加速度再乘以高度的变化。
And notice, only the height matters.
注意，只有高度才重要。
A steep climb and a gentle zig-zag to the same height store exactly the same energy.
陡直地爬升，和缓缓地走之字形上到同样的高度，储存的能量完全相同。
Two paths, same height.
两条路径，同一高度。
The steep straight climb and the long gentle zig-zag both end at the same level.
陡直的爬升和漫长平缓的之字形，终点在同一水平。
Because gravitational potential energy depends only on the change in height, both paths store exactly m g h.
因为重力势能只取决于高度的变化，两条路径储存的都是 m g h。
You do not need the path length, the speed, or the time — just mass, g, and how much higher you finished.
你不需要路径长度、速度或时间——只要质量、g，以及你升高了多少。
That is why GPE is so quick to use in exam questions.
这就是考试题里重力势能算得快的原因。
And kinetic energy?
那动能呢？
Push a resting mass with a force until it reaches a speed.
用一个力去推一个静止的质量，直到它达到某个速度。
Using the equations of motion, the work you do — force times distance — comes out to one half, times the mass, times the speed squared.
用运动学方程， 你做的功——力乘以距离——算出来等于二分之一乘以质量再乘以速度的平方。
That is the kinetic energy.
这就是动能。
There is also a neat shortcut: kinetic energy equals momentum squared, divided by twice the mass.
还有一个巧妙的捷径：动能等于动量的平方除以二倍的质量。
Useful when you know the momentum, but not the speed.
当你知道动量却不知道速度时，很有用。
Here is the derivation in words.
用文字把推导说一遍。
A mass starts at rest.
质量从静止开始。
A constant force F accelerates it over a displacement s, up to speed v.
恒力 F 在位移 s 上把它加速到速度 v。
From the equation v squared equals two a s, the distance is v squared over two a.
由速度平方等于二 a s，距离就是 v 的平方除以二 a。
Work done is F times s, and F is m a, so m a times v squared over two a cancels to one half m v squared.
做的功是 F 乘以 s，而 F 等于 m a， 所以 m a 乘以 v 平方再除以二 a，约掉后就是二分之一 m v 平方。
All that work becomes kinetic energy.
这份功全部变成动能。
That is why the formula is not m v, and not m v cubed — it is half m v squared.
所以公式不是 m v，也不是 m v 的立方——而是二分之一 m v 平方。
Combine p equals m v with the kinetic energy formula, and you get kinetic energy equal to momentum squared over two m.
把 p 等于 m v 与动能公式结合，就得到动能等于动量的平方除以二 m。
If the mass is constant and the momentum changes from p one to p two, the change in kinetic energy is p two squared minus p one squared, all over two m.
若质量不变，动量从 p 一变到 p 二，动能的变化就是 p 二的平方减去 p 一的平方，再除以二 m。
Reach for this when a question gives you momenta but never mentions the speeds — you know the momentum, and you can still find the energy.
当题目只给动量、从不提速度时，就用这个——你知道动量，仍然能求出能量。
Power is how fast you do work — the work done, divided by the time taken, measured in watts.
功率是你做功的快慢——做的功除以所用的时间，用瓦特来量度。
And here is a beautiful result.
这里有一个漂亮的结果。
If a force pushes an object moving at some velocity, the power delivered is simply the force, times the velocity.
如果一个力推动一个以某速度运动的物体，所提供的功率就等于力乘以速度。
A car cruising at twenty-five metres per second, against six hundred newtons of resistance, needs its engine to deliver fifteen thousand watts — fifteen kilowatts.
一辆车以每秒二十五米巡航，对抗六百牛顿的阻力，它的发动机需要输出一万五千瓦—— 也就是十五千瓦。
Power is work per second — or energy transferred per second.
功率是每秒做的功——或者说每秒传递的能量。
The same fifty joules done in one second is fifty watts; done in ten seconds it is only five watts.
同样五十焦耳，在一秒内做完就是五十瓦； 在十秒内做完就只有五瓦。
The unit is the watt, equal to one joule per second.
单位是瓦特，等于每秒一焦耳。
Like work, power is a scalar: it has size, but no direction of its own.
像功一样，功率是标量： 有大小，没有自己的方向。
Write P equals W over t, or delta E over delta t — both say the same thing.
写成 P 等于 W 除以 t，或能量变化除以时间变化——两者意思相同。
Where does P equals F v come from?
P 等于 F v 从何而来？
In a short time delta t, an object moving at velocity v covers a displacement of v times delta t.
在一小段时间德尔塔 t 内，以速度 v 运动的物体位移是 v 乘以德尔塔 t。
The work done by a force along the motion is F times that displacement.
沿运动方向的力做的功就是 F 乘以这段位移。
Divide by delta t, and the times cancel — power is simply force times velocity.
再除以德尔塔 t，时间约掉——功率就是力乘以速度。
This is one of the most useful results in mechanics.
这是力学里最有用的结果之一。
At constant speed on a flat road, the driving force equals the total resistive force, so engine power is resistance times speed.
在平路上匀速时，驱动力等于总阻力， 所以发动机功率就是阻力乘以速度。
One more graph reading worth naming: on a force-against-distance graph the work done is the area under the line, exactly as the area under a velocity-time graph is the displacement.
还有一种值得点名的读图方法： 在力—距离图上，所做的功是线下的面积， 就像速度—时间图下的面积是位移一样。
Reach for the area whenever the force is not constant.
只要力不是恒定的，就去求面积。
Three applications worth locking in.
三个值得牢记的应用。
On a flat road at constant speed, the engine balances drag — power equals resistive force times velocity.
平路上匀速时，发动机功率必须平衡阻力——P 等于阻力乘以速度。
If the drag grows with speed squared, doubling the speed needs eight times power, because the force quadruples AND the speed has doubled as well.
若阻力随速度的平方增大，速度加倍需要约八倍的功率，因为力变成四倍，速度又加倍。
To lift a weight straight up at a steady speed, the useful power is simply m g v.
竖直匀速提升重物时，有用功率就是 m g v。
A hovering aircraft needs large power too, because air must be pushed downwards all the time even though the height never changes.
悬停的飞机也需要很大功率， 因为即使高度不变，也必须不停地把空气向下推。
No machine is perfect.
没有机器是完美的。
Efficiency is the useful energy you get out, divided by the total energy you put in.
效率是你得到的有用能量，除以你投入的总能量。
It is always below one hundred percent, because some energy always leaks away — usually as heat.
它总是低于百分之百，因为总有一些能量漏掉——通常变成热。
Take a motor lifting a fifty-kilogram load at zero point four metres per second, while drawing two hundred and fifty watts.
举一台电动机， 以每秒零点四米提升一个五十千克的负载，同时消耗两百五十瓦。
Its useful output is about one hundred and ninety-six watts — so its efficiency is about seventy-eight percent.
它的有用输出约为一百九十六瓦—— 所以效率大约是百分之七十八。
Picture the energy flow.
想象能量的流动。
A thick arrow of total input splits into a useful output and a thinner waste stream — usually heat.
一条粗箭头表示总输入，分成有用输出和更细的浪费支流——通常是热。
Efficiency is the useful part divided by the total, times one hundred percent.
效率就是有用部分除以总量，再乘以百分之一百。
You can write the same idea with power instead of energy: useful power out over total power in.
你也可以用功率代替能量写同样的式子： 有用输出功率除以总输入功率。
Real machines always waste some input, so the answer is always less than one hundred percent.
真实机器总会浪费一些输入，所以答案总是低于百分之百。
For an electric motor the electrical input power is voltage times current.
对电动机来说，电功率输入是电压乘以电流。
If the efficiency is eta, the useful output power is efficiency times V I.
若效率是伊塔，有用输出功率就是效率乘以 V I。
From that useful power you can find a force, a lifting speed, or a tension — because useful power also equals force times velocity, or tension times velocity when a cable is winding up.
从这份有用功率，你可以求出一个力、一个提升速度，或一根绳的张力—— 因为有用功率也等于力乘以速度，在缆绳卷扬时就是张力乘以速度。
Energy gives you a powerful way to solve problems: just compare the start and the end.
能量给了你一个强大的解题方法：只需比较开始和结束的状态。
Take a moving block that slides into a spring on a smooth surface.
拿一个运动的方块，它在光滑表面上滑向一根弹簧。
At the moment of greatest squeeze, the block has stopped — so all of its kinetic energy has turned into elastic energy stored in the spring.
在压缩最大的那一刻，方块停了下来—— 所以它全部的动能都变成了储存在弹簧里的弹性能量。
No forces, no time — just energy in equals energy out.
没有力，没有时间——只有输入的能量等于输出的能量。
Here is a reliable plan.
这里有一套可靠的步骤。
Step one: find the start and end states — write the kinetic and potential energies in each.
第一步：找出始末状态——写出各自的动能和势能。
Step two: list any outside work.
第二步：列出任何外界做功。
Friction usually takes energy out; a push can add energy in.
摩擦通常取走能量；推力可以加入能量。
Step three: write the balance.
第三步：写出平衡。
Energy start plus work in equals energy end plus work lost as heat.
始态能量加输入的功，等于末态能量加热损。
Stick to this order and messy force diagrams often become one clean line of algebra.
按这个顺序走，杂乱的受力图往往变成一行干净的代数。
A block of kinetic energy one half m v squared slides into a spring.
一块动能为二分之一 m v 平方的方块滑向弹簧。
At the greatest compression x, the block is briefly at rest, so all that kinetic energy has become elastic potential energy one half k x squared, where k is the spring constant.
在最大压缩量 x 时，方块短暂静止， 所以全部动能都变成了弹性势能二分之一 k x 平方，其中 k 是劲度系数。
Set them equal and solve for x, or for v, or for k — three quantities, one energy statement.
令两边相等，就能解出 x、v 或 k——三个量，一条能量关系。
On a frictionless surface there is no heat term to track.
在无摩擦表面上，没有需要追踪的热项。
Another classic: a box is pushed at constant velocity up a rough ramp of length L that rises by height h.
另一个经典：箱子以恒定速度被推上粗糙斜坡，斜坡长 L、升高 h。
Because the speed does not change, the kinetic energy is the same at start and end.
因为速度不变，始末动能相同。
So the work done by the push goes into the gain in potential energy, plus the work done against friction.
所以推力做的功，全部进入势能的增加，加上克服摩擦做的功。
Write push work equals m g h plus friction work — and you can find any one of those if the others are known.
写成推力功等于 m g h 加摩擦功——知道其中几个，就能求出剩下的那个。
A ball dropped from height h one that bounces to height h two keeps only a fraction of its mechanical energy.
从高度 h 一落下又弹到高度 h 二的球，只保留一部分机械能。
That fraction is h two over h one, which also equals the square of the upward speed over the downward speed.
那个比例是 h 二比 h 一，也等于上升速度与下降速度之比的平方。
And for a projectile thrown to the same height at different angles: the final speed is the same, because only the height matters.
对于以不同角度抛到同一高度的抛体：末速度大小相同，因为只有高度才重要。
Use components if you need the direction of that final velocity.
若需要末速度的方向，再用分量去求。
Three marks to lock down.
三个要拿稳的分。
First, work is force times distance in the direction of the force — use the cosine when they are at an angle.
第一，功等于力乘以沿力方向的距离——两者成角度时用余弦。
Second, potential energy uses the vertical height gained, never the distance along a slope.
第二，势能用竖直方向升高的高度，绝不是沿斜坡的距离。
Third, power equals work over time, and also force times velocity.
第三，功率等于功除以时间， 也等于力乘以速度。
Get these, and energy problems become easy marks.
掌握这些，能量题就是轻松得分。
The fixed-wording definitions, one answer only.
固定措辞的定义，只给一个答案。
Work done by a force: the product of the force and the distance moved in the direction of the force.
力所做的功：力与沿力的方向移动的距离的乘积。
Conservation of energy: energy cannot be created or destroyed, only transferred from one form to another, so the total energy of a closed system is constant.
能量守恒：能量不能被创造或消灭，只能从一种形式转移到另一种形式， 所以封闭系统的总能量保持不变。
Efficiency: the ratio of useful energy or power output to the total input, usually as a percentage.
效率：有用的能量或功率输出与总输入之比，通常用百分数表示。
Power: the work done per unit time, or the rate at which energy is transferred.
功率：单位时间所做的功，或者能量转移的速率。
Gravitational potential energy: the energy an object has because of its position in a gravitational field.
重力势能：物体因其在重力场中的位置而具有的能量。
Kinetic energy: the energy it has because of its motion.
动能：物体因其运动而具有的能量。
Elastic potential energy: the energy stored in an object that has been stretched or compressed.
弹性势能：物体因被拉伸或压缩而储存的能量。
The traps.
陷阱。
Finish a change-in-energy calculation — the change in height is not the answer, m g delta h is.
能量变化的题要算完——高度的变化不是答案，m g Δh 才是。
Use the vertical height gained, not the distance along a slope, and check whether the question wants mass in kilograms or weight in newtons.
要用竖直方向上升的高度，不是沿斜面的距离， 而且要看清题目要的是以千克为单位的质量还是以牛顿为单位的重力。
In P equals F v, F is the DRIVING force; only at constant speed does it equal the resistive force.
在 P 等于 F v 里，F 是牵引力；只有在匀速时它才等于阻力。
Efficiency is useful output over TOTAL input, so it can never exceed one hundred per cent.
效率是有用输出除以总输入，所以永远不可能超过百分之百。
And on a force-extension graph the energy stored is the AREA under the line, not its gradient.
另外在力—伸长量图上，储存的能量是线下的面积，不是它的斜率。
One more exam habit.
再养成一个考试习惯。
Use conservation of energy in words first: loss in gravitational potential equals gain in kinetic, plus any work done against resistance.
先用文字写出能量守恒：重力势能的减少等于动能的增加， 再加上克服阻力做的功。
If the surface is smooth, drop the resistance term.
若表面光滑，就去掉阻力项。
If something is driven at constant speed, the kinetic change is zero and you are only balancing work in against potential and heat.
若某物被匀速驱动， 动能变化为零，你只需平衡输入的功与势能和热。
Efficiency is useful output divided by total input — keep the useful part clear in your mind before you reach for the calculator.
效率是有用输出除以总输入—— 动笔算之前，先把“有用”那部分在脑子里分清楚。