Work, Energy, and Power
AP Physics C: Mechanics Topic 3 7:46 English narration · English + 中文 subtitles burned in
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A block slides into a spring and squashes it until it stops.
一个滑块撞上弹簧，把它压缩到自己停下来。
How far does the spring compress?
弹簧被压缩了多少？
Try that with a kinematic equation. You cannot — the force grows the whole way in, so the acceleration is never constant.
用运动学公式试试看——你做不到，因为力一路都在增大，加速度从来不是常数。
<slow>Energy does not care.</slow> One line, one answer.
能量的方法则不在乎这一点：一行就能得到答案。
That is why this unit exists.
这正是本单元存在的意义。
This is Unit Three: work, energy and power.
这是第三单元：功、能量和功率。
Scalars, not vectors — which is exactly why they are easier.
它们是标量，不是矢量——这正是它们更好用的原因。
When the force varies, or the path is complicated, energy is the tool that still works.
当力在变化，或者路径很复杂时，能量方法依然有效。
Let's begin.
让我们开始吧。
Kinetic energy is the energy of motion. One half of the mass, times the speed squared, measured in joules.
动能是运动的能量：二分之一质量乘以速率的平方，单位是焦耳。
It is a scalar, so it has no direction — just a number.
它是标量，所以没有方向，只是一个数。
Notice the square. Double the speed and the kinetic energy does not double; it goes up four times.
注意那个平方：速率加倍，动能不是加倍， 而是变成四倍。
That single fact explains why stopping distances grow so fast with speed.
这一个事实就解释了刹车距离为什么随速度增长得这么快。
One subtlety, and the exam does ask it: kinetic energy depends on the observer's reference frame.
还有一个细节，考试确实会考：动能取决于谁在测量。
A passenger walking along a train has a small kinetic energy in the train's frame, and a huge one measured from the ground.
在火车上行走的乘客，在火车参考系里动能很小，从地面测量却非常大。
Both observers are right.
两位观察者都没有错。
Work is energy transferred by a force acting through a distance.
功是力通过一段距离所传递的能量。
For a constant force it is the force, times the distance, times the cosine of the angle between them.
对恒力，功等于力乘以距离， 再乘以两者夹角的余弦。
In general the force can change along the way, so the real definition is an integral: add up the force component along the path, over every tiny step.
一般情况下，力沿路径会变化， 所以真正的定义是一个积分：把沿路径方向的力分量在每一小段上加起来。
On a force-position graph, that integral is simply the area under the curve.
在力—位置图上，这个积分就是曲线下的面积。
If the exam gives you a graph instead of a formula, that area is the whole answer.
如果考试给你的是图像而不是公式，那块面积就是全部答案。
Work is a scalar, but it carries a sign, and that sign is physics, not bookkeeping.
功是标量，但它带有符号，而这个符号是物理，不是记账。
If the force has a component along the motion, the work is positive and the object speeds up.
如果力有沿运动方向的分量，功为正，物体加速。
If it opposes the motion, the work is negative — friction and air drag are always negative.
如果力与运动方向相反，功为负——摩擦力和空气阻力永远做负功。
And if the force is perpendicular to the motion, it does no work at all. The normal force on a sliding block does zero work. So does the tension in a string swinging a ball in a circle.
如果力垂直于运动方向，它完全不做功：滑块受到的法向力做功为零， 让小球做圆周运动的绳中张力也是如此。
Add up the work done by every force on an object, and you get the work-energy theorem: the net work equals the change in kinetic energy.
把作用在物体上每一个力所做的功加起来，就得到动能定理：合外力做的功等于动能的变化。
That one line replaces a whole page of kinematics.
这一行就能代替整整一页的运动学。
Try it. A force of three x squared newtons pushes a body along the x axis, from the origin out to two metres.
试一试：一个大小为三 x 平方牛顿的力沿 x 轴推动物体， 从原点到两米。
Integrate: the integral of three x squared is x cubed, so between zero and two the work is eight joules.
积分：三 x 平方的积分是 x 三次方，所以从零到二，功是八焦耳。
Or read it as the area under that curve.
也可以直接读作那条曲线下的面积。
Starting from rest, the body ends with exactly eight joules of kinetic energy.
如果物体从静止开始，它最后的动能正好是八焦耳。
Potential energy belongs to a system, not to a single object — it is stored in the positions of the parts.
势能属于系统，而不属于单个物体——它储存在各部分的相对位置里。
It only exists for conservative forces, the ones whose work does not depend on the path taken.
它只对保守力存在，也就是做功与路径无关的那些力。
Its definition is minus the work done by that force.
它的定义是该力所做功的负值。
You are free to choose where the potential energy is zero, because only changes in it ever matter.
你可以自由选择势能为零的位置，因为真正有意义的只是它的变化。
Pick the zero that makes your problem simplest.
选一个让问题最简单的零点。
Three results you should know by heart: near the ground, the change is mass times field strength times height. In general, gravity gives minus the constant times the two masses over the separation. And a spring stores one half times the spring constant times the extension squared.
三个必须背下来的结果：在地面附近， 重力势能的变化等于质量乘以场强再乘以高度；一般情况下， 引力势能等于负的引力常量乘以两质量再除以距离；弹簧储存二分之一劲度系数乘以形变量的平方。
Turn that definition around and something powerful appears: a conservative force is minus the slope of the potential energy curve.
把这个定义反过来用，就出现了一件很有力的事：保守力等于势能曲线斜率的负值。
The force always points downhill on that curve.
力总是沿着曲线「下坡」方向。
Where the slope is zero, the force is zero — that is an equilibrium.
斜率为零处，力为零——那就是平衡点。
A minimum is stable: push the object away and the force pushes it back.
极小值是稳定平衡：把物体推开，力会把它推回来。
A maximum is unstable: push it away and the force pushes it further.
极大值是不稳定平衡：把它推开，力会把它推得更远。
Take a potential energy of two x cubed minus six x. The force is six minus six x squared, which is zero at plus one and minus one.
取势能为二 x 三次方减六 x，力就是六减六 x 平方，在正一和负一处为零。
The second derivative is twelve x, positive at plus one — stable — and negative at minus one — unstable.
二阶导数是十二 x，在正一处为正——稳定；在负一处为负——不稳定。
Energy is conserved in every interaction. The only question is where it goes.
在一切相互作用中，能量都是守恒的，唯一的问题是它去了哪里。
Mechanical energy is kinetic plus potential.
机械能等于动能加势能。
If no outside work is done, and nothing inside the system rubs or drags, then that sum is the same at every moment.
如果没有外界做功，系统内部也没有摩擦或阻力， 那么这个和在每一时刻都相同。
Watch the two bars trade. Kinetic becomes potential, potential becomes kinetic, and the total never moves.
看这两根能量条来回交换： 动能变势能，势能变动能，而总量始终不变。
Now the single most useful picture in this unit.
现在来看本单元最有用的一张图。
Draw the potential energy against position, and draw the total energy as a horizontal line across it.
画出势能随位置的变化，再画一条水平线表示总能量。
The gap between the line and the curve is the kinetic energy at that position.
直线与曲线之间的间隔，就是该位置的动能。
Where the gap is widest, the object moves fastest.
间隔最大处，物体运动最快。
Where the line meets the curve, the kinetic energy is zero — the object stops for an instant and turns around.
直线与曲线相交处，动能为零——物体瞬间停下并折返。
Those are the turning points, and the object can never go beyond them.
那些点叫做转折点， 物体永远无法越过它们。
One picture, and the whole motion is mapped.
一张图，整个运动就被完全描绘出来了。
Friction and drag are different: they turn mechanical energy into thermal energy, so the sum of kinetic and potential falls.
摩擦力和阻力不一样：它们把机械能转化为热能，所以动能与势能之和会减少。
The energy is not lost — it is just no longer mechanical.
能量并没有消失，只是不再是机械能。
Here is the accounting. A two kilogram block starts from rest one point five metres up a ramp and reaches the bottom at four metres per second.
来做一次账：一个两千克的滑块从一点五米高处由静止下滑， 到达底部时速度为四米每秒。
Available at the top: mass times field strength times height, twenty-nine point four joules.
顶端可用的能量是质量乘以场强再乘以高度，等于二十九点四焦耳。
Arriving as kinetic energy: one half m v squared, sixteen joules.
到达底部的动能是二分之一 m v 平方，等于十六焦耳。
So thirteen joules went into heating the ramp and the block.
所以有十三焦耳变成了斜面和滑块的内能。
Power is the rate of energy transfer, measured in watts.
功率是能量传递的快慢，单位是瓦特。
Average power is the work divided by the time taken.
平均功率等于功除以所用的时间。
The instantaneous power is the dot product of force and velocity — so only the part of the force along the velocity delivers any power.
瞬时功率等于力与速度的点积——所以只有沿速度方向的那部分力才输出功率。
On a work-time graph, power is the slope.
在功—时间图上，功率就是斜率。
The same job done in half the time needs twice the power.
同样的工作用一半的时间完成，就需要两倍的功率。
A twelve hundred kilogram car climbs a hill at a steady fifteen metres per second.
一辆一千二百千克的汽车以恒定的十五米每秒爬坡。
The road rises one metre for every twenty metres travelled.
道路每前进二十米升高一米。
What power does the engine need, just to fight gravity?
仅仅为了克服重力，发动机需要多大功率？
Pause here and try it.
先暂停，自己试一试。
Steady speed means no change in kinetic energy, so the engine only has to supply the rate at which potential energy grows.
速度恒定说明动能不变，所以发动机只需提供势能增长的速率。
That is the weight, times the speed, times the sine of the slope angle — and here the sine is simply one over twenty.
那就是重力乘以速度，再乘以坡角的正弦——这里的正弦就是二十分之一。
Multiply them together: about eight point eight kilowatts.
把它们相乘，约为八点八千瓦。
Air resistance would add more on top.
空气阻力还会在此之上再增加一些。
Three habits that save marks.
三个能保住分数的习惯。
First, name your system before you write anything, and say which forces do work on it.
第一，动笔之前先说清楚你的系统，并说明哪些力对它做功。
Second, when the force varies, integrate — never reach for a constant-force formula.
第二，当力在变化时就用积分——绝不要套用恒力公式。
Third, if friction or any other nonconservative force acts, say the sentence: mechanical energy is not conserved, and the missing energy became thermal energy.
第三，如果有摩擦，就把那句话写出来：机械能不守恒，缺少的能量变成了热能。
Examiners give marks for that sentence.
阅卷人会为这句话给分。
Get those three, and this unit is yours.
做到这三点，这个单元就是你的了。