Kinematics
AP Physics C: Mechanics Topic 1 7:31 English narration · English + 中文 subtitles burned in
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A rocket lifts off.
一枚火箭升空。
Every second, its engine burns harder, so its acceleration keeps growing.
每一秒，它的发动机燃烧得更猛，所以加速度不断增大。
Now try to find its speed with the equations from your first physics course. You cannot.
现在试着用你在第一门物理课里学到的公式求它的速度——你做不到。
Those equations need a constant acceleration, and this one changes every instant.
那些公式要求加速度恒定，而这里的加速度每一瞬间都在变。
<slow>Physics C gives you the tool that never fails: calculus.</slow>
Physics C 给你一个永远不会失效的工具：微积分。
This is Unit One of Physics C Mechanics: kinematics.
这是 Physics C 力学的第一单元：运动学。
We describe motion without yet asking what causes it.
我们先描述运动，暂时不问是什么造成了运动。
One idea runs through everything: differentiate to go down the chain, integrate to climb back up.
有一条主线贯穿全部内容：求导沿着链条向下走，积分再爬回来。
Let's begin.
让我们开始吧。
Start with the two kinds of quantity.
先从两类量说起。
A scalar has size only: distance, speed, time, mass.
标量只有大小：距离、速率、时间、质量。
A vector has size and direction: displacement, velocity, acceleration, force.
矢量既有大小又有方向：位移、速度、加速度、力。
In this course you never add vectors as raw numbers.
在这门课里，你绝不能把矢量当作普通数字直接相加。
You break each vector into components along the axes, add the components, and rebuild at the end.
你要把每个矢量沿坐标轴分解成分量，把分量相加，最后再合成。
Physics C also writes a vector with unit vectors: i-hat points along x, j-hat points along y, and each one is one unit long.
Physics C 还用单位矢量来写矢量：i 帽指向 x 方向，j 帽指向 y 方向，每个长度都是一。
Written that way, calculus can act on each component on its own.
这样写以后，微积分就可以分别作用在每个分量上。
Here is the heart of the course.
这是整门课的核心。
Velocity is the derivative of position with respect to time.
速度是位置对时间的导数。
Acceleration is the derivative of velocity, so it is the second derivative of position.
加速度是速度的导数， 所以它是位置的二阶导数。
Each step down the chain asks the same question: how fast is this changing right now?
链条上的每一步都在问同一个问题：此刻它变化得有多快？
And keep two words apart.
还要把两个词分清楚。
Average velocity uses a whole interval — change in position divided by change in time.
平均速度用的是一整段区间——位置的变化除以时间的变化。
Instantaneous velocity is the value at a single moment: the slope of the tangent line, and what a speedometer shows.
瞬时速度是某一个时刻的值：切线的斜率，也就是速度表显示的值。
They agree only when the velocity never changes.
只有当速度始终不变时，两者才相等。
Let's use it.
我们来用一用。
A particle moves so that its position in metres is two t cubed minus three t squared.
一个质点的位置（以米为单位）是二 t 三次方减去三 t 平方。
Find the velocity and the acceleration at two seconds.
求它在两秒时的速度和加速度。
Pause here and try it.
先暂停，自己试一试。
Differentiate once and the velocity is six t squared minus six t.
求一次导数，速度是六 t 平方减去六 t。
Differentiate again and the acceleration is twelve t minus six.
再求一次导数，加速度是十二 t 减六。
Now put in t equals two.
现在把 t 等于二代进去。
The velocity is twenty-four minus twelve, which is twelve metres per second.
速度是二十四减十二，等于十二米每秒。
The acceleration is twenty-four minus six, which is eighteen metres per second squared.
加速度是二十四减六，等于十八米每二次方秒。
Now run the chain backwards.
现在把链条倒过来走。
If you know the acceleration and want the velocity, integrate.
如果你知道加速度、想求速度，就积分。
If you know the velocity and want the position, integrate again.
如果你知道速度、想求位置，就再积分一次。
Each integral needs one extra piece of information: the value at the start.
每一次积分都需要一条额外的信息： 起始时刻的值。
That is exactly what the initial velocity and the initial position are for.
初速度和初位置正是为此而存在的。
Differentiating throws that constant away. Integrating asks for it back.
求导会把这个常数丢掉，积分则要把它要回来。
Try the reverse.
再反过来做一遍。
A particle starts from rest at the origin, and its acceleration in metres per second squared is six t.
一个质点从原点由静止出发，它的加速度（以米每二次方秒为单位）是六 t。
Integrate the acceleration and the velocity is three t squared — the constant is zero because it starts from rest.
对加速度积分，速度是三 t 平方——常数为零，因为它从静止出发。
Integrate again and the position is t cubed — again zero, because it starts at the origin.
再积分一次，位置是 t 三次方——同样为零，因为它从原点出发。
At two seconds the velocity is twelve metres per second and the position is eight metres.
在两秒时，速度是十二米每秒，位置是八米。
So where do the familiar equations come from?
那么，那些熟悉的公式是从哪里来的呢？
They are that same integral, done once, for the special case where the acceleration never changes.
它们就是刚才那个积分做了一次， 只不过用在加速度始终不变这个特例上。
Integrate a constant acceleration and you get velocity equals initial velocity plus acceleration times time.
对恒定加速度积分， 就得到速度等于初速度加上加速度乘以时间。
Integrate that and you get the position equation.
再积分一次，就得到位置公式。
Use them freely — but only while the acceleration really is constant.
放心用它们——但前提是加速度确实恒定。
The most important constant case is free fall: near the Earth every object accelerates downward at nine point eight metres per second squared, once air resistance is small enough to ignore.
最重要的恒定加速度情形是自由落体： 在地球附近，只要空气阻力小到可以忽略，每个物体都以九点八米每二次方秒竖直向下加速。
Graphs say the same thing in pictures.
图像用另一种方式说同样的话。
On a position-time graph, the slope is the velocity.
在位置—时间图上，斜率就是速度。
On a velocity-time graph, the slope is the acceleration, and the area under the line is the displacement.
在速度—时间图上，斜率是加速度，而线下的面积是位移。
On an acceleration-time graph, the area is the change in velocity.
在加速度—时间图上，面积是速度的变化。
So slopes are derivatives, and areas are integrals.
所以斜率就是导数，面积就是积分。
Area below the time axis counts as negative, because the object moved the other way.
时间轴以下的面积记为负，因为物体朝相反方向运动。
Every velocity is measured against something — a reference frame.
每一个速度都是相对某个东西测量的——那就是参考系。
Read the subscripts as a chain: the velocity of A relative to C is the velocity of A relative to B, plus the velocity of B relative to C.
把下标当作一条链来读：A 相对 C 的速度，等于 A 相对 B 的速度，加上 B 相对 C 的速度。
A boat moves at four metres per second in still water and heads straight across a river that flows at three.
一条船在静水中速度为四米每秒，垂直指向对岸，而河水以三米每秒流动。
Relative to the ground it moves at five metres per second, angled downstream.
相对地面，它的速度是五米每秒，方向偏向下游。
But the crossing time uses only the across-stream part.
但过河时间只用垂直于河岸的那个分量。
For a river eighty metres wide that is twenty seconds — and the current carries the boat sixty metres downstream.
对于八十米宽的河，那就是二十秒—— 同时水流把船带向下游六十米。
In two or three dimensions, nothing new happens.
在二维或三维里，并没有出现新东西。
Position becomes a vector, and you differentiate each component on its own.
位置变成一个矢量， 你只要分别对每个分量求导。
Take a position vector of two t squared along x, plus four t minus t cubed along y.
取一个位置矢量：x 方向是二 t 平方， y 方向是四 t 减 t 三次方。
Differentiate each part. The velocity is four t along x, plus four minus three t squared along y.
分别求导，速度在 x 方向是四 t， 在 y 方向是四减三 t 平方。
At one second that is four and one, so the speed is the square root of seventeen, about four point one metres per second.
在一秒时，它们是四和一， 所以速率是十七的平方根，约为四点一米每秒。
Same physics — done once per axis.
物理还是同样的物理——只是每个轴各做一遍。
The most famous two-dimensional case is a projectile.
最著名的二维情形是抛体运动。
Horizontally there is no force, so the horizontal velocity never changes at all.
水平方向没有力，所以水平速度完全不变。
Vertically, gravity pulls down with a constant acceleration.
竖直方向，重力以恒定的加速度向下拉。
The two motions share exactly one thing: the clock.
两个方向的运动只共用一样东西：时间。
That is why the path is a parabola, and why a ball that is dropped and a ball that is fired sideways reach the ground together.
这就是轨迹为抛物线的原因，也是一个自由落下的球和一个水平射出的球同时落地的原因。
The classic problem.
经典题目。
A ball is launched at twenty metres per second, thirty degrees above the horizontal.
一个球以二十米每秒、与水平方向成三十度射出。
Pause here and try it.
先暂停，自己试一试。
Components first.
先求分量。
The horizontal part is twenty times the cosine of thirty, about seventeen point three.
水平分量是二十乘三十度的余弦，约为十七点三。
The vertical part is twenty times the sine of thirty, exactly ten.
竖直分量是二十乘三十度的正弦，正好是十。
The vertical part sets the clock: the flight lasts twice ten divided by nine point eight, about two seconds.
竖直方向决定时间： 飞行时间是二乘十再除以九点八，约为两秒。
The greatest height is ten squared over twice nine point eight, about five point one metres.
最大高度是十的平方除以二乘九点八， 约为五点一米。
And the range is seventeen point three times two — about thirty-five metres.
射程是十七点三乘以二——约三十五米。
Three habits that save marks.
三个能保住分数的习惯。
First, resolve every vector into components before you add anything — never add magnitudes at an angle.
第一，在相加之前，先把每个矢量分解成分量—— 绝不要把成角度的大小直接相加。
Second, check whether the acceleration is constant before you reach for a kinematic equation; if it varies, differentiate or integrate instead.
第二，在动用运动学公式之前， 先确认加速度是否恒定；如果它在变，就改用求导或积分。
Third, every integral needs its initial condition, so write down the value at time zero before you start.
第三，每一次积分都需要初始条件，所以动笔之前先写下时间为零时的值。
Get those three, and Unit One is yours.
做到这三点，第一单元就是你的了。