Resistive Forces · 阻力
| English | 中文 | Pinyin · 拼音 |
|---|---|---|
| resistive force/rɪˈzɪstɪv fɔːs/ | 阻力 | zǔ lì |
| differential equation/ˌdɪfəˈrenʃl ɪˈkweɪʒn/ | 微分方程 | wēi fēn fāng chéng |
| terminal velocity/ˈtɜːmɪnl vəˈlɒsɪti/ | 终极速度 | zhōng jí sù dù |
Why a skydiver stops speeding up
- Jump from a plane and you accelerate -- but not forever.
- Air pushes back harder the faster you fall, until it exactly balances gravity.
- From then on you fall at a steady, top speed.
- Understanding this needs a force that grows with velocity -- and a little calculus.
跳伞者为什么会停止加速
- 从飞机上跳下,你会加速——但不会永远加速。
- 你下落得越快,空气就把你顶得越厉害,直到它恰好平衡重力。
- 从那时起,你以一个稳定的最高速度下落。
- 要理解这一点,需要一个随速度增大的力——以及一点微积分。
A resistive force
- A resistive force 阻力 (air resistance, drag) opposes motion through a fluid.
- Unlike friction, it grows with speed -- move faster and it fights back harder.
- At rest there is no drag at all; at high speed it can be huge.
阻力
- 阻力(空气阻力、拽力)阻碍物体在流体中的运动。
- 与摩擦力不同,它随速度增大——动得越快,它反抗得越厉害。
- 静止时完全没有阻力;高速时它可以非常大。
A resistive force grows larger as the object moves faster. · 阻力随着物体运动速度的增加而变大。
Unlike ordinary friction, drag increases with speed -- that is why it can eventually balance gravity. · 与普通摩擦力不同，阻力随速度增加——这就是为什么它最终可以平衡重力的原因。
Modeling the drag
- We often model drag as proportional to speed, $F = -bv$, or to speed squared, $F = -cv^2$.
- The minus sign means it always points against the motion.
- Applying $\sum F = ma$ gives a differential equation 微分方程 for the velocity.
给阻力建模
- 我们常把阻力建模为与速度成正比 $F = -bv$,或与速度平方成正比 $F = -cv^2$。
- 负号表示它总是指向运动的反方向。
- 应用 $\sum F = ma$ 就得到关于速度的微分方程。
Applying Newton's second law to a velocity-dependent drag gives a ____ equation for v(t). · 对牛顿第二定律应用于速度相关的阻力进行积分，会得到关于 v(t) 的 ____ 方程。
$m\,dv/dt = mg - bv$ is a differential equation, solved to find how v approaches terminal velocity. · $m\,dv/dt = mg - bv$ 是一个微分方程，解此方程可求得 v 如何趋近终端速度。
Terminal velocity
- As you speed up, drag grows until it balances the driving force (gravity).
- Then the net force is zero, acceleration stops, and speed levels off.
- That steady top speed is the terminal velocity 终极速度.
终极速度
- 随着你加速,阻力增大,直到平衡驱动力(重力)。
- 这时合力为零,加速停止,速度趋于平稳。
- 那个稳定的最高速度就是终极速度。
Resistive forces and terminal velocity · 阻力和终端速度
As a falling body speeds up, drag grows until it balances weight and the body stops accelerating. · 当下落物体加速时，阻力增大直到与重力平衡，物体停止加速。
At terminal velocity, the object's acceleration is... · 在终端速度时，物体的加速度是...
Drag has grown to cancel gravity, so the net force -- and the acceleration -- is zero. · 阻力已增长到抵消重力，因此合外力——以及加速度——均为零。
Approaching the limit
- Solving $m\dfrac{dv}{dt} = mg - bv$ shows $v$ rising smoothly toward $v_t = mg/b$.
- It gets ever closer but never quite exceeds it.
- That is why a skydiver's speed flattens out instead of climbing without limit.
逼近极限
- 求解 $m\dfrac{dv}{dt} = mg - bv$ 表明 $v$ 平滑地升向 $v_t = mg/b$。
- 它越来越接近,却始终不会超过。
- 这就是为什么跳伞者的速度会趋于平缓,而不是无限攀升。
An object of weight $mg = 60\ \text{N}$ falls with drag $F = -bv$ where $b = 3\ \tfrac{\text{kg}}{\text{s}}$. Find its terminal velocity (in m/s). · 一个重量为 $mg = 60\ \text{N}$ 的物体下落，受到的阻力为 $F = -bv$，其中 $b = 3\ \tfrac{\text{kg}}{\text{s}}$。求其终端速度（单位：m/s）。
At terminal velocity $mg = bv_t$, so $v_t = 60/3 = 20\ \tfrac{\text{m}}{\text{s}}$. · 在终端速度时 $mg = bv_t$，因此 $v_t = 60/3 = 20\ \tfrac{\text{m}}{\text{s}}$。
Select all · 所有 true statements about terminal velocity. · 选择所有关于终端速度的正确陈述。
Drag acts the whole way down. Terminal velocity is just where it has grown enough to cancel gravity. · 阻力在整个下落过程中都存在。终端速度只是阻力增长到足以抵消重力的点。
With the same drag coefficient $b$, a heavier object has a terminal velocity that is... · 在相同的阻力系数 $b$ 下，较重的物体的终端速度是...
$v_t = mg/b$ grows with mass, so a heavier object falls faster at the top speed. · $v_t = mg/b$ 随质量增加，因此较重的物体在最高速度时下落得更快。
A falling object of mass $m$ feels drag $F = -bv$.
- Terminal velocity is where $\dfrac{dv}{dt} = 0$, so $mg = b v_t$.
- Solving, $v_t = \dfrac{mg}{b}$ -- heavier objects or weaker drag mean a higher top speed.
一个质量为 $m$ 的下落物体受到阻力 $F = -bv$。
- 终极速度出现在 $\dfrac{dv}{dt} = 0$ 处,所以 $mg = b v_t$。
- 求解得 $v_t = \dfrac{mg}{b}$——更重的物体或更弱的阻力意味着更高的最高速度。
Terminal velocity is not the moment drag appears -- drag acts the whole way down. It is the special speed where drag has grown enough to cancel gravity. Below it, you are still accelerating (just less and less).
终极速度不是阻力出现的那一刻——阻力在整个下落过程中都存在。它是阻力增长到足以抵消重力的那个特殊速度。在它之下,你仍在加速(只是越来越慢)。
A resistive force grows with speed ($F = -bv$ or $-cv^2$), so $\sum F = ma$ becomes a differential equation. As speed rises, drag builds until it cancels gravity -- the net force hits zero at the terminal velocity $v_t = mg/b$, which the object approaches smoothly but never exceeds.
阻力随速度增大($F = -bv$ 或 $-cv^2$),所以 $\sum F = ma$ 变成一个微分方程。随着速度上升,阻力增长直到抵消重力——合力在终极速度 $v_t = mg/b$ 处变为零,物体平滑地逼近它却永不超过。