Change in Momentum and Impulse · 动量变化与冲量
| English | 中文 | Pinyin · 拼音 |
|---|---|---|
| impulse/ˈɪmpʌls/ | 冲量 | chōng liàng |
| impulse-momentum theorem/ˈɪmpʌls məʊˈmentəm ˈθɪərəm/ | 冲量动量定理 | chōng liàng dòng liàng dìng lǐ |
| contact time/ˈkɒntækt taɪm/ | 接触时间 | jiē chù shí jiān |
Why airbags save lives
- In a crash, your body must stop -- there is no avoiding that.
- An airbag cannot change how much your motion changes, only how long it takes.
- By stretching the stop from a hundredth of a second to a tenth, it cuts the force tenfold.
- The physics behind this is impulse.
安全气囊为什么能救命
- 在碰撞中,你的身体必须停下——这无法避免。
- 安全气囊无法改变你的运动改变多少,只能改变它用多长时间。
- 把停止过程从百分之一秒拉长到十分之一秒,它就把力减小到十分之一。
- 这背后的物理就是冲量。
Impulse
- Impulse 冲量 is force applied over time:
- It is the area under a force-versus-time graph.
- A big force for a short time can equal a small force for a long time.
冲量
- 冲量是力作用一段时间:
- 它是力对时间图下的面积。
- 短时间的大力,可以等于长时间的小力。
A force of $50\ \text{N}$ acts for $0.4\ \text{s}$. What impulse does it deliver (in N·s)? · 一个 $50\ \text{N}$ 的力作用了 $0.4\ \text{s}$。它提供了多少冲量（单位：N·s）？
$J = F\Delta t = 50 \times 0.4 = 20\ \text{N}\cdot\text{s}$.
The impulse delivered by a varying force equals the area under its force-versus-time graph. · 变力提供的冲量等于其力-时间图线下方的面积。
$J = \int F\,dt$ is exactly that area. · $J = \int F\,dt$ 正好就是那个面积。
The impulse-momentum theorem
- The impulse-momentum theorem 冲量动量定理 says impulse equals the change in momentum:
- To change an object's momentum, apply a force for some time.
- The bigger the impulse, the bigger the change in motion.
冲量动量定理
- 冲量动量定理指出冲量等于动量的变化:
- 要改变物体的动量,就施加一个力一段时间。
- 冲量越大,运动的变化就越大。
An impulse of $12\ \text{N}\cdot\text{s}$ acts on a $3\ \text{kg}$ cart at rest. Its final speed (in m/s)? · 一个 $12\ \text{N}\cdot\text{s}$ 的冲量作用于静止的 $3\ \text{kg}$ 小车上。其最终速度是多少（单位：m/s）？
$J = \Delta p = m\Delta v$, so $\Delta v = 12/3 = 4\ \tfrac{\text{m}}{\text{s}}$ from rest. · $J = \Delta p = m\Delta v$，因此从静止开始达到 $\Delta v = 12/3 = 4\ \tfrac{\text{m}}{\text{s}}$。
Newton's law, deeper
- Differentiating gives Newton's second law in its truest form:
- Force is the rate of change of momentum.
- For constant mass this reduces to the familiar $F = ma$.
更深层的牛顿定律
- 求导给出牛顿第二定律最本质的形式:
- 力是动量的变化率。
- 对于质量不变的情形,这化简为熟悉的 $F = ma$。
Impulse and momentum change · 冲量与动量变化
Impulse is force times time, and it equals the change in momentum. Sort each case. · 冲量是力乘以时间，且等于动量的变化。对每种情况进行分类。
In its most general form, the net force equals the rate of change of ____. · 在最一般的形式中，合力等于____的变化率。
$\vec{F}_{net} = d\vec{p}/dt$ -- force is the time rate of change of momentum. · $\vec{F}_{net} = d\vec{p}/dt$ ——力是动量对时间的变化率。
Stretch the time, soften the blow
- For a fixed change in momentum, a longer contact time 接触时间 means a smaller average force ($F = \Delta p / \Delta t$).
- Airbags, crumple zones, bent knees on landing, and catching an egg gently all use this.
- Same momentum change, gentler force.
拉长时间,减轻冲击
- 对于固定的动量变化,更长的接触时间意味着更小的平均力($F = \Delta p / \Delta t$)。
- 安全气囊、溃缩区、着地时弯膝、轻轻接住鸡蛋,都利用了这一点。
- 同样的动量变化,更温和的力。
An airbag reduces injury by... · 安全气囊通过...减少伤害
The momentum change is fixed; the airbag stretches the time, so the average force $F = \Delta p/\Delta t$ drops. · 动量变化是固定的；安全气囊延长了时间，因此平均力 $F = \Delta p/\Delta t$ 下降。
A ball hitting a wall and bouncing back experiences a momentum change that is... a ball that hits and stops. · 一个球撞击墙壁并反弹所经历的动量变化... 一个撞击并停止的球相比...
A bounce reverses the velocity (e.g. $+20$ to · 到 $-20$), a change of $40$; stopping is only a change of $20$. · 反弹反转了速度（例如从 $+20$ 到 $-20$），变化量为 $40$；停止仅是 $20$ 的变化。
A $0.15\ \text{kg}$ ball hits a wall at $20\ \tfrac{\text{m}}{\text{s}}$ and bounces back at $20\ \tfrac{\text{m}}{\text{s}}$.
- Change in momentum: $\Delta p = m(v_f - v_i) = 0.15(-20 - 20) = -6\ \tfrac{\text{kg}\cdot\text{m}}{\text{s}}$.
- If the contact lasts $0.01\ \text{s}$, the average force is $|\Delta p|/\Delta t = 6/0.01 = 600\ \text{N}$.
一个 $0.15\ \text{kg}$ 的球以 $20\ \tfrac{\text{m}}{\text{s}}$ 撞墙,又以 $20\ \tfrac{\text{m}}{\text{s}}$ 弹回。
- 动量变化:$\Delta p = m(v_f - v_i) = 0.15(-20 - 20) = -6\ \tfrac{\text{kg}\cdot\text{m}}{\text{s}}$。
- 如果接触持续 $0.01\ \text{s}$,平均力是 $|\Delta p|/\Delta t = 6/0.01 = 600\ \text{N}$。
A bounce changes momentum more than a stop. Coming in at $20$ and leaving at $-20$ is a change of $40$ (in speed units), not $20$. Forgetting the sign flip on a bounce is a classic impulse mistake.
弹回比停止改变的动量更多。以 $20$ 进入、以 $-20$ 离开,变化是 $40$(速度单位),而不是 $20$。忘记弹回时的符号翻转是经典的冲量错误。
Impulse $\vec{J} = \int \vec{F}\,dt$ (the area under an $F$-$t$ graph) equals the change in momentum: $\vec{J} = \Delta\vec{p}$. Since $\vec{F} = d\vec{p}/dt$, stretching the contact time for a fixed momentum change lowers the average force -- the secret of every airbag.
冲量 $\vec{J} = \int \vec{F}\,dt$($F$-$t$ 图下的面积)等于动量的变化:$\vec{J} = \Delta\vec{p}$。由于 $\vec{F} = d\vec{p}/dt$,在动量变化固定时拉长接触时间会降低平均力——这是每个安全气囊的秘密。