Translational Kinetic Energy · 平动动能
| English | 中文 | Pinyin · 拼音 |
|---|---|---|
| kinetic energy/kɪˈnetɪk ˈenədʒi/ | 动能 | dòng néng |
| joule/dʒuːl/ | 焦耳 | jiāo ěr |
Twice the speed, four times the danger
- A car at $60\ \tfrac{\text{km}}{\text{h}}$ and the same car at $120\ \tfrac{\text{km}}{\text{h}}$ — twice the speed.
- But the fast car carries four times the energy of motion, not twice.
- That energy is kinetic energy 动能, and it grows with the square of speed.
- It is why crashes at high speed are so much more destructive.
速度快一倍,危险却是四倍
- 一辆车以 $60\ \tfrac{\text{km}}{\text{h}}$ 行驶,同一辆车以 $120\ \tfrac{\text{km}}{\text{h}}$——速度快了一倍。
- 但快车携带的运动能量是慢车的四倍,而不是两倍。
- 这种能量就是动能,它随速度的平方增长。
- 这就是高速碰撞破坏力大得多的原因。
The kinetic energy formula
- A moving mass stores kinetic energy $E_k = \tfrac12 m v^2$.
- It is a scalar (no direction) measured in joules 焦耳 ($\text{J}$).
- It depends on mass $m$ and speed $v$ — and always positive for a moving object.
- At rest ($v = 0$) the kinetic energy is zero.
动能公式
- 一个运动的质量储存动能 $E_k = \tfrac12 m v^2$。
- 它是一个标量(无方向),单位是焦耳($\text{J}$)。
- 它取决于质量 $m$ 和速度 $v$——对运动的物体总是正的。
- 静止时($v = 0$)动能为零。
What is the kinetic energy of a $2\ \text{kg}$ ball moving at $3\ \tfrac{\text{m}}{\text{s}}$, in joules? · 一个质量为 $2\ \text{kg}$ 的球以 $3\ \tfrac{\text{m}}{\text{s}}$ 的速度运动的动能是多少，单位为焦耳？
$E_k = \tfrac12 m v^2 = \tfrac12 (2)(9) = 9\ \text{J}$.
Kinetic energy is a vector, so it has a direction. · 动能是矢量，因此具有方向。
Kinetic energy is a scalar — a number of joules with no direction. · 动能是一个标量——一个无方向的焦耳数值。
Why $v^2$ matters so much
- Because of the $v^2$, doubling the speed makes $E_k$ four times larger.
- Tripling the speed makes it nine times larger.
- So a small speed increase is a large energy (and braking-distance) increase.
- Mass matters too, but only in direct proportion — double $m$, double $E_k$.
为什么 $v^2$ 如此重要
- 因为有 $v^2$,速度加倍使 $E_k$ 变为四倍。
- 速度变三倍使它变为九倍。
- 所以速度的小幅增加带来能量(和刹车距离)的大幅增加。
- 质量也有影响,但只是正比——$m$ 加倍,$E_k$ 加倍。
Energy grows as v squared · 能量随v的平方增长
E_k = a·v² (here a = ½m) · E_k = a·v² （此处 a = ½m）
Trace the quadratic curve of kinetic energy against speed and see how fast it climbs. · 追踪动能对速度的二次曲线，观察其上升速度之快。
If a car doubles its speed, its kinetic energy becomes: · 如果一辆汽车的速度加倍，其动能变为：
$E_k \propto v^2$, so doubling $v$ gives $2^2 = 4$ times the kinetic energy. · $E_k \propto v^2$，因此将 $v$ 加倍会使动能变为原来的 $2^2 = 4$ 倍。
The same $2\ \text{kg}$ ball now moves at $6\ \tfrac{\text{m}}{\text{s}}$. What is its kinetic energy, in joules? · 同一个 $2\ \text{kg}$ 的球现在以 $6\ \tfrac{\text{m}}{\text{s}}$ 的速度运动。它的动能是多少，单位为焦耳？
$E_k = \tfrac12 (2)(6^2) = 36\ \text{J}$ — four times the $9\ \text{J}$ at $3\ \tfrac{\text{m}}{\text{s}}$. · $E_k = \tfrac12 (2)(6^2) = 36\ \text{J}$ —— $9\ \text{J}$在$3\ \tfrac{\text{m}}{\text{s}}$处的四倍。
Select all · 所有 true statements about kinetic energy. · 选择所有关于动能的正确陈述。
$E_k = \tfrac12 mv^2 \ge 0$: it depends on $v^2$, is in joules, and is zero at rest. It is never negative. · $E_k = \tfrac12 mv^2 \ge 0$E_k=½mv²：它取决于$v^2$v²，单位是焦耳，且在静止时为零。它永远不会为负。
Work changes kinetic energy
- To speed something up you must do work on it (next lesson).
- The work–energy theorem: the net work equals the change in kinetic energy, $W_{\text{net}} = \Delta E_k$.
- Positive net work speeds it up; negative net work (like braking) slows it down.
- Kinetic energy is the "bank account" that work pays into or draws out of.
功改变动能
- 要让某物加速,你必须对它做功(下一节)。
- 功–能定理:合功等于动能的变化,$W_{\text{net}} = \Delta E_k$。
- 正的合功使它加速;负的合功(如刹车)使它减速。
- 动能就是功存入或取出的"银行账户"。
The work–energy theorem says the net work equals the change in ____ energy. · 功能原理指出净功等于____能量的变化。
$W_{\text{net}} = \Delta E_k$ — net work changes the kinetic energy. · $W_{\text{net}} = \Delta E_k$W_net=ΔE_k — 净功改变动能。
Kinetic energy depends on $v^2$, not $v$. A common error is to think doubling the speed doubles the energy. It quadruples it — which is why stopping distances grow so quickly with speed.
动能取决于**$v^2$,而不是 $v$。一个常见错误是认为速度加倍能量就加倍。它其实变为四倍**——这就是刹车距离随速度快速增长的原因。
Find the kinetic energy of a $2\ \text{kg}$ ball moving at $3\ \tfrac{\text{m}}{\text{s}}$.
- $E_k = \tfrac12 m v^2 = \tfrac12 (2)(3^2) = \tfrac12 (2)(9) = 9\ \text{J}$.
Double the speed to $6\ \tfrac{\text{m}}{\text{s}}$ and $E_k = \tfrac12(2)(36) = 36\ \text{J}$ — four times as much.
求一个 $2\ \text{kg}$ 的球以 $3\ \tfrac{\text{m}}{\text{s}}$ 运动的动能。
- $E_k = \tfrac12 m v^2 = \tfrac12 (2)(3^2) = \tfrac12 (2)(9) = 9\ \text{J}$。
速度加倍到 $6\ \tfrac{\text{m}}{\text{s}}$,$E_k = \tfrac12(2)(36) = 36\ \text{J}$——是原来的四倍。
Kinetic energy is the energy of motion, $E_k = \tfrac12 m v^2$ (a scalar, in joules). It depends on the square of speed, so doubling $v$ quadruples $E_k$. The work–energy theorem links it to work: $W_{\text{net}} = \Delta E_k$.
动能是运动的能量,$E_k = \tfrac12 m v^2$(标量,单位焦耳)。它取决于速度的平方,所以 $v$ 加倍,$E_k$ 变为四倍。功–能定理把它与功联系起来:$W_{\text{net}} = \Delta E_k$。