Translational Kinetic Energy · 平动动能
| English | 中文 | Pinyin · 拼音 |
|---|---|---|
| kinetic energy/kɪˈnetɪk ˈenədʒi/ | 动能 | dòng néng |
| scalar/ˈskeɪlə/ | 标量 | biāo liàng |
| work-energy theorem/wɜːk ˈenədʒi ˈθɪərəm/ | 功能定理 | gōng néng dìng lǐ |
| net work/net wɜːk/ | 净功 | jìng gōng |
Why speed is so dangerous
- A car at $40\ \text{km/h}$ does far more than double the damage of one at $20\ \text{km/h}$.
- It does four times the damage -- because energy depends on the square of speed.
- That hidden square is why small speed increases matter so much.
- The energy of motion has a name and a formula.
速度为什么如此危险
- $40\ \text{km/h}$ 的车造成的破坏,远不止 $20\ \text{km/h}$ 那辆车的两倍。
- 而是四倍——因为能量取决于速度的平方。
- 正是这个隐藏的平方,使得速度的小幅增加变得如此重要。
- 运动的能量有一个名字和一个公式。
Kinetic energy
- Kinetic energy 动能 is the energy an object has because it is moving:
- It is a scalar 标量 -- just a number, with no direction.
- Bigger mass or bigger speed means more kinetic energy.
动能
- 动能是物体因为运动而具有的能量:
- 它是一个标量——只是一个数,没有方向。
- 质量更大或速度更大,动能就更多。
A $4\ \text{kg}$ object moves at $3\ \tfrac{\text{m}}{\text{s}}$. What is its kinetic energy (in J)? · 一个 $4\ \text{kg}$ 的物体以 $3\ \tfrac{\text{m}}{\text{s}}$ 运动。其动能是多少（单位：J）？
$K = \tfrac{1}{2}mv^2 = \tfrac{1}{2}(4)(9) = 18\ \text{J}$.
Kinetic energy can be negative if an object moves backward. · 如果物体向后运动，动能可能为负。
The $v^2$ makes $K$ positive regardless of direction -- kinetic energy is a scalar and never negative. · $v^2$ 使得 $K$ 为正，无论方向如何——动能是标量且永远不为负。
The square is everything
- Because of the $v^2$, doubling the speed quadruples the kinetic energy.
- Triple the speed and you get nine times the energy.
- Mass matters too, but only linearly -- speed is the powerful one.
平方就是一切
- 由于 $v^2$,速度加倍会使动能变为四倍。
- 速度变为三倍,能量就变为九倍。
- 质量也有影响,但只是线性的——速度才是厉害的那个。
If a car's speed doubles, its kinetic energy becomes... · 如果汽车的速度加倍，其动能变为...
$K \propto v^2$, so doubling $v$ multiplies $K$ by $2^2 = 4$. · $K \propto v^2$，因此将 $v$ 加倍会使 $K$ 乘以 $2^2 = 4$。
Select all · 所有 true statements about kinetic energy. · 选择所有关于动能的正确陈述。
Kinetic energy is a positive scalar that scales with $v^2$; it is never negative. · 动能是一个正标量，按 $v^2$ 缩放；它永远不为负。
The work-energy theorem
- The work-energy theorem 功能定理 links force to energy:
- The net work done on an object equals its change in kinetic energy.
- Positive net work speeds it up; negative net work slows it down.
功能定理
- 功能定理把力和能量联系起来:
- 作用在物体上的净功等于它动能的变化。
- 正的净功使它加速;负的净功使它减速。
Kinetic energy grows with speed squared · 动能随速率的平方增长
Double the speed and the kinetic energy quadruples - it depends on the square of speed. · 速率加倍，动能变为四倍——它取决于速率的平方。
A $3\ \text{kg}$ cart goes from $2$ to · 到 $4\ \tfrac{\text{m}}{\text{s}}$. How much net work was done on it (in J)? · 一个 $3\ \text{kg}$ 的小车从 $2$ 变为 $4\ \tfrac{\text{m}}{\text{s}}$。对其做的净功是多少（单位：J）？
$\Delta K = \tfrac{1}{2}(3)(4^2) - \tfrac{1}{2}(3)(2^2) = 24 - 6 = 18\ \text{J}$.
The work-energy theorem says the net work equals the change in ____ energy. · 动能定理指出，净功等于 ____ 的变化量。
$W_{net} = \Delta K$ -- net work in, kinetic energy up. · $W_{net} = \Delta K$ —— 净功输入，动能增加。
For a whole system
- For a group of particles, add up each one's $\tfrac{1}{2}mv^2$ to get the total kinetic energy.
- The work-energy theorem still holds for the system as a whole.
- This lets energy handle complicated motions a single force diagram cannot.
对于整个系统
- 对于一组粒子,把每个粒子的 $\tfrac{1}{2}mv^2$ 相加,得到总动能。
- 功能定理对整个系统仍然成立。
- 这让能量能处理单个受力图无法处理的复杂运动。
A $2\ \text{kg}$ ball speeds up from $3$ to $5\ \tfrac{\text{m}}{\text{s}}$.
- $K_i = \tfrac{1}{2}(2)(3^2) = 9\ \text{J}$; $K_f = \tfrac{1}{2}(2)(5^2) = 25\ \text{J}$.
- By the theorem, the net work done was $\Delta K = 25 - 9 = 16\ \text{J}$.
一个 $2\ \text{kg}$ 的球从 $3$ 加速到 $5\ \tfrac{\text{m}}{\text{s}}$。
- $K_i = \tfrac{1}{2}(2)(3^2) = 9\ \text{J}$;$K_f = \tfrac{1}{2}(2)(5^2) = 25\ \text{J}$。
- 根据定理,所做的净功是 $\Delta K = 25 - 9 = 16\ \text{J}$。
Kinetic energy is never negative -- the $v^2$ makes it positive whichever way the object moves. And doubling speed does not double the energy; it quadruples it. Reaching for "twice as fast, twice the energy" is the classic trap.
动能永远不会为负——$v^2$ 使它无论物体朝哪个方向运动都为正。而且速度加倍不会使能量加倍;它会使能量变为四倍。想当然地认为"快一倍、能量一倍"是经典的陷阱。
Kinetic energy $K = \tfrac{1}{2}mv^2$ is a scalar that grows with the square of speed -- double the speed, quadruple the energy. The work-energy theorem says the net work equals the change in kinetic energy, $W_{net} = \Delta K$.
动能 $K = \tfrac{1}{2}mv^2$ 是一个随速度平方增长的标量——速度加倍,能量变四倍。功能定理指出净功等于动能的变化,$W_{net} = \Delta K$。