Displacement, Velocity, and Acceleration · 位移、速度与加速度
| English | 中文 | Pinyin · 拼音 |
|---|---|---|
| displacement/dɪˈspleɪsmənt/ | 位移 | wèi yí |
| velocity/vəˈlɒsɪti/ | 速度 | sù dù |
| acceleration/əkˌseləˈreɪʃn/ | 加速度 | jiā sù dù |
| instantaneous/ˌɪnstənˈteɪnɪəs/ | 瞬时 | shùn shí |
Walk around the block, end up nowhere
- Walk all the way around a city block and you have covered plenty of ground.
- Yet you finish exactly where you started -- so your net change in position is zero.
- Physics needs to tell these two apart: how far you travelled versus how far you ended up.
- The distinction runs through all of motion.
绕街区走一圈,却哪也没到
- 绕着城市街区走一整圈,你走了不少路。
- 可你恰好回到了出发的地方——所以你位置的净变化为零。
- 物理需要把这两者区分开:你走了多远,与你最终到了多远。
- 这个区别贯穿整个运动学。
Displacement versus distance
- Displacement 位移 is the vector from start to finish -- straight-line, with direction.
- Distance is the scalar total path length -- always positive, no direction.
- Around a loop, distance is large but displacement is zero.
位移与距离
- 位移是从起点到终点的矢量——直线,带方向。
- 距离是标量的总路径长度——总是正的,没有方向。
- 绕一圈时,距离很大,但位移为零。
You jog once around a $400\ \text{m}$ track and stop where you began. What is your displacement (in m)? · 你慢跑一圈$400\ \text{m}$跑道并在起点停下。你的位移是多少（单位：m）？
Displacement is start-to-finish. You end where you began, so it is $0$ -- though the distance was $400\ \text{m}$. · 位移是从起点到终点的距离。你回到了起点，所以是$0$——尽管路程是$400\ \text{m}$。
Velocity versus speed
- Velocity 速度 is the rate of change of displacement, a vector: $\vec{v} = \dfrac{d\vec{x}}{dt}$.
- Speed is its magnitude -- how fast, with no direction.
- Average velocity uses total displacement; average speed uses total distance.
速度与速率
- 速度是位移的变化率,一个矢量:$\vec{v} = \dfrac{d\vec{x}}{dt}$。
- 速率是它的大小——有多快,没有方向。
- 平均速度用总位移;平均速率用总距离。
The vector rate of change of displacement is called the . · 位移的矢量变化率称为。
$\vec{v} = d\vec{x}/dt$ -- velocity is the (vector) rate of change of displacement. · $\vec{v} = d\vec{x}/dt$——速度是位移的（矢量）变化率。
Acceleration
- Acceleration 加速度 is the rate of change of velocity: $\vec{a} = \dfrac{d\vec{v}}{dt}$.
- Speeding up, slowing down, or turning all count as accelerating.
- Its direction need not match the velocity -- braking points it backward.
加速度
- 加速度是速度的变化率:$\vec{a} = \dfrac{d\vec{v}}{dt}$。
- 加速、减速或转弯都算作加速。
- 它的方向不必与速度一致——刹车时它指向后方。
Motion graphs · 运动图像
Set an initial velocity and acceleration, then read displacement, velocity and acceleration off the graphs. · 设定初始速度和加速度，然后从图像中读取位移、速度和加速度。
A car moving at constant speed around a curve is accelerating. · 汽车以恒定速率沿曲线行驶时正在加速。
Its direction changes, so its velocity changes -- that is acceleration, even at constant speed. · 其方向改变，因此速度改变——这就是加速，即使速率恒定。
Select all · 所有 situations that count as accelerating. · 选出所有属于加速的情况。
Any change in velocity -- faster, slower, or a change of direction -- is an acceleration. · 任何速度的变化——变快、变慢或方向改变——都是加速。
The calculus chain
- Position, velocity, and acceleration form a derivative chain: $x \xrightarrow{d/dt} v \xrightarrow{d/dt} a$.
- Differentiate to go down the chain; integrate to climb back up.
- Know any one as a function of time and calculus gives you the others.
微积分链条
- 位置、速度和加速度构成一条求导链:$x \xrightarrow{d/dt} v \xrightarrow{d/dt} a$。
- 求导沿链向下;积分沿链爬回。
- 只要知道其中任一个关于时间的函数,微积分就能给出其余的。
If position is $x(t)$, how do you get the acceleration? · 如果位置是$x(t)$，如何求加速度？
$v = dx/dt$ and $a = dv/dt$, so $a$ is the second derivative of position. · $v = dx/dt$和$a = dv/dt$，因此$a$是位置的二阶导数。
For · 支持 $x(t) = 2t^2$, find the velocity at $t = 3$ (in m/s). · 对于$x(t) = 2t^2$，求在$t = 3$处的速度（单位：m/s）。
$v = dx/dt = 4t$, so $v(3) = 12\ \tfrac{\text{m}}{\text{s}}$. · $v = dx/dt = 4t$，所以 $v(3) = 12\ \tfrac{\text{m}}{\text{s}}$。
A particle's position is $x(t) = t^3 - 6t^2 + 9t$ (metres).
- Velocity: $v = \dfrac{dx}{dt} = 3t^2 - 12t + 9$.
- Acceleration: $a = \dfrac{dv}{dt} = 6t - 12$. At $t = 2$, $a = 0$ -- the velocity is momentarily not changing.
一个粒子的位置是 $x(t) = t^3 - 6t^2 + 9t$(米)。
- 速度:$v = \dfrac{dx}{dt} = 3t^2 - 12t + 9$。
- 加速度:$a = \dfrac{dv}{dt} = 6t - 12$。在 $t = 2$ 时,$a = 0$——速度此刻不变。
An instantaneous 瞬时 velocity and an average velocity are not the same. Average velocity is total displacement over total time; instantaneous velocity is the derivative at one moment. On a curved position graph they can differ a lot.
瞬时速度和平均速度不是一回事。平均速度是总位移除以总时间;瞬时速度是某一时刻的导数。在弯曲的位置图上,两者可能相差很大。
Displacement (vector, start-to-finish) differs from distance (scalar path length). Velocity $= d\vec{x}/dt$ and acceleration $= d\vec{v}/dt$ form a derivative chain $x \to v \to a$. Turning counts as accelerating, and instantaneous values come from derivatives, not averages.
位移(矢量,起点到终点)不同于距离(标量路径长度)。速度 $= d\vec{x}/dt$ 和加速度 $= d\vec{v}/dt$ 构成求导链 $x \to v \to a$。转弯算作加速,而瞬时值来自导数,不是平均值。