Motion in Two or Three Dimensions · 二维或三维运动
| English | 中文 | Pinyin · 拼音 |
|---|---|---|
| independent/ˌɪndɪˈpendənt/ | 独立的 | dú lì de |
| projectile motion/prəˈdʒektaɪl ˈməʊʃn/ | 抛体运动 | pāo tǐ yùn dòng |
A thrown ball does two things at once
- Throw a ball forward and it sails ahead and falls to the ground.
- Amazingly, these two motions do not interfere -- they happen side by side.
- Drop one ball and throw another horizontally: they hit the floor at the same time.
- Splitting motion into independent directions is the key to two dimensions.
抛出的球同时做两件事
- 向前抛出一个球,它既向前飞又向地面落下。
- 神奇的是,这两种运动互不干扰——它们并肩发生。
- 落下一个球,同时水平抛出另一个:它们同时落地。
- 把运动拆分成互相独立的方向,是二维问题的关键。
Components are independent
- Horizontal and vertical motion are independent 独立的 -- each obeys its own equations.
- Gravity pulls only down, so it changes the vertical motion but not the horizontal.
- Solve each direction as its own 1-D problem, then combine.
分量互相独立
- 水平运动和竖直运动是独立的——各自遵守自己的方程。
- 重力只向下拉,所以它改变竖直运动,却不改变水平运动。
- 把每个方向当作各自的一维问题求解,再组合。
A ball dropped from a height and one thrown horizontally from the same height hit the ground at the same time. · 从高度下落的球和从同一高度水平抛出的球同时落地。
Horizontal and vertical motion are independent -- both have the same vertical drop, so the same fall time. · 水平和垂直运动相互独立——两者有相同的垂直下落距离，因此有相同的下落时间。
Projectile motion
- Projectile motion 抛体运动 is the classic case: constant horizontal velocity, constant downward acceleration $g$.
- Horizontally: $x = v_x t$ (no acceleration).
- Vertically: it accelerates just like a dropped object.
抛体运动
- 抛体运动是经典情形:水平速度恒定,竖直向下的加速度恒为 $g$。
- 水平方向:$x = v_x t$(无加速度)。
- 竖直方向:它就像一个落体一样加速。
A ball is thrown horizontally from a $5\ \text{m}$ table ($g = 10$). How long until it lands (in s)? · 球从$5\ \text{m}$桌子水平抛出（$g = 10$）。落地需要多长时间（单位：s）？
Looking at the vertical direction, $5 = \tfrac{1}{2}(10)t^2$, so $t^2 = 1$ and $t = 1\ \text{s}$. The horizontal speed does not matter for the time. · 看垂直方向，$5 = \tfrac{1}{2}(10)t^2$，因此$t^2 = 1$和$t = 1\ \text{s}$。水平速度不影响时间。
During projectile flight (ignoring air resistance), the horizontal velocity... · 在抛体飞行期间（忽略空气阻力），水平速度...
Gravity acts only vertically, so there is no horizontal acceleration -- $v_x$ is constant. · 重力仅垂直作用，因此无水平加速度——$v_x$保持恒定。
Vector kinematics
- In 2-D and 3-D, position, velocity, and acceleration are all vectors: $\vec{r}$, $\vec{v}$, $\vec{a}$.
- Each component follows the same calculus chain, $x \to v_x \to a_x$.
- The full motion is just several 1-D motions bundled together.
矢量运动学
- 在二维和三维中,位置、速度和加速度都是矢量:$\vec{r}$、$\vec{v}$、$\vec{a}$。
- 每个分量都遵循同样的微积分链 $x \to v_x \to a_x$。
- 完整的运动只是几个一维运动捆在一起。
Motion in two dimensions · 二维运动
Launch a projectile and see the horizontal and vertical motions play out independently. · 发射抛体并观察水平和垂直运动如何独立展开。
The horizontal and vertical parts of a projectile's motion share only the . · 抛体运动的水平和垂直部分仅共享。
Time is the only shared variable; otherwise each direction is solved on its own. · 时间是唯一的共享变量；否则每个方向独立求解。
Putting it together
- Find the time from one direction (often the vertical), then use it in the other.
- The horizontal range depends on both the launch speed and how long the projectile is in the air.
- Recombine the component velocities to get the true speed and direction.
把它们组合起来
- 从一个方向(通常是竖直方向)求出时间,再把它用到另一个方向。
- 水平射程既取决于发射速度,也取决于抛体在空中停留多久。
- 把各分量速度重新组合,得到真实的速率和方向。
From that $5\ \text{m}$ table, the ball's horizontal speed is $4\ \tfrac{\text{m}}{\text{s}}$. How far from the base does it land (in m)? · 从该$5\ \text{m}$桌子，球的水平速度为$4\ \tfrac{\text{m}}{\text{s}}$。它落在离底座多远的地方（单位：m）？
Using $t = 1\ \text{s}$ from before, $x = v_x t = 4 \times 1 = 4\ \text{m}$. · 使用之前的$t = 1\ \text{s}$，$x = v_x t = 4 \times 1 = 4\ \text{m}$。
Select all · 所有 true statements about projectile motion. · 选出关于抛体运动的所有正确陈述。
The two directions are independent -- constant $v_x$, downward $g$ -- and do not affect each other. · 两个方向相互独立——恒定的$v_x$，向下的$g$——且互不影响。
A ball is thrown horizontally at $5\ \tfrac{\text{m}}{\text{s}}$ from a $20\ \text{m}$ cliff ($g = 10$).
- Vertical: $20 = \tfrac{1}{2}gt^2$ gives $t = 2\ \text{s}$ to land.
- Horizontal: $x = v_x t = 5 \times 2 = 10\ \text{m}$ from the base.
一个球从 $20\ \text{m}$ 的悬崖上以 $5\ \tfrac{\text{m}}{\text{s}}$ 水平抛出($g = 10$)。
- 竖直方向:$20 = \tfrac{1}{2}gt^2$ 给出落地时间 $t = 2\ \text{s}$。
- 水平方向:$x = v_x t = 5 \times 2 = 10\ \text{m}$,离崖底 $10\ \text{m}$。
The horizontal and vertical parts share only time. Do not mix a horizontal velocity into a vertical equation. The vertical motion decides when it lands; the horizontal motion decides how far -- keep them in separate columns.
水平部分和竖直部分只共享时间。不要把水平速度混进竖直方程。竖直运动决定它何时落地;水平运动决定它落多远——把它们放在各自的一栏里。
In two dimensions, horizontal and vertical motion are independent and share only the time. Projectile motion has constant horizontal velocity and downward acceleration $g$. Treat $\vec{r}$, $\vec{v}$, $\vec{a}$ component by component -- each is its own 1-D problem, then recombine.
在二维中,水平运动和竖直运动是独立的,只共享时间。抛体运动具有恒定的水平速度和向下的加速度 $g$。把 $\vec{r}$、$\vec{v}$、$\vec{a}$ 逐分量处理——每个都是各自的一维问题,再重新组合。