Parametric Functions and Planar Motion · 参数函数与平面运动
| English | 中文 | Pinyin · 拼音 |
|---|---|---|
| parametric function/ˌpærəˈmetrɪk ˈfʌŋkʃn/ | 参数方程函数 | cān shù fāng chéng hán shù |
| parameter/pəˈræmɪtə/ | 参数 | cān shù |
Where is it, right now?
- Throw a ball and it follows a graceful arc through the air.
- To describe that flight, we track its horizontal and vertical positions over time.
- Each is its own function of time — and together they give the ball's location at every moment.
- This is exactly what parametric functions were made for.
它此刻在哪里?
- 抛出一个球,它在空中划出一道优美的弧线。
- 要描述这段飞行,我们跟踪它随时间变化的水平和竖直位置。
- 每一个都是自己的时间函数——两者合起来,给出球在每一刻的位置。
- 这正是参数方程被创造出来做的事。
Position over time
- A parametric function 参数方程函数 models motion by giving position $(x(t), y(t))$ at each time.
- The parameter 参数 $t$ is the clock; plug in a time and read off the location.
- Horizontal motion lives in $x(t)$; vertical motion lives in $y(t)$.
- The pair traces the object's path across the plane.
随时间变化的位置
- 参数方程函数(parametric function)通过给出每一时刻的位置 $(x(t), y(t))$ 来给运动建模。
- 参数(parameter)$t$ 是那只钟;代入一个时间,读出位置。
- 水平运动存在于 $x(t)$ 里;竖直运动存在于 $y(t)$ 里。
- 这一对描出物体在平面上的路径。
When parametric functions model motion, the parameter $t$ usually represents… · 当参数函数模拟运动时，参数$t$通常代表...
In a · 在 parametric function modeling motion, $t$ is time, and $(x(t), y(t))$ is the object's position then. · 在模拟运动的参数函数中，$t$是时间，$(x(t), y(t))$是物体当时的位置。
Splitting the motion
- The great trick is that $x$ and $y$ can be analysed independently.
- A thrown ball moves at a steady horizontal speed while gravity curves its vertical motion.
- So $x(t)$ might be linear while $y(t)$ is a downward parabola.
- Combine them and the ball follows its familiar arc.
拆分运动
- 最妙的一招是,$x$ 和 $y$ 可以独立分析。
- 一个抛出的球以稳定的水平速度前进,而重力让它的竖直运动弯曲。
- 所以 $x(t)$ 可能是线性的,而 $y(t)$ 是一条向下的抛物线。
- 把它们合起来,球就沿着它熟悉的弧线飞行。
A ball has $x = 3t$ and $y = 20t - 5t^2$. What is its height $y$ at $t = 2$? · 一个球有 $x = 3t$ 和 $y = 20t - 5t^2$。它的高度是 $y$ 在 $t = 2$?
$y = 20(2) - 5(2)^2 = 40 - 20 = 20$; and $x = 3(2) = 6$, so the ball is at $(6, 20)$. · $y = 20(2) - 5(2)^2 = 40 - 20 = 20$；以及$x = 3(2) = 6$，因此球位于$(6, 20)$。
The object's starting position is found by evaluating $x$ and $y$ at $t =$ ____. · 物体的初始位置通过评估$x$和$y$在$t =$处____获得。
At $t = 0$ the motion begins, so $(x(0), y(0))$ is the initial position. · 在$t = 0$时运动开始，所以$(x(0), y(0))$是初始位置。
Select all · 所有 true statements about parametric motion. · 选择关于参数运动的所有正确陈述。
A parametric path can curve (like a thrown ball's arc), not just be a line. The other three are correct. · 参数路径可以是曲线（如投掷球的轨迹），而不仅仅是直线。其他三项是正确的。
Reading the motion
- The starting point is at $t = 0$: evaluate $x(0)$ and $y(0)$.
- Later times give later positions along the path.
- Where the path is highest, the vertical motion has momentarily stopped rising.
- Following $t$ upward shows the direction of travel.
读出运动
- 起点在 $t = 0$:求 $x(0)$ 和 $y(0)$。
- 更晚的时间给出路径上更晚的位置。
- 路径最高处,竖直运动一时停止上升。
- 沿 $t$ 增大的方向看,就看出运动的方向。
A path traced by a parameter · 参数描绘的路径
As the parameter runs, the point moves and traces out a planar curve. · 随着参数的运行，点移动并描绘出一条平面曲线。
The horizontal and vertical motions can be analysed separately as $x(t)$ and $y(t)$. · 水平和垂直运动可以分别作为$x(t)$和$y(t)$进行分析。
Splitting motion into independent $x$ and $y$ parts is the great strength of the parametric view. · 将运动分解为独立的$x$和$y$部分是参数视图的强大之处。
Beyond position
- From the position functions you can ask how fast each coordinate is changing.
- That leads to velocity — the next lesson's rates of change.
- You can also find when the object lands (where $y = 0$ again) or how far it travels.
- Parametric motion turns geometry into a moving story.
超越位置
- 从位置函数,你可以问每个坐标变化得多快。
- 那就引向速度——下一课的变化率。
- 你也能求出物体何时落地(何处 $y$ 再次为 $0$)或它走了多远。
- 参数运动把几何变成一个运动的故事。
Do not confuse the path with the motion. The picture of the arc shows the shape, but the parametrization also encodes when the ball is at each point and how fast. Two balls can share a path yet move along it very differently.
不要把路径和运动搞混。弧线的图显示了形状,但参数化还编码了球何时在每个点、多快。两个球可以共享一条路径,却沿着它以非常不同的方式运动。
A ball has $x = 3t$ and $y = 20t - 5t^2$ (metres, seconds).
- At $t = 0$: position $(0, 0)$ — it starts at the origin.
- At $t = 2$: $x = 6$, $y = 40 - 20 = 20$ — it is $6$ m across and $20$ m high.
- It lands when $y = 0$ again: $20t - 5t^2 = 0 \Rightarrow t = 4$ s.
一个球有 $x = 3t$、$y = 20t - 5t^2$(米,秒)。
- 在 $t = 0$:位置 $(0, 0)$——它从原点出发。
- 在 $t = 2$:$x = 6$,$y = 40 - 20 = 20$——它横向 $6$ 米、高 $20$ 米。
- 它在 $y$ 再次为 $0$ 时落地:$20t - 5t^2 = 0 \Rightarrow t = 4$ 秒。
A parametric function models planar motion by giving position $(x(t), y(t))$ at each time, with the parameter $t$ as the clock. Horizontal and vertical motion split into $x(t)$ and $y(t)$, which can be analysed separately and recombined into the path.
参数方程函数通过给出每一时刻的位置 $(x(t), y(t))$ 来给平面运动建模,参数 $t$ 就是那只钟。水平和竖直运动拆成 $x(t)$ 和 $y(t)$,可以分开分析,再重新组合成路径。