Representing Motion · 运动的表示
| English | 中文 | Pinyin · 拼音 |
|---|---|---|
| graph/ɡræf/ | 图像 | tú xiàng |
| slope/sləʊp/ | 斜率 | xié lǜ |
| velocity/vəˈlɒsɪti/ | 速度 | sù dù |
| acceleration/əkˌseləˈreɪʃn/ | 加速度 | jiā sù dù |
| area/ˈeərɪə/ | 面积 | miàn jī |
| displacement/dɪˈspleɪsmənt/ | 位移 | wèi yí |
One motion, four disguises
- A ball rolls down a ramp. That single motion can be told four ways.
- In words ("it speeds up steadily"), in a table of times and positions, in an equation, and in a graph 图像.
- Physicists switch between these fluently — each reveals something the others hide.
- The graph is often the most powerful: its shape is the story.
同一段运动,四副面孔
- 一个球滚下斜坡。这一段运动可以用四种方式讲述。
- 用文字("它稳稳地加速")、用时间与位置的表格、用方程,以及用图像。
- 物理学家在这几种之间自如切换——每一种都揭示了别的方式所隐藏的东西。
- 图像往往最有力:它的形状本身就是故事。
Position–time graphs
- Plot position against time: the vertical value is where the object is.
- The slope 斜率 (gradient) at any point is the velocity 速度.
- A straight, sloping line ⇒ constant velocity; a curve ⇒ velocity is changing.
- A horizontal line ⇒ the object is at rest (slope zero).
位置–时间图
- 把位置对时间作图:纵坐标是物体所在的位置。
- 任意一点的斜率(梯度)就是速度。
- 一条倾斜的直线 ⇒ 匀速;一条曲线 ⇒ 速度在变化。
- 一条水平线 ⇒ 物体静止(斜率为零)。
On a position–time graph, what does the slope of the line represent? · 在位置–时间图上，直线的斜率代表什么？
Slope is rise over run $= \Delta x / \Delta t$, which is the velocity. · 斜率是纵坐标变化除以横坐标变化$= \Delta x / \Delta t$，即速度。
A horizontal line on a position–time graph means the object is at rest. · 位置–时间图上的水平线意味着物体静止。
A horizontal line has zero slope, so the velocity is zero — the object is at rest. · 水平线斜率为零，因此速度为零——物体静止。
Velocity–time graphs
- Now plot velocity against time — a different graph of the same trip.
- The slope is the acceleration 加速度.
- The area 面积 between the line and the time axis is the displacement 位移.
- A line below the axis means negative velocity; its area counts as negative displacement.
速度–时间图
- 现在把速度对时间作图——同一段旅程的另一幅图。
- 斜率是加速度。
- 直线与时间轴之间的面积是位移。
- 位于轴线以下的线表示速度为负;它的面积算作负位移。
Slope and area on the graphs · 图线上的斜率和面积
Set a negative acceleration and see the velocity line fall while the position graph curves over. · 设置负加速度，观察速度线下降，位置图线弯曲。
On a velocity–time graph, what does the area between the line and the time axis represent? · 在速度–时间图上，直线与时间轴之间的面积代表什么？
Area $=$ velocity $\times$ time · 时间 $= \Delta x$, the displacement · 位移. · 面积$=$速度$\times$时间$= \Delta x$，即位移。
Reading area and slope
- Slope needs a rise over run; area needs the shape under the line.
- Under a straight v–t line, the area is a triangle or trapezium — use $\tfrac12 \times \text{base} \times \text{height}$.
- Below-axis area is subtracted; that is how an object can return toward its start.
- Always read the axis labels first: slope and area mean different things on different graphs.
读斜率与面积
- 斜率要看纵向变化 ÷ 横向变化;面积要看直线下方的形状。
- 在一条 v–t 直线下,面积是三角形或梯形——用 $\tfrac12 \times$ 底 $\times$ 高。
- 轴线以下的面积要减去;物体正是这样能够朝出发点折返。
- 计算前先读坐标轴标签:在不同的图上,斜率和面积含义不同。
On a v–t graph the velocity rises in a straight line from $0$ to · 到 $12\ \tfrac{\text{m}}{\text{s}}$ over $6\ \text{s}$. What is the displacement, in metres? · 在速度-时间图中，速度从 $0$ 到 $12\ \tfrac{\text{m}}{\text{s}}$ 呈直线上升，历时 $6\ \text{s}$。位移是多少米？
Area of the triangle $= \tfrac12 \times 6 \times 12 = 36\ \text{m}$. · 三角形面积$= \tfrac12 \times 6 \times 12 = 36\ \text{m}$。
When acceleration is constant: SUVAT
- With constant acceleration, four equations link $s,\ u,\ v,\ a,\ t$ (the "SUVAT" set):
- $v = u + at$
- $s = ut + \tfrac12 a t^2$
- $v^2 = u^2 + 2as$
- Pick the equation that contains your three knowns and the one unknown you want.
当加速度恒定:SUVAT
- 在匀加速下,四个方程把 $s,\ u,\ v,\ a,\ t$ 联系起来("SUVAT"组):
- $v = u + at$
- $s = ut + \tfrac12 a t^2$
- $v^2 = u^2 + 2as$
- 选那个包含你三个已知量和一个待求未知量的方程。
Match each graph feature to what it represents. · 将每个图线特征与其代表的含义匹配。
Slope of x–t is velocity; slope of v–t is acceleration; area under v–t is displacement. · x–t图的斜率是速度；v–t图的斜率是加速度；v–t图下的面积是位移。
Which SUVAT equation has no time $t$ in it? $v^2 = u^2 + 2a\_\_$. · 哪个SUVAT方程不包含时间$t$？$v^2 = u^2 + 2a\_\_$。
$v^2 = u^2 + 2as$ links velocity, acceleration and displacement without needing the time. · $v^2 = u^2 + 2as$在不需时间的情况下关联了速度、加速度和位移。
Slope and area are not interchangeable. On a velocity–time graph the slope is acceleration and the area is displacement. On a position–time graph the slope is velocity and the area has no physical meaning. Read the axes before you calculate.
斜率和面积不能互换。在速度–时间图上,斜率是加速度,面积是位移。在位置–时间图上,斜率是速度,而面积没有物理意义。计算前先看坐标轴。
A cart starts from rest and its velocity rises in a straight line to $12\ \tfrac{\text{m}}{\text{s}}$ over $6\ \text{s}$.
- The v–t graph is a triangle: base $6\ \text{s}$, height $12\ \tfrac{\text{m}}{\text{s}}$.
- Displacement $=$ area $= \tfrac12 \times 6 \times 12 = 36\ \text{m}$.
一辆小车从静止出发,速度沿直线在 $6\ \text{s}$ 内升到 $12\ \tfrac{\text{m}}{\text{s}}$。
- v–t 图是一个三角形:底 $6\ \text{s}$,高 $12\ \tfrac{\text{m}}{\text{s}}$。
- 位移 $=$ 面积 $= \tfrac12 \times 6 \times 12 = 36\ \text{m}$。
Read motion from graphs: on a position–time graph, slope = velocity; on a velocity–time graph, slope = acceleration and area = displacement. When acceleration is constant, the SUVAT equations ($v=u+at$, $s=ut+\tfrac12at^2$, $v^2=u^2+2as$) do the rest.
从图像读运动:在位置–时间图上,斜率 = 速度;在速度–时间图上,斜率 = 加速度、面积 = 位移。当加速度恒定时,SUVAT 方程($v=u+at$、$s=ut+\tfrac12at^2$、$v^2=u^2+2as$)搞定其余。