IB Physics: Analyzing Motion Graphs | IB物理：运动图像的分析方法

📚 IB Physics: Analyzing Motion Graphs | IB物理：运动图像的分析方法

Motion graphs are among the most frequently tested topics in IB Physics Paper 1 and Paper 2. A single graph can encode displacement, velocity, and acceleration simultaneously, and the skill of extracting quantitative information from them is essential for both SL and HL students.

运动图像是IB物理Paper 1和Paper 2中最常考查的主题之一。一张图像可以同时包含位移、速度和加速度的信息，掌握从图像中提取定量信息的能力对SL和HL学生都至关重要。

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1. The Three Core Motion Graphs | 三种核心运动图像

IB Physics focuses on three types of motion graph: displacement–time (s–t), velocity–time (v–t), and acceleration–time (a–t) graphs. Each graph has a unique physical meaning, and the relationships between them are governed by differentiation and integration.

IB物理主要关注三类运动图像：位移–时间（s–t）、速度–时间（v–t）和加速度–时间（a–t）图像。每类图像都有独特的物理意义，它们之间的关系由微分和积分决定。

The key idea is that the gradient (slope) of one graph gives the value of the next quantity up, while the area under a graph gives the next quantity down in the chain.

核心思想是：某一图像切线的斜率（梯度）给出链条中的下一个物理量，而图像下方的面积则给出链条中的上一个物理量。

The chain can be remembered as:

这条链条可以记忆为：

displacement → velocity → acceleration
位移 → 速度 → 加速度

Gradient of s–t gives v; gradient of v–t gives a; area under v–t gives displacement; area under a–t gives change in velocity.

s–t图像的斜率给出v；v–t图像的斜率给出a；v–t图像下方的面积给出位移；a–t图像下方的面积给出速度变化量。

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2. Displacement–Time Graphs: Reading Velocity from the Slope | 位移–时间图像：从斜率读取速度

On a displacement–time graph, the horizontal axis is time t and the vertical axis is displacement s (not distance). The gradient at any point equals the instantaneous velocity at that instant.

在位移–时间图像中，横轴为时间t，纵轴为位移s（注意不是路程）。图像上任意一点的斜率等于该时刻的瞬时速度。

For a straight-line s–t graph, the motion has constant velocity. The slope is calculated as:

对于直线型的s–t图像，物体做匀速运动。斜率计算如下：

v = Δs / Δt = (s₂ − s₁) / (t₂ − t₁)

A positive slope indicates motion in the positive direction, a negative slope indicates motion in the negative direction, and a zero slope means the object is at rest.

斜率为正表示沿正方向运动，斜率为负表示沿负方向运动，斜率为零表示物体静止。

When the graph is curved, the instantaneous velocity is found by drawing a tangent line at the point of interest and measuring its gradient. The average velocity between two times is found from the gradient of the secant (chord) joining the two corresponding points.

当图像为曲线时，瞬时速度通过在所关注的点处作切线并测量其斜率获得。某两时刻之间的平均速度则通过连接两对应点的割线（弦）的斜率求得。

• Key point: distance is the total path length; displacement is the straight-line change in position. A s–t graph that goes below the time axis shows motion in the opposite direction, but distance is always accumulated.

  关键点：路程是路径总长度；位移是位置沿直线方向的变化量。s–t图像落到时间轴下方表示反方向运动，但路程始终是累加的。

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3. Velocity–Time Graphs: Slopes and Areas | 速度–时间图像：斜率与面积

The velocity–time graph is the most information-rich graph in IB Physics. Its gradient gives acceleration, and the area between the graph and the time axis gives displacement.

速度–时间图像是IB物理中包含信息最丰富的图像。其斜率给出加速度，图像与时间轴之间围成的面积给出位移。

For uniform acceleration, the v–t graph is a straight line. The acceleration is constant and equals the gradient:

对于匀加速运动，v–t图像是一条直线。加速度恒定且等于斜率：

a = Δv / Δt = (v₂ − v₁) / (t₂ − t₁)

The displacement over a time interval is found by calculating the area under the graph. For a trapezium-shaped region:

某时间间隔内的位移通过计算图像下方的面积得到。对于梯形区域：

s = ½ (u + v) × t

where u is the initial velocity and v is the final velocity.

其中u是初速度，v是末速度。

Areas below the time axis count as negative displacement. For example, if a ball is thrown upward and returns to its starting point, the total displacement (net area) is zero, even though the total distance travelled is not zero.

时间轴下方的面积计为负位移。例如，竖直上抛的小球落回出发点时，总位移（净面积）为零，尽管总路程不为零。

When the acceleration is not uniform, the v–t graph is curved. In that case, the displacement can be estimated by counting squares or by using graphical integration.

当加速度不均匀时，v–t图像为曲线。此时位移可通过数方格或图形积分法估算。

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4. Acceleration–Time Graphs: Areas and Jumps | 加速度–时间图像：面积与突变

An acceleration–time graph shows how acceleration changes with time. The area under an a–t graph equals the change in velocity Δv over that interval.

加速度–时间图像表示加速度随时间的变化。a–t图像下方的面积等于该时间间隔内的速度变化量Δv。

Δv = area under a–t graph = ∫ a dt

If the acceleration is constant, the a–t graph is a horizontal straight line. The area is simply a × Δt, which matches the kinematic equation v = u + aΔt.

如果加速度恒定，a–t图像是一条水平直线。面积简化为a × Δt，这与运动学方程v = u + aΔt一致。

Sharp jumps in the a–t graph correspond to sudden changes in acceleration, such as at the moment a dropped ball bounces off the floor. In reality, these jumps are never perfectly instantaneous, but idealised graphs treat them as vertical lines.

a–t图像中的陡峭跳跃对应加速度的突然变化，例如小球落地弹起的瞬间。在现实中，这种跳跃不可能是完全瞬时的，但理想化图像将其处理为竖直线。

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5. Converting Between Graphs | 图像之间的转换

IB exam questions frequently ask you to construct one type of graph from another. The conversion rules are entirely based on slopes and areas.

IB考试题经常要求你根据一种图像构造另一种图像。转换规则完全基于斜率和面积。

To convert from s–t to v–t: take the gradient at every point. A straight-line s–t segment becomes a horizontal v–t segment; a parabolic s–t segment becomes a linear (sloped) v–t segment.

由s–t图转换为v–t图：取每一点的斜率。s–t图中的直线段变为v–t图中的水平段；s–t图中的抛物线段变为v–t图中的倾斜直线段。

To convert from v–t to a–t: take the gradient at every point. A horizontal v–t segment means zero acceleration; a sloped v–t segment means constant non-zero acceleration.

由v–t图转换为a–t图：取每一点的斜率。v–t图中的水平段表示加速度为零；倾斜段表示非零的恒定加速度。

To convert from v–t to s–t: integrate the area under the graph step by step, starting from the initial displacement s₀. Each new section of the s–t graph will have a gradient equal to the current velocity.

由v–t图转换为s–t图：从初始位移s₀开始逐步计算图像下方的面积。s–t图中每一段新曲线的斜率等于当前速度。

A common exam trap: when the v–t graph crosses the time axis (velocity changes sign), the s–t graph reaches a local maximum or minimum at that instant, because the object momentarily stops before reversing direction.

一个常见的考试陷阱：当v–t图像穿过时间轴（速度改变符号）时，s–t图像在该时刻达到局部极大值或极小值，因为物体在反向前瞬间停止。

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6. Curved Graphs: Instantaneous vs Average Values | 弯曲图像：瞬时值与平均值

For non-uniform motion, graphs become curved, and the distinction between instantaneous and average quantities becomes crucial.

对于非匀变速运动，图像变为曲线，此时区分瞬时量与平均值变得至关重要。

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