📚 IB Physics: Core Concepts of Kinematics and Graph Analysis Methods | IB物理：运动学核心概念与图像分析方法
Kinematics is the foundation of IB Physics. It describes motion without considering its causes, and mastering it is essential for understanding mechanics, energy, and even fields. This article covers the core definitions, equations, and graph interpretation skills you need for your IB exams.
运动学是IB物理的基础。它描述运动而不考虑运动的原因，掌握运动学对于理解力学、能量甚至场论都至关重要。本文将涵盖IB考试所需的核心定义、方程和图像解读技巧。
1. Scalars and Vectors | 标量与矢量
In kinematics, every physical quantity is either a scalar or a vector. A scalar has magnitude only, while a vector has both magnitude and direction. Distance and speed are scalars; displacement and velocity are vectors.
在运动学中，每一个物理量要么是标量，要么是矢量。标量只有大小，而矢量既有大小又有方向。路程和速率是标量；位移和速度是矢量。
- Distance / 路程: total length of path travelled, always positive. / 物体运动轨迹的总长度，始终为正。
- Displacement / 位移: straight-line distance from start to finish with direction. / 从起点到终点的直线距离，并带有方向。
- Speed / 速率: distance per unit time, scalar. / 单位时间内通过的路程，标量。
- Velocity / 速度: displacement per unit time, vector. / 单位时间内的位移，矢量。
Average speed = total distance ÷ total time
平均速率 = 总路程 ÷ 总时间
Average velocity = displacement ÷ time = Δs / Δt
平均速度 = 位移 ÷ 时间 = Δs / Δt
A common exam trap is confusing distance and displacement when an object changes direction. For example, if a ball rolls 3 m east then 4 m west, the distance is 7 m but the displacement is only 1 m west.
一个常见的考试陷阱是当物体改变方向时分不清路程和位移。例如，如果一个小球向东滚3米再向西滚4米，路程是7米，但位移只有1米（向西）。
2. Instantaneous Velocity and Acceleration | 瞬时速度与加速度
Instantaneous velocity is the velocity of an object at a specific moment in time. It is found by taking the limit of average velocity as the time interval approaches zero. Acceleration is the rate of change of velocity.
瞬时速度是物体在某一特定时刻的速度，它通过让时间间隔趋近于零时取平均速度的极限来获得。加速度是速度的变化率。
a = Δv / Δt = (v − u) / t
a = Δv / Δt = (v − u) / t
Here, u is initial velocity and v is final velocity. Acceleration is a vector. Negative acceleration does not always mean slowing down; it depends on the direction of the velocity. If velocity and acceleration point in opposite directions, the object slows down. If they point in the same direction, the object speeds up.
这里，u是初速度，v是末速度。加速度是矢量。负加速度并不总是意味着减速，这取决于速度的方向。如果速度与加速度方向相反，物体减速；如果方向相同，物体加速。
Deceleration occurs when v and a have opposite signs.
当v与a符号相反时，物体做减速运动。
3. The SUVAT Equations | 运动学四大公式（SUVAT）
e: s = displacement, u = initial velocity, v = final velocity, a = acceleration, t = time.
对于匀变速直线运动，IB物理要求你掌握四个SUVAT公式。各变量为：s = 位移，u = 初速度，v = 末速度，a = 加速度，t = 时间。
v = u + at
s = ut + ½at²
v² = u² + 2as
s = ½(u + v)t
These equations only apply when acceleration is constant. In free-fall near Earth' a = g = 9.8 m s⁻² downward. Always choose a positive direction and keep signs consistent.
这些公式仅适用于加速度恒定的情况。在地球表面附近的自由落体中，a = g = 9.8 m s⁻²，方向向下。务必选定正方向并保持符号一致。
| Suvat variable | Meaning | SI unit |
| s | displacement / 位移 | m |
| u | initial velocity / 初速度 | m s⁻¹ |
| v | final velocity / 末速度 | m s⁻¹ |
| a | acceleration / 加速度 | m s⁻² |
| t | time / 时间 | s |
When using SUVAT equations, always list what you know and what you need. Choose the equation that contains all known variables and only one unknown.
使用SUVAT公式时，先列出已知量和待求量，然后选择包含所有已知量且只有一个未知量的公式。
4. Free Fall and Projectile Motion Basics | 自由落体与抛体运动基础
Free fall is motion under the influence of gravity only. Air resistance is neglected in IB standard-level analysis. The acceleration is always 9.8 m s⁻² downward, regardless of the object's mass.
自由落体是仅受重力作用的运动。在IB标准分析中忽略空气阻力。无论物体质量如何，加速度始终为9.8 m s⁻²向下。
s no horizontal acceleration. The vertical motion follows the SUVAT equations with a = −g.
对于抛体运动，将水平运动和竖直运动分开处理。水平速度恒定，因为没有水平加速度。竖直运动遵循SUVAT方程，其中a = −g。
Horizontal: vₓ = uₓ, sₓ = uₓt
水平方向：vₓ = uₓ，sₓ = uₓt
Vertical: vᵧ = uᵧ − gt, sᵧ = uᵧt − ½gt²
竖直方向：vᵧ = uᵧ − gt，sᵧ = uᵧt − ½gt²
The time of flight is determined by the vertical motion only. The range is the horizontal distance travelled during that time.
飞行时间仅由竖直运动决定。射程是在这段时间内水平方向运动的距离。
5. Reading Displacement-Time Graphs | 位移-时间图像解读
A displacement-time graph shows how position changes over time. The slope of an s-t graph gives the velocity. A straight line means constant velocity; a curved line means changing velocity, i.e., acceleration.
位移-时间图像显示位置随时间的变化。s-t图像的斜率给出速度。直线表示匀速运动；曲线表示变速运动，即存在加速度。
- Slope = velocity / 斜率 = 速度
- Zero slope → object at rest / 斜率零 → 物体静止
- Positive slope → moving in positive direction / 斜率为正 → 向正方向运动
- Negative slope → moving in negative direction / 斜率为负 → 向负方向运动
- Increasing slope → speeding up / 斜率增大 → 加速
- Decreasing slope → slowing down / 斜率减小 → 减速
To find instantaneous velocity from an s-t graph, draw a tangent line at the point of interest and calculate its gradient. The steepness of the tangent tells you the speed at that instant.
要从s-t图像中求瞬时速度，应在感兴趣的点处画切线并计算其斜率。切线的陡峭程度表示该时刻的速率。
6. Reading Velocity-Time Graphs | 速度-时间图像解读
Velocity-time graphs are among the most important in IB Physics. The slope gives acceleration, and the area under the graph gives displacement.
速度-时间图像是IB物理中最重要的图像之一。斜率表示加速度，图像下方的面积表示位移。
Slope of v-t graph = acceleration
v-t图像的斜率 = 加速度
Area under v-t graph = displacement
v-t图像下方的面积 = 位移
Area above the time axis represents positive displacement. Area below the time axis represents negative displacement. The net displacement is the difference between the two areas.
时间轴上方的面积表示正位移，时间轴下方的面积表示负位移。净位移等于两者之差。
| Feature of v-t graph | Physical meaning |
| Horizontal line / 水平线 | Constant velocity, a = 0 / 匀速运动，a = 0 |
| Straight diagonal line / 倾斜直线 | Constant acceleration / 匀变速运动 |
| Curved line / 曲线 | Changing acceleration / 变加速运动 |
| Line crosses time axis / 穿过时间轴 | Direction reversal / 运动方向反转 |
To find displacement from a curved v-t graph, you may need to count squares or use integration. In IB exams, you will often be given simple shapes such as triangles, rectangles, and trapezoids.
要从弯曲的v-t图像中求位移，你可能需要数方格或使用积分。在IB考试中，通常会给出三角形、矩形和梯形等简单形状。
7. Reading Acceleration-Time Graphs | 加速度-时间图像解读
An acceleration-time graph shows how acceleration changes over time. The area under an a-t graph gives the change in velocity, Δv.
加速度-时间图像显示加速度随时间的变化。a-t图像下方的面积表示速度的变化量Δv。
Area under a-t graph = change in velocity = v − u
a-t图像下方的面积 = 速度变化量 = v − u
A horizontal line at a positive value means constant positive acceleration. A horizontal line at zero means constant velocity. The slope of an a-t graph has no direct physical meaning in standard IB kinematics, so do not calculate it unless asked.
在正值的水平线表示恒定正加速度。在零处的水平线表示匀速运动。在标准IB运动学中，a-t图像的斜率没有直接的物理意义，除非题目要求，否则不要计算它。
8. Converting Between Graphs | 图像之间的转换
IB exam questions often ask you to sketch one graph given another. For example, given a v-t graph, you can determine the a-t graph by taking the slope at every point, and determine the s-t graph by considering the area.
IB考试题经常要求你根据一个图像画出另一个图像。例如，给定v-t图像，你可以通过求各点斜率得到a-t图像，通过考虑面积得到s-t图像。
s-t graph → v-t graph: differentiate (slope). / s-t图像 → v-t图像：求导（斜率）。
v-t graph → a-t graph: differentiate (slope). / v-t图像 → a-t图像：求导（斜率）。
a-t graph → v-t graph: integrate (area). / a-t图像 → v-t图像：积分（面积）。
v-t graph → s-t graph: integrate (area). / v-t图像 → s-t图像：积分（面积）。
When converting, pay attention to the initial conditions. The initial displacement and initial velocity are needed to determine the integration constants.
转换时注意初始条件。初始位移和初始速度用于确定积分常数。
9. Common Exam Errors and Tips | 常见考试错误与技巧
Many students lose marks in kinematics questions because of simple sign errors or unit mistakes. Always define the positive direction before solving a problem, and always check whether the acceleration is constant before using SUVAT equations.
许多学生在运动学题目中因为简单的符号错误或单位错误而失分。解题前务必定义正方向，并检查加速度是否恒定，然后才能使用SUVAT公式。
- Tip 1 / 技巧1: Read whether the question asks for distance or displacement. They are different quantities. / 仔细阅读题目问的是路程还是位移，两者不同。
- Tip 2 / 技巧2: At the highest point of projectile motion, vertical velocity is 0, but vertical acceleration is still g. / 在抛体运动的最高点，竖直速度为0，但竖直加速度仍为g。
- Tip 3 / 技巧3: On a velocity-time graph, a line below the time axis means the object is moving in the negative direction, not necessarily slowing down. / 在速度-时间图像中，时间轴下方的线表示物体向负方向运动，并不一定在减速。
- Tip 4 / 技巧4: antities are in SI units: metres, seconds, metres per second, metres per second squared. / 使用v = u + at时，确保所有量均为SI单位：米、秒、米每秒、米每二次方秒。
- Tip 5 / 技巧5: Always sketch a small diagram for motion problems. It clarifies directions and helps avoid confusion. / 运动学问题一定要画示意图，它有助于明确方向，避免混淆。
10. Worked Example | 例题精讲
maximum height reached; (b) the total time of flight; (c) the velocity just before hitting the ground.
一个球从地面以20 m s⁻¹的初速度竖直上抛。取g = 10 m s⁻²。求：(a)能达到的最大高度；(b)总飞行时间；(c)落地前瞬间的速度。
Solution / 解答:
Choose upward as positive. At the maximum height, v = 0.
取向上为正方向。在最大高度处，v = 0。
(a) Using v² = u² + 2as:
(a) 使用v² = u² + 2as：
0² = 20² + 2(−10)s → s = 20 m
0² = 20² + 2(−10)s → s = 20 m
(b) Using v = u + at for the upward journey:
(b) 上升阶段使用v = u + at：
0 = 20 + (−10)t → t = 2 s
0 = 20 + (−10)t → t = 2 s
ownward journey takes the same time as the upward journey because the motion is symmetric.
总飞行时间 = 2 × 2 = 4 s。由于运动是对称的，下落阶段与上升阶段所用时间相同。
(c) Using v = u + at for the whole journey:
(c) 全程使用v = u + at：
v = 20 + (−10)(4) = −20 m s⁻¹
v = 20 + (−10)(4) = −20 m s⁻¹
The negative sign shows the ball is moving downward just before impact, with the same speed as the initial speed.
负号表示球在落地前瞬间向下运动，速度大小与初速度相同。
11. Graphical Analysis for a Bouncing Ball | 弹跳球的图像分析
Bouncing ball questions are classic IB kinematics problems. When a ball bounces, its velocity changes sign suddenly at impact. On a v-t graph, this appears as a sharp vertical jump. On an s-t graph, the slope changes sign continuously because the position curve is smooth.
弹跳球问题是经典的IB运动学问题。当球弹跳时，其速度在碰撞瞬间突然改变方向。在v-t图像中，这表现为一条陡峭的竖直跳跃。在s-t图像中，斜率连续改变符号，因为位置曲线是光滑的。
During free flight between bounces, the acceleration is constant at −g, so the v-t graph has the same slope for all segments. The maximum height decreases after each bounce if the ball is not perfectly elastic.
在两次弹跳之间的自由飞行阶段，加速度恒为−g，因此v-t图像的所有线段斜率相同。如果球不是完全弹性的，每次弹跳后的最大高度会减小。
12. Summary | 总结
Kinematics is a highly visual topic in IB Physics. You must be able to interpret displacement-time, velocity-time, and acceleration-time graphs fluently, and switch between them. Mastering the SUVAT equations, understanding vector direction, and avoiding common sign errors will boost your exam performance significantly.
运动学是IB物理中极具图像化特征的主题。你必须熟练解读位移-时间、速度-时间和加速度-时间图像，并能在它们之间自如转换。掌握SUVAT公式、理解矢量方向并避免常见的符号错误，将极大提升你的考试成绩。
| Graph / 图像 | Slope / 斜率 | Area / 面积 |
| s-t | velocity / 速度 | not used / 不使用 |
| v-t | acceleration / 加速度 | displacement / 位移 |
| a-t | not used / 不使用 | change in velocity / 速度变化量 |
Remember that every graph tells a story. Read the story carefully, and the equations will follow naturally. Practice converting between graphs and solving multi-stage motion problems; these skills appear frequently in both paper 1 and paper 2 of IB Physics.
记住，每个图像都在讲述一个运动的故事。仔细读懂这个故事，方程就会自然浮现。多练习图像转换和多阶段运动问题的求解，这些技能在IB物理试卷1和试卷2中经常出现。
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