IB Physics: Kinematics Key Points | IB物理：运动学考点精讲

📚 IB Physics: Kinematics Key Points | IB物理：运动学考点精讲

Kinematics is the branch of mechanics that describes motion without considering its causes. In IB Physics, kinematics forms the foundation for mechanics, circular motion, and even fields. Mastering the key definitions, graphs, and equations is essential for exam success.

运动学是力学中描述运动而不考虑其成因的分支。在IB物理中，运动学是力学、圆周运动乃至场论的基础。掌握核心定义、图像和方程，是考试成功的关键。

1. Scalars and Vectors | 标量与矢量

In kinematics, every 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，每个物理量不是标量就是矢量。标量只有大小，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 (s) is the total length of the path travelled, regardless of direction. It is always positive.

路程（s）是物体运动轨迹的总长度，与方向无关，始终为正。
Displacement (s) is the straight-line distance from the initial to the final position, including direction. It can be positive, negative, or zero.

位移（s）是从初位置到末位置的直线距离，包含方向，可以为正、负或零。
Speed is the rate of change of distance: \(v = \frac{\Delta s}{\Delta t}\). Velocity is the rate of change of displacement: \(\vec{v} = \frac{\Delta \vec{s}}{\Delta t}\).

速率是路程的变化率：v = Δs/Δt。速度是位移的变化率：v = Δs/Δt（矢量）。

Average speed = total distance / total time
Average velocity = displacement / time

In IB exams, always check whether the question asks for speed or velocity. Using the wrong one is a common mistake.

在IB考试中，务必看清题目问的是速率还是速度。用错概念是常见错误。

2. Distance–Time and Position–Time Graphs | 路程-时间图与位置-时间图

Graphs are powerful tools for analysing motion. On a position-time graph, the slope at any point gives the instantaneous velocity.

图像是分析运动的强大工具。在位置-时间图中，任意一点的斜率给出瞬时速度。

A straight line with positive slope on a position-time graph means constant positive velocity.

位置-时间图上斜率为正的直线表示恒定正速度。
A curved line means the velocity is changing. The tangent at a point gives the instantaneous velocity.

曲线表示速度在变化。某点的切线给出瞬时速度。
A horizontal line means the object is at rest; displacement is constant.

水平线表示物体静止，位移不变。

For distance-time graphs, the slope gives speed, and distance never decreases. Position-time graphs can show negative positions as well.

对于路程-时间图，斜率给出速率，路程永不减少。位置-时间图则可以显示负的位置。

3. Velocity–Time Graphs | 速度-时间图

The velocity-time graph is one of the most important tools in IB kinematics. Its slope gives acceleration, and the area under the graph gives displacement.

速度-时间图是IB运动学中最重要的工具之一。其斜率给出加速度，图像下方的面积给出位移。

Slope = acceleration \(a = \frac{\Delta v}{\Delta t}\). A positive slope means speeding up in the positive direction; a negative slope means slowing down or moving in the negative direction.

斜率 = 加速度 a = Δv/Δt。正斜率表示沿正方向加速；负斜率表示减速或沿负方向运动。
The area between the graph and the time axis equals displacement. Areas above the axis are positive; areas below are negative.

图像与时间轴之间的面积等于位移。轴上方的面积为正，下方的面积为负。
A horizontal line on a velocity-time graph means constant velocity (zero acceleration).

速度-时间图上的水平线表示匀速运动（加速度为零）。

Displacement = area under v-t graph
Acceleration = slope of v-t graph

Be careful: total distance is the sum of the magnitudes of all areas, while displacement is the signed sum.

注意：总路程是所有面积的绝对值之和，而位移是带符号面积之和。

4. Acceleration–Time Graphs | 加速度-时间图

Acceleration-time graphs show how acceleration changes over time. The area under an a-t graph gives the change in velocity.

加速度-时间图表示加速度随时间的变化。a-t图下的面积给出速度的变化量。

A constant positive acceleration appears as a horizontal line above the time axis.

恒定正加速度表现为时间轴上方的水平线。
The change in velocity \(\Delta v\) equals the area under the a-t graph.

速度变化量 Δv 等于 a-t 图下的面积。
If the a-t graph has a negative region, the velocity decreases during that interval.

如果a-t图出现负区域，则在该时间段内速度减小。

In IB data analysis questions, you may need to sketch an a-t graph from a v-t graph. Remember: the slope of v-t becomes the value of a-t.

在IB数据分析题中，你可能需要根据v-t图画出a-t图。记住：v-t图的斜率就是a-t图的值。

5. Equations of Motion for Constant Acceleration | 匀变速直线运动方程

The four kinematic equations, also called suvat equations, apply only when acceleration is constant. They connect displacement (s), initial velocity (u), final velocity (v), acceleration (a), and time (t).

四个运动学方程（又称suvat方程）仅在加速度恒定时适用。它们联系位移（s）、初速度（u）、末速度（v）、加速度（a）和时间（t）。

v = u + at
s = ut + ½at²
s = ½(u + v)t
v² = u² + 2as

Each equation uses four of the five variables. Choose the one that omits the variable you do not know and do not need.

每个方程使用五个变量中的四个。选择省略你不知道也不需要知道的变量的方程。
Always define a positive direction. Then take displacement, velocity, and acceleration as positive or negative consistently.

务必先规定正方向。然后一致地将位移、速度和加速度取正或负。
These equations are vector equations in one dimension. In IB, we usually treat them algebraically with signed scalars.

这些方程是一维矢量方程。在IB中，我们通常用带符号的标量进行代数运算。

Common traps: using the equations when acceleration is not constant, or forgetting to convert units (e.g., km/h to m/s).

常见陷阱：在加速度不恒定时使用这些方程，或忘记换算单位（如km/h换算为m/s）。

6. Projectile Motion Basics | 抛体运动基础

Projectile motion is motion in two dimensions under constant gravitational acceleration. The horizontal and vertical motions are independent.

抛体运动是在恒定重力加速度下的二维运动。水平运动和竖直运动相互独立。

Horizontal motion: constant velocity, \(a_x = 0\). Thus \(x = u_x t\).

水平方向：匀速运动，aₓ = 0。因此 x = uₓt。
Vertical motion: constant acceleration \(a_y = -g\), typically \(g = 9.8 \text{ m s}^{-2}\) downward.

竖直方向：匀加速运动，a_y = -g，通常g = 9.8 m/s²，方向向下。
The initial velocity components are \(u_x = u\cos\theta\) and \(u_y = u\sin\theta\), where \(\theta\) is the launch angle above the horizontal.

初速度分量为 uₓ = u cosθ，u_y = u sinθ，其中θ是相对于水平面的抛射角。

Time of flight, maximum height, and range can be derived from the suvat equations. For example, the time to reach maximum height is \(t = u_y/g\), and the range is \(R = u^2\sin(2\theta)/g\) when landing at the same height.

飞行时间、最大高度和射程可由suvat方程推导。例如，达到最大高度的时间为 t = u_y/g，当落回同一高度时，射程为 R = u²sin(2θ)/g。

7. Independence of Horizontal and Vertical Motion | 水平与竖直运动的独立性

A key concept in projectile motion is that the horizontal motion is unaffected by the vertical motion. A bullet dropped and a bullet fired horizontally from the same height will hit the ground at the same time.

抛体运动的一个关键概念是水平运动不受竖直运动影响。从同一高度同时释放的子弹和水平射出的子弹会同时落地。

In the absence of air resistance, the horizontal velocity remains constant.

无空气阻力时，水平速度保持不变。
The vertical acceleration is always \(g\) downward, regardless of the horizontal velocity.

竖直加速度始终为向下的g，与水平速度无关。
This independence allows us to solve projectile problems by treating the two axes separately.

这种独立性使我们能够将抛体问题分解为两个互相独立的方向来求解。

In IB Paper 2, you may be asked to describe or prove this independence using a strobe photograph or video analysis.

在IB Paper 2中，你可能会被要求用频闪照片或视频分析来描述或证明这种独立性。

8. Effect of Air Resistance | 空气阻力的影响

Real projectiles experience drag, which opposes motion and depends on speed. Air resistance reduces both range and maximum height, and the trajectory is no longer a perfect parabola.

真实抛体会受到空气阻力，阻力与运动方向相反，且随速度变化。空气阻力会减小射程和最大高度，运动轨迹不再是完美的抛物线。

During the upward motion, both gravity and drag act downward, so the deceleration is greater than \(g\).

在上升阶段，重力和阻力都向下，因此减速的加速度大于g。
During the downward motion, gravity acts downward but drag acts upward, so the acceleration is less than \(g\).

在下降阶段，重力向下但阻力向上，因此加速度小于g。
The maximum height and range are smaller, and the angle for maximum range is no longer exactly 45°.

最大高度和射程更小，最大射程对应的角度也不再恰好是45°。

IB questions often ask you to sketch the projectile path with air resistance compared to the ideal one. The real path is lower and steeper on descent.

IB题目常要求你画出有空气阻力时的轨迹并与理想轨迹比较。真实轨迹更低，下落阶段更陡。

9. Relative Motion | 相对运动

Relative velocity describes the velocity of one object as observed from another moving object. In IB, this is usually treated in one or two dimensions.

相对速度描述的是一个物体相对于另一个运动物体的速度。在IB中，通常在一维或二维中处理。

For objects moving along the same line, relative velocity is \(v_{AB} = v_A – v_B\).

对于沿同一直线运动的物体，相对速度为 v_AB = v_A – v_B。
For two-dimensional relative velocity, use vector subtraction: \(\vec{v}_{AB} = \vec{v}_A – \vec{v}_B\).

对于二维相对速度，使用矢量减法：v_AB = v_A – v_B（矢量）。
A classic example is a boat crossing a river. To land directly opposite, the boat must head upstream at an angle such that its velocity perpendicular to the flow cancels the river current.

一个经典例子是小船渡河。若要对岸正对出发点，船必须朝上游倾斜一个角度，使得垂直于水流方向的速度分量抵消水流的影响。

Remember: when two objects move toward each other, their closing speed is the sum of their speeds; when moving in the same direction, it is the difference.

记住：两物体相向运动时，接近速度是两者速率之和；同向运动时，接近速度是两者速率之差。

10. Uniform Circular Motion as Kinematics | 匀速圆周运动中的运动学

Circular motion is also part of kinematics. In uniform circular motion, the speed is constant, but the velocity changes because the direction changes continuously.

圆周运动也属于运动学范畴。在匀速圆周运动中，速率恒定，但由于方向不断改变，速度时刻在变化。

The period \(T\) is the time for one complete revolution. Frequency \(f = 1/T\).

周期T是完成一整圈所需的时间。频率 f = 1/T。
Angular speed \(\omega = 2\pi/T = 2\pi f\), measured in rad/s.

角速度 ω = 2π/T = 2πf，单位rad/s。
Linear speed \(v = \omega r\), where \(r\) is the radius.

线速度 v = ωr，其中r为半径。
The centripetal acceleration is \(a_c = v^2/r = \omega^2 r\), directed toward the centre of the circle.

向心加速度 a_c = v²/r = ω²r，方向指向圆心。

Although centripetal acceleration arises from forces, the kinematic description of circular motion is required for many IB mechanics questions.

虽然向心加速度源于力，但许多IB力学问题需要用到圆周运动的运动学描述。

11. Data Analysis and Graphing Skills | 数据分析与作图技巧

IB Physics requires you to analyse motion data, draw graphs, and calculate slopes and areas. Pay attention to uncertainties and significant figures.

IB物理要求你分析运动数据、作图并计算斜率和面积。注意不确定度和有效数字。

When drawing a position-time graph from a table, choose scales so that the graph fills at least half of the grid.

根据数据表绘制位置-时间图时，选择合适的标度，使图像至少占据网格的一半。
To find instantaneous velocity from a curved position-time graph, draw a tangent at the point and calculate its slope using a large triangle.

要从弯曲的位置-时间图中求瞬时速度，需在该点画切线，并用大三角形计算斜率。
On a velocity-time graph, use a ruler to estimate the area for displacement. Count grid squares or use trapezoids.

在速度-时间图上，用直尺辅助估算面积以求位移。可数方格或用梯形法。

Always include units in your final answer. A graph without labelled axes with units will lose marks.

最终答案务必包含单位。坐标轴未标注单位和物理量的图像会被扣分。

12. Common Pitfalls in IB Kinematics | IB运动学常见误区

Many students lose marks due to small but repeated errors. Here are the most common pitfalls to avoid.

许多学生因为细小但重复的错误而失分。以下是最常见的误区，应当避免。

Confusing distance with displacement: distance is a scalar, displacement is a vector. Example: a round trip has zero displacement but non-zero distance.

混淆路程与位移：路程是标量，位移是矢量。例如：往返一次位移为零，但路程不为零。
Using suvat equations when acceleration is not constant, such as when air resistance is significant.

在加速度不恒定时使用suvat方程，例如空气阻力显著时。
Forgetting the negative sign for downward motion. Define upward as positive and keep \(g = -9.8 \text{ m s}^{-2}\) consistently.

忘记向下运动的负号。规定向上为正，并始终使用 g = -9.8 m/s²。
Misreading the area under a v-t graph as displacement when the graph is below the axis; the area must be signed.

当v-t图在轴下方时，误将面积当作位移；面积必须带符号。
Not converting units, especially km/h to m/s. Divide by 3.6 to convert km/h to m/s.

不换算单位，尤其是km/h转m/s。km/h除以3.6即为m/s。

Practise past-paper questions and always write down the known variables before choosing an equation.

多练习真题，并在选择方程前先列出已知量。

13. Exam-style Worked Example | 考试风格例题精解

Let us apply the concepts to a typical IB question: A particle is projected vertically upward with an initial speed of 20 m/s from the ground. Find the maximum height and the time to return to the ground. Take \(g = 9.8 \text{ m s}^{-2}\).

让我们将概念应用于一道典型IB题：一个粒子从地面以20 m/s的初速度竖直上抛。求最大高度和落回地面的时间。取 g = 9.8 m/s²。

Solution / 解答:

1. At maximum height, \(v = 0\). Use \(v^2 = u^2 + 2as\): \(0 = 20^2 + 2(-9.8)s\), so \(s = 400 / 19.6 = 20.4 \text{ m}\).

1. 在最大高度处，v = 0。使用 v² = u² + 2as：0 = 20² + 2(-9.8)s，得 s = 400 / 19.6 = 20.4 m。

2. Time to maximum height from \(v = u + at\): \(0 = 20 + (-9.8)t\), so \(t = 2.04 \text{ s}\). Total time = \(2 \times 2.04 = 4.08 \text{ s}\).

2. 由 v = u + at 求达到最大高度的时间：0 = 20 + (-9.8)t，得 t = 2.04 s。总时间 = 2 × 2.04 = 4.08 s。

Alternatively, use symmetry: the time up equals the time down. This saves time in the exam.

或者利用对称性：上升时间等于下降时间。考试中这样可以节省时间。

14. Quick Revision Checklist | 快速复习清单

Before the exam, ensure you can do the following without referencing notes.

考试前，请确保你能不参考笔记完成以下任务。

Define distance, displacement, speed, velocity, and acceleration accurately.

准确定义路程、位移、速率、速度和加速度。
Draw and interpret position-time, velocity-time, and acceleration-time graphs.

绘制并解释位置-时间图、速度-时间图和加速度-时间图。
Use all four suvat equations correctly with consistent sign convention.

使用统一符号规则，正确应用四个suvat方程。
Analyse projectile motion by resolving into horizontal and vertical components.

通过将抛体运动分解为水平和竖直分量进行分析。
Explain the effect of air resistance on trajectory.

解释空气阻力对轨迹的影响。
Calculate relative velocities in one and two dimensions.

计算一维和二维相对速度。
Describe uniform circular motion using period, frequency, angular speed, and centripetal acceleration.

用周期、频率、角速度和向心加速度描述匀速圆周运动。

Mastering these 14 areas will give you a solid foundation for IB Physics Paper 1 and Paper 2 mechanics questions.

掌握这14个方面，将为你在IB物理Paper 1和Paper 2中的力学题打下坚实基础。

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IB Physics: Kinematics Key Points | IB物理：运动学考点精讲