📚 GCSE WJEC Mathematics: Mechanics Revision Notes | GCSE WJEC 数学：力学考点精讲
Mechanics in GCSE WJEC Mathematics bridges the gap between algebra, graphs, and real-world motion. You will learn to describe how objects move using speed, velocity, acceleration, and vector quantities. Mastering travel graphs and vector work is essential for the non-calculator and calculator papers alike, helping you solve problems involving journeys, forces, and displacement without leaving the maths classroom.
在 WJEC GCSE 数学中，力学内容将代数、图表与现实世界的运动联系起来。你将学习用速率、速度、加速度以及向量来描述物体的运动。掌握行程图和向量运算对非计算器卷和计算器卷都至关重要，能帮助你解决涉及行程、力和位移的问题，而不超出数学课堂的范畴。
1. Speed, Distance and Time | 速度、距离与时间
Speed is a measure of how fast something moves. The basic formula is speed = distance ÷ time, usually written as v = d / t.
速度是物体运动快慢的量度。基本公式为：速度 = 距离 ÷ 时间，通常写作 v = d / t。
You can rearrange the equation to find distance (d = v × t) or time (t = d ÷ v). Always work in consistent units: metres and seconds give m/s, kilometres and hours give km/h.
你可以变形公式来求距离（d = v × t）或时间（t = d ÷ v）。始终用一致的单位：米和秒得到米/秒，千米和小时得到千米/时。
For a two-part journey, add the distances from each leg. Break the problem into stages, calculate each stage separately, then sum the results.
对于两段行程，应将每段距离相加。把问题分解成阶段，分别计算每一阶段，然后汇总。
2. Distance-Time Graphs: Interpreting Motion | 距离-时间图：解读运动
A distance-time graph plots distance on the vertical axis against time on the horizontal axis. A straight horizontal line means the object is stationary.
距离-时间图把距离标在纵轴，时间标在横轴。一条水平直线表示物体静止不动。
A straight sloping line indicates constant speed. The steeper the slope, the greater the speed. A curved line shows acceleration or deceleration.
一条倾斜直线表示匀速运动。斜率越陡，速度越大。曲线则表示加速或减速。
Always read the axes carefully: the gradient is not speed if the graph shows distance–time – it is exactly speed, calculated as rise ÷ run.
务必仔细阅读坐标轴：距离–时间图的斜率本身就是速度，由纵向增量除以横向增量求得。
3. Calculating Speed from a Distance-Time Graph | 由距离-时间图计算速度
To find the speed at a particular instant, draw a tangent to the curve if the graph is curved, then calculate its gradient. For a straight section, simply use gradient = (change in distance) ÷ (change in time).
若要求某一瞬间的速度，可在曲线图上画出该点的切线，再计算切线斜率。对于直线段，直接用斜率 = (距离变化量) ÷ (时间变化量)。
speed = Δd ÷ Δt
Label the units: a gradient of 20 metres per 4 seconds gives a speed of 5 m/s. Exam questions often ask for the speed in km/h, so multiply m/s by 3.6 to convert.
标明单位：梯度为20米/4秒，得出速度为5米/秒。考试常要求以千米/时为单位，此时将米/秒乘以3.6即可换算。
Remember that the gradient of a distance-time graph cannot be negative; distance always increases (or stays constant) over time. A downward slope would mean returning to the start, which is represented by a decrease in distance, still giving a positive speed when considering the magnitude of the gradient.
注意距离-时间图的斜率不会为负；距离随时间总是增加（或保持不变）。若出现下行线段，表示返回起点，此时距离减少，但斜率的绝对值仍给出速度的大小。
4. Understanding Acceleration | 理解加速度
Acceleration is the rate of change of velocity. If u is the initial velocity, v the final velocity, and t the time taken, the average acceleration is a = (v − u) ÷ t.
加速度是速度变化的快慢。若 u 为初速度，v 为末速度，t 为所用时间，则平均加速度为 a = (v − u) ÷ t。
a = (v − u) ÷ t
Deceleration is simply negative acceleration. If the final speed is less than the initial speed, the acceleration value is negative. In WJEC questions you might also see the equation v² = u² + 2ad where d is displacement; this formula is useful when time is not given.
减速只是负加速度。若末速度小于初速度，加速度值为负。在WJEC考题中，你可能也会看到 v² = u² + 2ad 的公式，其中 d 表示位移；当不给出时间时这个公式很有用。
Always use the correct units: if velocities are in m/s, time in seconds, acceleration is in m/s². Convert as needed before substituting numbers.
始终使用正确的单位：若速度用米/秒，时间用秒，加速度单位就是米/秒²。代入数值前应先换算。
5. Velocity-Time Graphs: Gradient and Area | 速度-时间图：斜率与面积
A velocity-time graph records velocity on the vertical axis and time on the horizontal. The gradient of the line gives acceleration.
速度-时间图将速度标在纵轴，时间标在横轴。直线的斜率表示加速度。
The area between the graph line and the time axis represents the distance travelled. This is true even if the graph dips below the axis; then you must consider absolute areas to find total distance, or a signed area for displacement.
图线与时间轴之间的面积代表行进的距离。即使图线落到轴下方，这个规则仍成立；此时要用面积的绝对值求总路程，而带符号的面积则表示位移。
For constant acceleration, the area can be split into a rectangle and a triangle, using area = ½ × base × height for triangles. For varying acceleration, count squares or use given approximations.
对匀加速运动，可将面积分割为矩形和三角形，三角形面积用 ½ × 底 × 高来计算。对变加速运动，则通过数格子或使用给定的近似方法来求面积。
6. Finding Distance from a Velocity-Time Graph | 利用速度-时间图求距离
The distance travelled is the area under the velocity-time graph. In an exam, you may be asked to calculate this area precisely using geometry.
行进的距离等于速度-时间图下所围的面积。考试中可能要求你用几何方法精确计算该面积。
| Shape | When to use | Area formula |
|---|---|---|
| Rectangle | Constant velocity | base × height |
| Triangle | Uniform acceleration from rest | ½ × base × height |
| Trapezium | Acceleration when u ≠ 0 | ½ × (a + b) × height |
Be careful with units: if velocity is in m/s and time in seconds, the area is in metres. If the graph uses km/h and hours, distance is in km.
注意单位：若速度取米/秒，时间取秒，则面积单位为米。如果图线使用千米/时和小时，距离单位即为千米。
7. Introduction to Vectors | 向量入门
In mechanics, many quantities are vectors, having both magnitude and direction. Displacement, velocity, and force are vectors, while distance and speed are scalars.
力学中，许多量是向量，既有大小又有方向。位移、速度和力是向量，而距离和速率是标量。
Vectors are often written in bold, like v, or as column vectors: a displacement of 3 metres east and 4 metres north can be written as v = (3, 4) or as 3i + 4j.
向量常用粗体表示，如 v，或写为列向量：向东3米、向北4米的位移可记作 v = (3, 4) 或 3i + 4j。
The magnitude (size) of a vector v = (x, y) is found using Pythagoras: |v| = √(x² + y²). The direction can be given as a bearing or an angle from a reference direction.
向量 v = (x, y) 的大小用勾股定理求得：|v| = √(x² + y²)。方向可以用方位角或与参考方向的夹角来表示。
8. Vector Addition and Resultant | 向量的加法与合向量
To add two vectors, add their components: if a = (a₁, a₂) and b = (b₁, b₂), then a + b = (a₁ + b₁, a₂ + b₂). This is the parallelogram law.
两向量相加，对应分量相加：若 a = (a₁, a₂)，b = (b₁, b₂)，则 a + b = (a₁ + b₁, a₂ + b₂)。此即平行四边形法则。
The resultant vector is the single vector that has the same effect as the original ones acting together. In force problems, the resultant force determines the direction of acceleration.
合向量是一个单独的向量，其作用效果与原来几个向量共同作用时一样。在力的问题中，合力决定了加速度的方向。
Subtracting vectors: a − b = a + (−b). You reverse the direction of b and then add. Draw a diagram to avoid sign errors.
向量减法：a − b = a + (−b)。将 b 反向后再相加。建议画图以避免符号错误。
9. Using Vectors in Mechanics Problems | 力学问题中的向量应用
In WJEC exam questions, vector methods often describe relative velocity or displacement. For example, a boat crossing a river has its own velocity and is also affected by the current.
在 WJEC 考题中，向量方法常用来描述相对速度或位移。例如，一艘船过河时有自己的速度，同时也受水流影响。
To find the actual velocity, add the boat’s velocity vector and the current’s velocity vector using component addition. The magnitude of the resultant gives the boat’s speed over ground, while the direction shows the actual path.
要求实际速度，需将船的速度向量与水流速度向量按分量相加。合向量的大小给出船对地的速率，方向则显示实际航迹。
Always set up a coordinate system: let i be east and j be north. Then any vector can be written in component form. This makes it easier to solve simultaneous motion problems.
务必建立坐标系：设 i 为东，j 为北，则任一向量都可用分量形式写出。这样更易解决同时运动问题。
10. Common Pitfalls and Exam Tips | 常见错误与应考技巧
Many students confuse distance-time and velocity-time graphs. Always check the vertical-axis label: if it says ‘distance’, the gradient is speed; if it says ‘velocity’, the gradient is acceleration and the area under the graph is distance.
许多学生混淆距离-时间图和速度-时间图。务必检查纵轴标签：若写“距离”，则斜率代表速率；若写“速度”，则斜率代表加速度，图下面積才是距离。
A common mistake is forgetting to convert units. If a speed is given in km/h but time in seconds, you must change either km/h to m/s or seconds to hours. Use the conversion factor 1 m/s = 3.6 km/h.
常见错误是忘记换算单位。若速度单位是千米/时而时间用秒，必须将千米/时转为米/秒或将秒转为小时。用换算关系 1 米/秒 = 3.6 千米/时。
When adding vectors, never add magnitudes directly unless they are in the same direction. Always use component addition or drawing a scale diagram.
向量相加时，除非两向量方向相同，否则绝不直接将大小相加。始终采用分量加法或按比例画图。
Finally, practise reading questions carefully. Underline key information such as ‘constant speed’, ‘from rest’, ‘deceleration’, and note which quantity is being asked for: distance, displacement, speed, or velocity.
最后，要练习仔细读题。在关键词如“匀速”“从静止开始”“减速”下划线，并留意题目所问的量是路程、位移、速率还是速度。
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