A-Level Physics: Resolving Velocity Addition Problems | A-Level 物理：速度合成问题解析

📚 A-Level Physics: Resolving Velocity Addition Problems | A-Level 物理：速度合成问题解析

In A-Level mechanics, velocity addition is a fundamental skill that connects relative motion, vector algebra and real-world applications such as boats crossing rivers, rain falling on a pedestrian, and aircraft flying in wind. Mastering this topic requires a clear understanding of frames of reference and a reliable method for combining vector velocities.

在A-Level力学中，速度合成是一项基本技能，它将相对运动、矢量代数以及渡河、雨滴落在行人身上、飞机在风中飞行等实际应用联系在一起。掌握这一主题需要清晰理解参考系，并掌握合成矢量速度的可靠方法。

1. Frames of Reference and Relative Velocity | 参考系与相对速度

In physics, there is no absolute motion. Every velocity measurement depends on the observer's frame of reference. A frame of reference is a coordinate system attached to a chosen origin, which may be stationary relative to the ground or moving uniformly.

在物理中，不存在绝对运动。每一次速度测量都依赖于观察者所处的参考系。参考系就是附着于选定原点的坐标系，它相对于地面可以是静止的，也可以做匀速运动。

Consider two objects A and B moving with velocities v_A and v_B relative to the ground. The velocity of A relative to B is defined as v_AB = v_A − v_B. Equivalently, the velocity of B relative to A is v_BA = v_B − v_A = −v_AB.

设物体A和B相对于地面的速度分别为v_A和v_B。则A相对于B的速度定义为v_AB = v_A − v_B。同理，B相对于A的速度为v_BA = v_B − v_A = −v_AB。

v_AB = v_A − v_B

2. One-Dimensional Velocity Addition | 一维速度合成

When all motion is along a single straight line, velocity addition reduces to simple algebraic addition or subtraction. Choose a positive direction; velocities in that direction are positive, and velocities in the opposite direction are negative.

当所有运动都沿同一条直线时，速度合成简化为简单的代数加减。先选定一个正方向，沿正方向的速度为正，反方向的速度为负。

Example: A train moves east at 30 m s⁻¹. A passenger walks east at 2 m s⁻¹ relative to the train. The passenger's velocity relative to the ground is 30 + 2 = 32 m s⁻¹. If the passenger walks west, it is 30 − 2 = 28 m s⁻¹.

例如：一列火车以30 m s⁻¹向东行驶。车内乘客相对火车以2 m s⁻¹向东行走。乘客相对于地面的速度为30 + 2 = 32 m s⁻¹。若乘客向西走，则为30 − 2 = 28 m s⁻¹。

v_PG = v_PT + v_TG

3. Two-Dimensional Velocity Addition | 二维速度合成

In two dimensions, velocities are vectors. The resultant of two velocities is obtained by vector addition: place the vectors nose-to-tail, or resolve each velocity into perpendicular components.

在二维情况下，速度是矢量。两个速度的合速度通过矢量加法得到：将矢量首尾相接，或将每个速度分解为垂直分量。

If the two component velocities are perpendicular, the magnitude of the resultant is R = √v_x² + v_y², and its direction is given by θ = tan⁻¹(v_y / v_x), measured from the x-axis.

若两个分速度相互垂直，则合速度的大小为R = √v_x² + v_y²，其方向为θ = tan⁻¹(v_y / v_x)，从x轴量起。

R = √v_x² + v_y², θ = tan⁻¹(v_y / v_x)

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4. River Crossing Problems | 渡河问题

A classic application of two-dimensional velocity addition is a boat crossing a river. Let v_bw be the boat's velocity relative to the water, and v_wb the water's velocity relative to the bank. The boat's velocity relative to the bank is the vector sum: v_bb = v_bw + v_wb.

二维速度合成的经典应用是小船渡河。设v_bw为船相对于水的速度，v_wb为水相对于岸的速度。则船相对于岸的速度为矢量和：v_bb = v_bw + v_wb。

The vector diagram is a triangle: the boat's heading is one side, the current is another, and the resultant path is the third side. To land at a point directly opposite the starting point, the resultant velocity must point straight across the river, so the boat must head upstream at angle θ satisfying sin θ = v_wb / v_bw.

矢量图是一个三角形：船头方向是一条边，水流速度是另一条边，合路径是第三条边。若要在起点正对岸的某点靠岸，合速度必须正指对岸，因此船必须朝上游偏转，偏转角θ满足sin θ = v_wb / v_bw。

sin θ = v_wb / v_bw

5. Shortest Path vs Shortest Time | 最短路径与最短时间

For a boat crossing a river, two optimisation questions are common in CIE exams: shortest path and shortest time.

渡河问题中，

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