Relative Velocity | 相对速度

📚 Relative Velocity | 相对速度

Relative velocity is a crucial concept in the IB Mathematics curriculum, forming a bridge between pure vector theory and real‑world motion problems. It describes how the velocity of one object appears when observed from another moving object, and it is applied extensively in kinematics, navigation, interception and optimisation questions. Understanding relative velocity requires a solid grasp of vector addition and subtraction, as well as the ability to interpret motion in both one and two dimensions.

相对速度是IB数学课程中一个关键概念，它将纯向量理论与现实运动问题联系起来。相对速度描述了一个物体在另一个运动物体上观察时所呈现的速度，被广泛应用于运动学、导航、拦截和优化问题中。理解相对速度需要扎实掌握向量的加法与减法，并能对一维和二维运动进行解释。

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1. Introduction to Relative Velocity | 相对速度简介

If two objects A and B are moving with velocities v_A and v_B measured in a common fixed frame of reference (e.g. the ground), then the velocity of A relative to B is given by the vector difference v_A/B = v_A – v_B. This relative velocity tells us how fast and in which direction A appears to be moving from the perspective of B. For IB exam questions, it is essential to identify the correct reference frames and to use vector subtraction rigorously.

如果两个物体A和B在共同的固定参照系（例如地面）中分别以速度v_A和v_B运动，那么A相对于B的速度由向量差v_A/B = v_A – v_B给出。这个相对速度告诉我们从B的视角观察，A看起来在向哪个方向以多快的速度运动。在IB考题中，正确识别参照系并严格使用向量减法至关重要。

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2. Relative Velocity in One Dimension | 一维相对速度

When two objects move along the same straight line, relative velocity simplifies to a scalar difference with a sign convention. Let the positive direction be defined, for example, to the right. If car A travels at 20 m s⁻¹ and car B at 15 m s⁻¹ in the same direction, the velocity of A relative to B is v_A/B = 20 – away from B at 5 m s⁻¹. If the cars move towards each other, with A at +20 m s⁻¹ and B at –15 m s⁻¹, then v_A/B = 20 – (–15) = 35 m s⁻¹, giving a high closing speed.

当两个物体沿同一直线运动时，相对速度可简化为带符号约定的标量差。规定正方向（例如向右），若汽车A以20 m s⁻¹、汽车B以15 m s⁻¹同向行驶，A相对于B的速度为v_A/B = 20 – 15 = +5 m s⁻¹，即A以5 m s⁻¹远离B。若两车相向而行，A为+20 m s⁻¹，B为–15 m s⁻¹，则v_A/B = 20 – (–15) = 35 m s⁻¹，表现出很高的接近速率。

v_A/B = v_A – v_B

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3. Vector Subtraction and Relative Velocity | 向量减法与相对速度

Graphically, subtracting vectors can be visualised by adding the negative of the second vector: v_A – v_B = v_A + (–v_B). Place the two vectors tail‑to‑tail; the relative velocity vector is the vector that connects the tip of v_B to the tip of v_A when both originate from the same point. This geometric approach is especially helpful when dealing with direction angles, such as in navigation problems where velocities are given as magnitude and bearing.

在图形上，向量减法可以通过加上第二个向量的负向量来实现：v_A – v_B = v_A + (–v_B)。将两个向量尾对尾放置；若两者始于同一点，则相对速度向量就是从v_B的末端指向v_A末端的向量。在处理带有方位角的速度（如导航问题）时，这种几何方法尤其有用。

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4. Relative Velocity in Two Dimensions | 二维相对速度

In two‑dimensional motion, the relative velocity vector v_AB = v_A – v_B is computed component‑wise. Suppose v_A = (u₁, v₁) and v_B = (u₂, v₂), then v_AB = (u₁ – u₂, v₁ – v₂). Its magnitude |v_AB| gives the speed of A relative to B, and its direction can be expressed as an angle relative to a reference axis. IB questions often require the use of i‑j notation or column vectors and may ask for the relative bearing or the time until the distance is minimised.

在二维运动中，相对速度向量v_AB = v_A – v_B通过分量计算。设v_A = (u₁, v₁)、v_B = (u₂, v₂)，则v_AB = (u₁ – u₂, v₁ – v₂)。其模长|v_AB|表示A相对于B的速率，方向可用与参考轴的夹角表示。IB题目常使用i‑j记法或列向量，并要求计算相对方位角或达到最小距离的时间。

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5. Relative Position and Closest Approach | 相对位置与最近接近距离

For two moving objects with constant velocities, the closest distance of approach is a classic application of relative velocity. Let r_A(t) = r_A0 + tv_A and r_B(t) = r_B0 + tv_B. The relative position vector is r(t) = r₀ + tv_rel, where r₀ = r_A0 – r_B0 and v_rel = v_A – v_B e of the distance is minimised when the derivative with respect to t is zero, yielding t = –(r₀ · v_rel) / |v_rel|². The minimum distance can then be found from |r(t)| or by the perpendicular formula d_min = |r₀ × v_rel| / |v_rel| (in 2D the cross product magnitude gives the area).

对于两个匀速运动的物体，最近接近距离是相对速度的经典应用。设r_A(t) = r_A0 + tv_A，r_B(t) = r_B0 + tv_B。相对位置向量为r(t) = r₀ + tv_rel，其中r₀ = r_A0 – r_B0，v_rel = v_A – v_B。将距离平方对t求导并令其为零，可得t = –(r₀ · v_rel) / |v_rel|²。最小距离可由|r(t)|计算，也可通过垂直距离公式d_min = |r₀ × v_rel| / |v_rel|求得（二维中叉积大小给出面积）。

In IB problems, a common task is: 'Find the shortest distance between the two ships and the time at which it occurs.' ch happened in the past, so the minimum distance for t ≥ 0 is simply the initial distance.

在IB试题中，常见的问题是：“求两船之间的最短距离及其发生的时间”。考生需计算相对速度、初始相对位置，然后运用微分或几何上的垂直条件。务必检查时间t是否为正；若t为负，说明最近距离发生在过去，此时对于t ≥ 0的最小距离即为初始距离。

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6. Interception and Pursuit Problems | 拦截与追及问题

To intercept a moving target, the relative velocity of the pursuer with respect to the target must be directed exactly along the line connecting them at the initial instant. If a pursuer P must intercept a target Q moving with velocity v_Q, and P can travel at a fixed speed |v_P|, then the direction of v_P is found by solving v_P – v_Q = k (r_Q0 – r_P0 own direction angle or a tangent ratio, and the intercept time is t = initial separation / |v_rel|.

要拦截一个运动目标，拦截者相对于目标的相对速度必须沿着初始时刻两者连线方向。若拦截者P需拦截以速度v_Q运动的目标Q，而P能以固定速率|v_P|航行，则需通过解v_P – v_Q = k (r_Q0 – r_P0) （k为正标量）来求出v_P的方向。这个向量方程可转化为关于未知方向角的二次方程或正切比例式，拦截时间t = 初始距离 / |v_rel|。

Many IB problems set a known speed for P and ask for the required bearing. Use the fact that the relative velocity must be parallel to the initial relative displacement. Draw a velocity triangle and apply the sine rule or cosine rule. For example, given v_Q, the speed of P, and the angle of the line PQ, the magnitude of the relative velocity and the direction of v_P can be determined.

许多IB题目给定P的速率并要求计算所需方位角。利用相对速度必须与初始相对位移平行的条件，画出速度三角形并运用正弦定理或余弦定理。例如，给定v_Q、P的速率以及连线PQ的角度，即可求出相对速度的大小和v_P的方向。

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7. Riverboat Problems | 小船渡河问题

A classic application of relative velocity is a boat crossing a river with a current. The boat's velocity relative to the water, v_B/W, combines with the water's velocity relative to the ground, v_W/G, to give the boat's actual velocity relative to the ground: v_B/G = v_B/W + v_W/G. If the boat points directly across the river, the downstream drift can be calculated; if the boat must land directly opposite, the heading must be adjusted upstream so that the resultant velocity has no component along the current.

相对速度的一个经典应用是小船渡河问题。船相对于水的速度v_B/W与水流相对于地面的速度v_W/G合成，得到船相对于地面的实际速度：v_B/G = v_B/W + v_W/G。若船正对河岸行驶，可计算下游漂移量；若船必须在对岸正对面靠岸，则需向上游调整船头方向，使得合速度在沿河方向上没有分量。

Time to cross = river width / (|v_B/W| nsated. IB exam questions often ask the swimmer to find the direction to minimise the crossing time or to land at a specific point, testing both vector addition and trigonometric problem‑solving.

渡河时间 = 河宽 / (|v_B/W| cos θ 垂直分量)。若不加以补偿，下游漂移距离 = （水流速度）×（渡河时间）。IB考题常要求游泳者找到使渡河时间最小或到达指定地点的方向，综合考查向量加法和三角解题能力。

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8. Aircraft and Wind Problems | 飞机与风问题

Similar to riverboat problems, an aircraft's velocity relative to the air, v_A/Air (airspeed vector), adds to the wind velocity relative to the ground, v_Wind/G, to produce the aircraft's ground velocity v_A/G. Pilots must adjust their heading to compensate for wind drift in order to follow a desired track. This is a direct application of vector addition in a real‑world context, frequently examined in IB Math as a 'navigation problem'.

类似于小船渡河问题，飞机相对于空气的速度v_A/Air（空速向量）与风相对于地面的速度v_Wind/G相加，得到飞机的地速v_A/G。飞行员必须调整航向以补偿风的影响，从而沿预定航线飞行。这是向量加法在真实情境中的直接应用，IB数学中常以“导航问题”出现。

Given the airspeed, wind velocity and the desired track, the required heading can be found by constructing a vector triangle. The wind correction angle is the angle between the heading and the track. Ground speed is the magnitude of v_A/G. Students may be asked to calculate the time for a flight or to determine whether a certain fuel range is sufficient.

给定空速、风速和预定航线，可通过构建向量三角形求得所需航向。偏流角即为航向与航线之间的夹角。地速是v_A/G的大小。学生可能需要计算飞行时间或判断燃油航程是否足够。

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9. Summary and Exam Tips | 总结与考试技巧

Mastering relative velocity in IB Mathematics boils down to a few core skills: defining velocity vectors clearly, performing accurate vector subtraction or addition, and applying geometric reasoning through vector triangles. Always draw a clear diagram showing all velocities and their directions. Use consistent notation, such as subscripts to indicate reference frames (e.g. v_A/B). When calculating closest approach, remember that the relative velocity must be perpendicular to the relative displacement vector at the instant of minimum separation.

在IB数学中掌握相对速度可归结为几个核心技能：清晰地定义速度向量，准确地完成向量的减法或加法，并运用向量三角形的几何推理。始终绘制清晰的示意图，标出所有速度及其方向。使用一致的符号，如下标表示参照系（如v_A/B）。在计算最近接近距离时，记住在最小间距的瞬间相对速度必须与相对位移向量垂直。

For interception, the relative velocity must be parallel to the initial line joining the objects. In riverboat and wind problems, the key is to identify the correct reference frames and to apply the vector addition equation. Practice by solving past paper questions, paying close attention to whether the question expects an algebraic derivation or a geometric solution using the sine/cosine rules.

对于拦截问题，相对速度必须平行于初始连线。在小船和风的问题中，关键在于识别正确的参照系并应用向量加法公式。通过练习往年试题来巩固，注意题目要求的是代数推导还是使用正弦/余弦定理的几何解法。

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