Differentiating Vectors: From Position to Acceleration | 向量微分：从位置到加速度

📚 Differentiating Vectors: From Position to Acceleration | 向量微分：从位置到加速度

In Edexcel A-Level Mathematics, differentiating vectors is a key skill that links pure calculus with mechanics. A vector can change in both magnitude and direction as time goes on, and differentiation allows you to examine that change component by component.

在爱德思 A-Level 数学中，向量微分是连接纯数学微积分与力学的关键技能。向量会随着时间改变大小和方向，而求导可以让你逐个分量地分析这种变化。

This article covers position, velocity and acceleration vectors, differentiation rules, integration back to position, and the common errors that cost marks in exams.

本文涵盖位置向量、速度向量、加速度向量、求导法则、由加速度积分回位置，以及考试中常见的失分错误。

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1. Why Differentiate a Vector? | 为什么要对向量求导？

A scalar derivative measures how fast one number changes with respect to another. A vector derivative measures how fast each component of a vector changes with respect to time, while direction and magnitude may both change.

标量导数测量一个数相对于另一个数的变化快慢。向量导数测量向量每个分量相对于时间的变化快慢，同时方向和大小可能都在变化。

This is essential in mechanics: velocity is the rate of change of position, and acceleration is the rate of change of velocity.

这在力学中至关重要：速度是位置的变化率，加速度是速度的变化率。

In kinematics problems, the motion is often described by a position vector r(t), where t is time. Differentiating r(t) gives the velocity vector, and differentiating again gives the acceleration vector.

在运动学问题中，运动通常由位置向量 r(t) 描述，其中 t 是时间。对 r(t) 求导得到速度向量，再求导得到加速度向量。

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2. Vector Functions and Position Vectors | 向量函数与位置向量

A two-dimensional position vector is usually written in terms of the unit vectors i and j, where i points horizontally and j points vertically.

二维位置向量通常用单位向量 i 和 j 表示，其中 i 指向水平方向，j 指向垂直方向。

r(t) = x(t)i + y(t)j

Here x(t) and y(t) are scalar functions of time t. For example, r(t) = (t² − 3t)i + (2t + 1)j describes a particle whose horizontal position is t² − 3t and whose vertical position is 2t + 1.

这里 x(t) 和 y(t) 是时间 t 的标量函数。例如，r(t) = (t² − 3t)i + (2t + 1)j 描述了一个质点，它的水平位置为 t² − 3t，垂直位置为 2t + 1。

The unit vectors i and j are constant: they do not change with time. This is why vector differentiation can be carried out by differentiating the scalar components separately.

单位向量 i 和 j 是恒定的，不随时间变化。这就是为什么向量求导可以分别对标量分量求导。

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3. Component-by-Component Differentiation | 逐分量求导

If i and j are fixed unit vectors, their derivatives are zero. Therefore you differentiate a vector by differentiating each scalar component separately.

如果 i 和 j 是固定的单位向量，它们的导数为零。因此你对向量求导时，只需分别对其每个标量分量求导。

dr/dt = x′(t)i + y′(t)j

For r(t) = (t² − 3t)i + (2t + 1)j, the derivative is dr/dt = (2t − 3)i + 2j.

对于 r(t) = (t² − 3t)i + (2t + 1)j，其导数为 dr/dt = (2t − 3)i + 2j。

The second derivative is found by differentiating again: d²r/dt² = 2i + 0j = 2i.

二阶导数再次求导得到：d²r/dt² = 2i + 0j = 2i。

This works in exactly the same way for three-dimensional vectors. If r(t) = x(t)i + y(t)j + z(t)k, then dr/dt = x′(t)i + y′(t)j + z′(t)k.

这对三维向量也完全适用。如果 r(t) = x(t)i + y(t)j + z(t)k，那么 dr/dt = x′(t)i + y′(t)j + z′(t)k。

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4. Geometric Interpretation: The Tangent Vector | 几何意义：切向量

The derivative vector dr/dt points along the tangent to the path of the particle. Its direction is the direction of motion at that instant.

导数向量 dr/dt 指向质点运动路径的切线方向。它的方向就是该时刻的运动方向。

For a curve defined by a vector function r(t), dr/dt is called the tangent vector. This idea is useful in pure mathematics as well as mechanics.

对于由向量函数 r(t) 定义的曲线，dr/dt 被称为切向量。这个思想在纯数学和力学中都很有用。

Tangent vector = dr/dt = x′(t)i + y′(t)j

If the tangent vector is zero at some time, the particle may be momentarily at rest or changing direction sharply. In most A-Level questions, however, the tangent vector is nonzero.

如果切向量在某个时刻为零，质点可能瞬间静止或发生急剧转向。不过在大多数 A-Level 题目中，切向量通常不为零。

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5. Velocity as the First Derivative | 速度作为一阶导数

If r(t) is the position vector of a particle, then the velocity vector v(t) is the first derivative of r with respect to time.

如果 r(t) 是质点的位置向量，那么速度向量 v(t) 就是 r 对时间的一阶导数。

v(t) = dr/dt = v_x(t)i + v_y(t)j

The velocity vector gives both the speed of the particle and its direction of motion at time t. The components v_x(t) and v_y(t) are the horizontal and vertical components of velocity.

速度向量既给出质点在

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