Tangent Vectors, Velocity and Acceleration | 切向量、速度与加速度

📚 Tangent Vectors, Velocity and Acceleration | 切向量、速度与加速度

In IB Mathematics, vector calculus is used to describe the motion of particles in a plane or in space. The position vector r(t) gives the location at time t, while its derivatives produce velocity and acceleration. Tangent vectors describe the direction of motion, which is essential for understanding curves and motion.

在IB数学中，向量微积分用于描述平面或空间中粒子的运动。位置向量r(t)给出t时刻的位置，其导数产生速度和加速度。切向量描述运动方向，对于理解曲线和运动至关重要。

1. Position Vectors and Parametric Curves | 位置向量与参数曲线

A particle moving in space is described by a position vector r(t) that depends on a parameter, usually time t. In 2D, we write r(t) = x(t) i + y(t) j, where i and j are unit vectors along the x- and y-axes. In 3D, we write r(t) = x(t) i + y(t) j + z(t) k.

运动中的粒子由位置向量r(t)描述，它依赖于参数，通常是时间t。在二维中，我们写作r(t) = x(t) i + y(t) j，其中i和j分别是沿x轴和y轴的单位向量。在三维中，我们写作r(t) = x(t) i + y(t) j + z(t) k。

The set of all possible positions as t varies is called a parametric curve. The functions x(t), y(t) and z(t) are called parametric equations.

当t变化时，所有可能的位置集合称为参数曲线。函数x(t)、y(t)和z(t)称为参数方程。

2. Velocity as the Derivative of Position | 速度：位置向量的导数

The velocity vector v(t) is the instantaneous rate of change of position with respect to time. It is found by differentiating each component of r(t):

速度向量v(t)是位置关于时间的瞬时变化率。通过对r(t)的每个分量求导得到：

v(t) = r′(t) = dx/dt i + dy/dt j + dz/dt k

The velocity vector is always tangent to the path of the particle. Its direction is the direction of motion, and its magnitude is the speed.

速度向量始终与粒子路径相切。其方向就是运动方向，其大小就是速率。

3. Acceleration as the Second Derivative | 加速度：二阶导数

The acceleration vector a(t) describes how the velocity changes with time. It is obtained by differentiating the velocity vector again:

加速度向量a(t)描述速度随时间的变化。通过对速度向量再次求导得到：

a(t) = v′(t) = r″(t)

Acceleration can have components both parallel and perpendicular to the velocity, so it is not necessarily pointing in the same direction as the motion.

加速度可以同时具有平行和垂直于速度的分量，因此它不一定与运动方向相同。

4. Tangent Vectors and the Unit Tangent Vector | 切向量与单位切向量

For a smooth curve r(t), the derivative r′(t) is a tangent vector at that point, provided r′(t) ≠ 0. It points in the direction of increasing parameter t.

对于光滑曲线r(t)，只要r′(t) ≠ 0，导数r′(t)就是该点处的一个切向量，它指向参数t增大的方向。

To find a unit tangent vector, we divide the tangent vector by its own length:

为了得到单位切向量，我们将切向量除以其长度：

T(t) = r′(t) / |r′(t)|

This unit tangent vector is often written as T and has length 1.

这个单位切向量常写作T，其长度为1。

5. Speed and Magnitude of Velocity | 速率与速度大小

The speed of the particle is the magnitude of the velocity vector, denoted |v(t)| or ||r′(t)||. In 2D:

粒子的速率是速度向量的大小，记作|v(t)|或‖r′(t)‖。在二维中：

speed = |v(t)| = √((dx/dt)² + (dy/dt)²)

Speed is a scalar and is always non-negative. It tells us how fast the particle is moving, regardless of direction.

速率是标量，始终非负。它告诉我们粒子运动得多快，而不涉及方向。

6. Displacement, Distance and Path Length | 位移、距离与路程

Displacement over a time interval [a, b] is the vector change in position:

在时间区间[a, b]上，位移是位置向量的改变量：

Δr = r(b) − r(a)

This is different from the distance travelled. The distance along the path is found by integrating the speed:

这不同于经过的路程。沿着路径的路程通过对速率积分得到：

distance = ∫ₐᵇ |v(t)| dt

Displacement depends only on the endpoints, while distance depends on the entire path.

位移只取决于起点和终点，而路程取决于整条路径。

7. Tangent Lines to Parametric Curves | 参数曲线的切线

Given a point r(t₀) on the curve, the tangent line can be written using the tangent vector r′(t₀):

给定曲线上一点r(t₀)，可以利用切向量r′(t₀)写出切线方程：

L(λ) = r(t₀) + λ r′(t₀)

Here λ is a scalar parameter. This line is the best linear approximation to the curve at that point.

这里λ是标量参数。该直线是曲线在该点处的最佳线性近似。

8. Worked Example 1: Position to Velocity and Acceleration | 例题1：由位置求速度和加速度

Let r(t) = t² i + (t³ − t) j. Find v(t), a(t), the speed at t = 1, and the unit tangent vector at t = 1.

设r(t) = t² i + (t³ − t) j。求v(t)、a(t)、在t = 1时的速率，以及t = 1时的单位切向量。

Differentiating component by component:

逐分量求导：

v(t) = 2t i + (3t² − 1) j

a(t) = 2 i + 6t j

At t = 1:

当t = 1时：

v(1) = 2 i + 2 j, speed = |v(1)| = √(2² + 2²) = √8 = 2√2

The unit tangent vector is:

单位切向量为：

T(1) = (2 i + 2 j) / (2√2) = (1/√2) i + (1/√2) j

9. Worked Example 2: Acceleration to Position with Initial Conditions | 例题2：由加速度和初始条件求位置

A particle moves with acceleration a(t) = 2t i + 3 j. Given that at t = 0, v = 0 and r = i, find v(t) and r(t).

一粒子以加速度a(t) = 2t i + 3 j运动。已知t = 0时，v = 0且r = i，求v(t)和r(t)。

Integrate the acceleration to get the velocity:

对加速度积分得到速度：

v(t) = t² i + 3t j + C

Using v(0) = 0 gives C = 0, so v(t) = t² i + 3t j.

利用v(0) = 0得C = 0，因此v(t) = t² i + 3t j。

Integrate again to get the position:

再次积分得到位置：

r(t) = (t³/3) i + (3t²/2) j + D

Using r(0) = i gives D = i, hence:

利用r(0) = i得D = i，因此：

r(t) = (1 + t³/3) i + (3t²/2) j

10. Kinematics in Vector Form: Circular Motion | 向量形式下的运动学：圆周运动

Consider uniform circular motion described by r(t) = R cos(ωt) i + R sin(ωt) j, where R is the radius and ω is the angular speed.

考虑匀速圆周运动，由r(t) = R cos(ωt) i + R sin(ωt) j描述，其中R是半径，ω是角速度。

The velocity is

速度为

v(t) = −Rω sin(ωt) i + Rω cos(ωt) j

and the acceleration is

加速度为

a(t) = −Rω² cos(ωt) i − Rω² sin(ωt) j = −ω² r(t)

Notice that in uniform circular motion, the acceleration is always directed towards the centre, and it is perpendicular to the velocity.

注意在匀速圆周运动中，加速度始终指向圆心，并且垂直于速度。

11. Common Errors and Exam Tips | 常见错误与考试要点

Confusing speed with velocity: speed is a scalar |v|, while velocity is a vector v.

混淆速率与速度：速率是标量|v|，而速度是向量v。
Forgetting that the unit tangent vector must be divided by its magnitude; T is not simply r′(t).

忘记单位切向量必须除以其模长；T并不是简单的r′(t)。
When integrating, always include the constant vector C; use initial conditions to find it.

积分时，不要忘记常数向量C；利用初始条件求它。
Remember that the acceleration vector may have a tangential component and a normal component, so it is not always parallel to the velocity.

记住加速度向量可以有切向分量和法向分量，所以它并不总是与速度平行。
When asked for the distance travelled, you must integrate |v(t)|, not |Δr|.

当要求路程时，必须对|v(t)|积分，而不是对|Δr|积分。

12. Summary | 总结

Velocity is the derivative of the position vector, and acceleration is the derivative of the velocity vector. The tangent vector r′(t) points along the motion, and the unit tangent vector T = r′(t)/|r′(t)| gives the direction of motion. Speed is the magnitude of the velocity, and the distance along a curve is the integral of speed. These ideas connect vectors, calculus and geometry, and they appear regularly in IB Mathematics exams.

速度是位置向量的导数，加速度是速度向量的导数。切向量r′(t)指向运动方向，单位切向量T = r′(t)/|r′(t)|给出运动方向。速率是速度的大小，曲线上的路程是速率的积分。这些概念将向量、微积分和几何联系起来，在IB数学考试中经常出现。

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Tangent Vectors, Velocity and Acceleration | 切向量、速度与加速度