Kinematics Vectors and Projectile Motion | 运动学矢量与抛体运动

📚 Kinematics Vectors and Projectile Motion | 运动学矢量与抛体运动

In AP Physics C: Mechanics, the study of kinematics extends to two and three dimensions using vector descriptions. A solid command of vectors—addition, subtraction, scalar multiplication, and decomposition into components—is essential for analyzing motion such as projectile motion. This article revisits displacement, velocity, and acceleration as vector quantities and applies them to the classic case of a projectile moving under constant gravitational acceleration, neglecting air resistance. The calculus-based methods required for AP Physics C are woven throughout the discussion.

在 AP 物理 C：力学中，运动学的研究通过矢量的描述扩展到了二维和三维。牢固掌握矢量的加法、减法、数乘和分量分解，对于分析比如抛体运动这类问题是必不可少的。本文将重新审视位移、速度和加速度这些矢量量，并将其应用于在恒定重力加速度下运动的经典抛体案例（忽略空气阻力）。AP 物理 C 所要求的基于微积分的方法将贯穿全文的讨论。

1. Scalars and Vectors | 标量与矢量

A scalar is a physical quantity completely specified by a single number (magnitude) and its unit; examples are distance, speed, mass, and temperature. A vector requires both magnitude and direction for a full description. Displacement, velocity, acceleration, and force are vectors. In print, vectors are denoted by boldface (v) or by an arrow above the symbol (v⃗). In calculations it is often convenient to express a vector in terms of its components along perpendicular axes: A = A_x î + A_y ĵ, where î and ĵ are unit vectors in the x- and y‑directions.

标量是一个完全由一个数字（大小）及其单位确定的物理量；例如距离、速率、质量和温度。矢量则需要大小和方向才能完整描述。位移、速度、加速度和力都是矢量。在印刷体中，矢量用粗体（v）或符号上方的箭头（v⃗）表示。计算中，通常把矢量用垂直坐标轴上的分量表示：A = A_x î + A_y ĵ，其中 î 和 ĵ 分别是 x 方向和 y 方向的单位矢量。

Vector addition follows the triangle or parallelogram rule; components add algebraically. If C = A + B, then C_x = A_x + B_x and C_y = A_y + B_y. Subtraction is addition of the negative vector. The magnitude of a vector is given by the Pythagorean theorem: |A| = √(A_x² + A_y²).

矢量加法遵循三角形或平行四边形法则；各分量进行代数相加。若 C = A + B，则 C_x = A_x + B_x 且 C_y = A_y + B_y。减法即为加上负矢量。矢量的大小由勾股定理给出：|A| = √(A_x² + A_y²)。

2. Position, Displacement, and Distance | 位置、位移与路程

The position vector r locates a particle relative to the origin of a chosen coordinate system. As the particle moves from position r₁ to r₂, the displacement is Δr = r₂ － r₁. Displacement is a vector; it depends only on the initial and final positions, not on the path taken. The total distance traveled is the length of the actual path and is a positive scalar.

位置矢量 r 描述质点相对于选定坐标系原点的位置。当质点从位置 r₁ 运动到 r₂ 时，位移为 Δr = r₂ － r₁。位移是矢量，它只依赖于初末位置，与所经过的路径无关。通过的总路程是实际路径的长度，是一个正标量。

Δr = r(t + Δt) － r(t)

Δr 的大小 ≠ 路程（除非直线运动且方向不变）

In two‑dimensional motion, the displacement vector can be resolved as Δr = Δx î + Δy ĵ. Understanding the distinction between displacement and distance is crucial when setting up kinematic equations.

在二维运动中，位移矢量可分解为 Δr = Δx î + Δy ĵ。建立运动学方程时，理解位移与路程的区别至关重要。

3. Velocity and Acceleration Vectors | 速度与加速度矢量

The average velocity vector over a time interval Δt is v_av = Δr/Δt. Instantaneous velocity is the limit of the average velocity as Δt → 0, which is the first derivative of the position vector: v = dr/dt. Velocity is tangent to the trajectory at every instant. Similarly, average acceleration is Δv/Δt, and instantaneous acceleration is a = dv/dt = d²r/dt².

在时间间隔 Δt 内的平均速度矢量为 v_av = Δr/Δt。瞬时速度是当 Δt → 0 时平均速度的极限，也就是位置矢量的一阶导数：v = dr/dt。速度在每一时刻都与轨迹相切。类似地，平均加速度为 Δv/Δt，瞬时加速度为 a = dv/dt = d²r/dt²。

v = v_x î + v_y ĵ ; v_x = dx/dt, v_y = dy/dt

a = a_x î + a_y ĵ ; a_x = dv_x/dt, a_y = dv_y/dt

Because acceleration is also a vector, a particle can accelerate even if its speed is constant—for instance, in uniform circular motion the direction of velocity changes, giving a centripetal acceleration. The ability to treat v_x, v_y, a_x, and a_y independently is the backbone of two‑dimensional kinematics.

因为加速度也是矢量，即便质点的速率不变，它仍可能有加速度——例如在匀速圆周运动中，速度的方向变化便产生了向心加速度。能够独立处理 v_x、v_y、a_x 和 a_y，是二维运动学的基础。

4. Kinematic Equations for Constant Acceleration | 恒加速度的运动学方程

When the acceleration vector a is constant in both magnitude and direction, the motion in each perpendicular direction obeys the familiar one‑dimensional kinematic formulas, written in vector form or component form.

当加速度矢量 a 的大小和方向都恒定时，每个垂直方向上的运动都遵循熟悉的一维运动学公式，可以写成矢量形式或分量形式。

Vector equation	Component equations (x and y)
v = v₀ + a t	v_x = v_0x + a_x t, v_y = v_0y + a_y t
r = r₀ + v₀ t + ½ a t²	x = x₀ + v_0x t + ½ a_x t² y = y₀ + v_0y t + ½ a_y t²
v·v = v₀·v₀ + 2 a·Δr	v_x² = v_0x² + 2 a_x Δx, v_y² = v_0y² + 2 a_y Δy

The third equation uses the dot product because v² = v_x² + v_y². It is valid only when the acceleration is constant. These component equations are the primary tools for solving projectile problems.

第三个方程使用了点积，因为 v² = v_x² + v_y²。它仅在加速度恒定时成立。这些分量方程是求解抛体问题的主要工具。

5. Projectile Motion: Independence of Components | 抛体运动：分量的独立性

Projectile motion is a combination of constant‑velocity horizontal motion and constant‑acceleration vertical motion, provided air resistance is negligible and the gravitational field is uniform. The two motions are independent: the horizontal component of velocity never changes, while the vertical component experiences a constant downward acceleration a_y = –g (taking upward as positive).

抛体运动是水平方向的匀速运动与竖直方向的匀加速运动的组合，前提是空气阻力可忽略且重力场均匀。这两个运动是独立的：水平速度分量永不改变，而竖直分量经历恒定的向下加速度 a_y = –g（取向上为正）。

a_x = 0 , a_y = –g = –9.8 m s⁻² (on Earth)

The trajectory of a projectile is parabolic. By eliminating time between the x and y equations of motion, one obtains the equation of the path: y = (tanθ₀) x – [g/(2 v₀² cos²θ₀)] x², where θ₀ is the launch angle above the horizontal.

抛体的轨迹是一条抛物线。将 x 和 y 运动方程之间的时间消去，便得到轨迹方程：y = (tanθ₀) x – [g/(2 v₀² cos²θ₀)] x²，其中 θ₀ 是发射时与水平面的夹角。

Understanding this independence allows us to treat the vertical motion like a particle thrown straight upward while the horizontal motion proceeds steadily.

理解这种独立性，我们就可以像处理竖直上抛的质点那样处理竖直运动，而水平运动则匀速进行。

6. Horizontal Projection | 水平抛射

When an object is launched horizontally from a height H with initial speed v₀, the initial velocity components are v_0x = v₀, v_0y = 0. The vertical motion starts from rest and falls under gravity: it determines the time of flight.

当物体从高度 H 处以初速度 v₀ 水平抛出时，初速度分量为 v_0x = v₀，v_0y = 0。竖直运动从静止开始，在重力作用下下落：它决定了飞行时间。

Time of flight: t_flight = √(2H / g)

This time is derived from the kinematic equation y = H – ½ g t², setting y = 0 at impact. During this time, the horizontal range is simply R = v₀ t_flight = v₀ √(2H/g). The final velocity has both a horizontal component v₀ and a vertical component v_y = –g t_flight; the impact speed is therefore v = √(v₀² + 2gH).

这个时间由运动学方程 y = H – ½ g t² 推导得出，并令撞击时 y = 0。在这段时间内，水平射程只是 R = v₀ t_flight = v₀ √(2H/g)。末速度同时具有水平分量 v₀ 和竖直分量 v_y = –g t_flight；因此撞击速率为 v = √(v₀² + 2gH)。

The path is a half‑parabola opening downward. AP problems often ask for the time, range, or the angle at which the projectile strikes the ground.

其轨迹是一条开口向下的半抛物线。AP 考试常会要求计算时间、射程或抛体落地时的角度。

7. Angled Projectile Motion | 斜抛运动

For a projectile launched from ground level with speed v₀ at an angle θ₀ above the horizontal, the initial components are v_0x = v₀ cosθ₀ and v_0y = v₀ sinθ₀. The object returns to the same vertical level (y = y₀) at the end of the flight.

对于从地面以速率 v₀、与水平面夹角 θ₀ 斜抛出的物体，其初速度分量为 v_0x = v₀ cosθ₀ 和 v_0y = v₀ sinθ₀。物体在飞行末了回到同一竖直高度（y = y₀）。

Time of flight: T = 2 v₀ sinθ₀ / g

Maximum height: H_max = (v₀ sinθ₀)² / (2g)

Range: R = v₀² sin(2θ₀) / g

The range formula shows that, for a given initial speed, the maximum range is achieved at θ₀ = 45°. Complementary angles (e.g., 30° and 60°) give the same range, but the higher angle yields a longer flight

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Kinematics Vectors and Projectile Motion | 运动学矢量与抛体运动