Horizontal Projectile Motion: Laws and Applications | 水平抛体运动的规律与应用

📚 Horizontal Projectile Motion: Laws and Applications | 水平抛体运动的规律与应用

A projectile launched with a purely horizontal initial velocity from a height follows a curved path called a trajectory. The motion is a classic example of two-dimensional kinematics, and it can be fully understood by treating the horizontal and vertical components separately. This article explains the governing equations, the parabolic trajectory, the time of flight, the range, and several practical applications, in line with the A-level Mathematics and Physics curriculum.

水平抛体运动是指物体以完全水平的初速度从高处抛出后，沿弯曲轨迹运动的现象。它是二维运动学中的经典模型。理解水平抛体运动，需要将水平与竖直两方向分别处理。本文将系统讲解其运动方程、抛物轨迹、飞行时间、射程及实际应用，紧扣 A-level 数学与物理课程要求。

---

1. Mathematical Model and Assumptions | 数学模型与基本假设

In the simplest model, air resistance is neglected. The only force acting on the projectile is gravity, directed vertically downward with magnitude g = 9.81 m s⁻². The projectile starts at height h above a reference level, with initial velocity u₀ horizontally. Once released, its horizontal acceleration is zero, and its vertical acceleration is −g, taking upward as positive. The motion is therefore completely determined by the initial height and the horizontal speed.

在最简模型中，忽略空气阻力。物体只受竖直向下的重力，大小为 g = 9.81 m s⁻²。物体在参考水平面上方高度 h 处以水平初速度 u₀ 抛出。一旦抛出，水平方向的加速度为零，竖直方向加速度为 −g（设向上为正）。因此运动完全由初始高度和水平速度决定。

---

2. Independence of Horizontal and Vertical Motions | 水平与竖直运动的独立性

Galileo’s principle of independence states that the horizontal and vertical components of motion do not interfere with each other. The horizontal component continues at constant speed, while the vertical component behaves like a free-falling object released from rest. This property allows us to analyse the motion with separate equations and then combine the results.

伽利略运动独立性原理指出，水平方向与竖直方向的运动互不干扰。水平方向保持匀速，竖直方向则相当于从静止开始自由落体。借助这一性质，我们可以分别用方程分析两个方向，再综合结果。

---

3. Horizontal Motion Equations | 水平运动方程

Take the launch point as the origin, with the x-axis horizontal and the y-axis vertical downward. This choice often simplifies the mathematics. Since the horizontal acceleration aₓ = 0, the horizontal displacement after time t is

x = u₀t

and the horizontal velocity remains constant: vₓ = u₀.

以抛出点为原点，x 轴水平，y 轴竖直向下，这样的选取往往能简化计算。由于水平加速度 aₓ = 0，经过时间 t 后的水平位移为

x = u₀t

水平速度保持不变：vₓ = u₀。

---

4. Vertical Motion Equations | 竖直运动方程

The vertical component starts at rest, so the initial vertical velocity is u_y = 0, and it accelerates at g downward. Taking y as positive downward, the displacement and velocity are

竖直方向初速度为零，即 u_y = 0，并以大小为 g 的加速度向下加速。取 y 向下为正，则竖直位移和速度为

a_y = g = 9.81 m s⁻²

v_y = gt

y = ½gt²

---

5. Trajectory Equation | 轨迹方程

Eliminating t between x = u₀t and y = ½gt² gives the equation of the trajectory:

由 x = u₀t 与 y = ½gt² 消去 t，得到轨迹方程：

y = (g / (2u₀²)) x²

Because y is proportional to x², the path is a parabola whose vertex is at the launch point. The coefficient g/(2u₀²) decreases as the horizontal launch speed increases, so a faster projectile travels farther for the same drop in height.

由于 y 与 x² 成正比，轨迹是以抛出点为顶点的抛物线。系数 g/(2u₀²) 随水平初速度增大而减小，因此在相同下落高度下，速度越快，水平射程越远。

---

6. Time of Flight and Range | 飞行时间与射程

When the projectile falls to a vertical distance h below the launch level, y = h. From y = ½gt², the time of flight T is

当物体下落到发射点下方高度 h 时，y = h。由 y = ½gt² 得飞行时间

T = √(2h/g)

Notice that T depends only on the initial height h, not on the horizontal speed u₀. The horizontal range R is the horizontal distance travelled during this time:

注意 T 只取决于初始高度 h，与水平速度 u₀ 无关。水平射程 R 为这段时间内水平方向走过的距离：

R = u₀T = u₀√(2h/g)

---

7. Velocity at Any Time | 任意时刻的速度

The velocity has two components: a constant horizontal component vₓ = u₀ and an increasing vertical component v_y = gt. The resultant speed v is therefore

任意时刻速度有两个分量：恒定的水平分量 vₓ = u₀，以及不断增大的竖直分量 v_y = gt。合速度大小为

v = √(u₀² + (gt)²)

The direction of motion, measured as the angle below the horizontal, satisfies

速度方向与水平方向的夹角 θ 满足

tan θ = v_y / vₓ = gt / u₀

Thus the falling body is always moving increasingly steeply downward.

因此物体下落时，速度方向越来越陡地向下倾斜。

---

8. Worked Example | 例题分析

A ball is rolled off a cliff of height 45 m with a horizontal speed of 12 m s⁻¹. Taking g = 10 m s⁻², find the time of flight, the horizontal range, and the velocity just before impact.

一小球以 12 m s⁻¹ 的水平速度从 45 m 高的悬崖滚下。取 g = 10 m s⁻²，求飞行时间、水平射程以及落地前的速度。

Using T = √(2h/g) = √(90/10) = 3.0 s.

由 T = √(2h/g) = √(90/10) = 3.0 s。

The range is R = u₀T = 12 × 3 = 36 m.

射程 R = u₀T = 12 × 3 = 36 m。

The vertical velocity at impact is v_y = gT = 30 m s⁻¹. The resultant speed is v = √(12² + 30²) = √(144 + 900) = √1044 = 32.3 m s⁻¹. The direction is tan θ = 30/12 = 2.5, giving θ = 68.2° below the horizontal.

落地时的竖直分速度 v_y = gT = 30 m s⁻¹。合速度 v = √(12² + 30²) = √(144 + 900) = √1044 = 32.3 m s⁻¹。方向满足 tan θ = 30/12 = 2.5，故 θ = 68.2°，即与水平方向成 68.2° 向下。

---

9. Effect of Changing Parameters | 参数变化的影响

We can summarise the effect of changing the initial height h and the horizontal speed u₀ on the key quantities: time of flight T, range R, and speed at impact v.

我们可以总结初始高度 h 与水平速度 u₀ 的变化对关键物理量的影响：飞行时间 T、射程 R 和落地速度 v。

Physical quantity | 物理量	Double u₀ | 初速度加倍	Double h | 高度加倍
Time of flight T | 飞行时间	Unchanged | 不变	Multiply by √2 | 变为 √2 倍
Range R | 射程	Multiply by 2 | 变为 2 倍	Multiply by √2 | 变为 √2 倍
Speed at impact v | 落地速度	Increases | 增大	Increases | 增大
Impact angle θ | 落地角度	Decreases | 减小	Increases | 增大

The table above shows how the main results change when the initial speed or height is doubled, all other factors being fixed. Doubling the horizontal speed doubles the range but does not change the time of flight. Doubling the height increases the time of flight and the range by a factor of √2, and it also increases the final velocity and makes the impact direction steeper.

上表展示了初速度或高度加倍时主要结果的变化，其他条件不变。水平初速度加倍时，射程加倍，但飞行时间不变。高度加倍时，飞行时间和射程都变为原来的 √2 倍，同时落地速度增大，落地方向也更陡。

---

10. Practical Applications | 实际应用

Horizontal projectile motion appears in many real situations. In sports, a basketball passed horizontally from above the hoop, or a ball falling off a moving table, both follow this model. In engineering, water jets emerging horizontally from a tank, or debris launched from a conveyor belt, are analysed in the same way. In aviation, cargo dropped from a horizontal flight path has the same initial velocity as the aeroplane and lands ahead of the release point; pilots must account for the time of fall and airspeed.

水平抛体运动广泛存在于实际生活中。在体育运动中，从篮圈上方水平传出的篮球、从运动桌面上滑落的球体，都遵循这一模型。在工程领域，水平射出的水射流、从传送带末端飞出的物料，也用同样方法分析。在航空运输中，从水平飞行的飞机上投放的货物具有与飞机相同的水平初速度，会落在释放点前方；飞行员必须综合考虑下落时间与空速。

---

11. Common Mistakes and Problem-Solving Tips | 常见错误与解题技巧

A frequent error is to use different times for the horizontal and vertical components, or to treat the horizontal motion as accelerated. Another error is to ignore the initial vertical velocity when the launch is horizontal. Remember these points:

常见错误包括：对水平方向和竖直方向使用了不同的时间，或者误认为水平方向有加速度；在水平抛出时忽略了竖直初速度为零的条件。请记住以下要点：

• Read the origin and positive directions at the beginning.

  开始时先明确原点与正方向。

• Write the equations for x and y separately.

  分别写出 x 与 y 方向的运动方程。

• Use the vertical motion to find the time.

  用竖直方向求时间。

• Use the horizontal motion to find the range.

  用水平方向求射程。

• When finding the final velocity, combine components with Pythagoras, not by simply adding speeds.

  求合速度时，用勾股定理合成两个分量，而不是简单相加。

---

12. Summary and Revision Points | 总结与复习要点

Horizontal projectile motion is governed by two independent motions: uniform horizontal motion and uniformly accelerated vertical motion. The key equations are x = u₀t, y = ½gt², T = √(2h/g), R = u₀√(2h/g), and v = √(u₀² + (gt)²). The trajectory is a parabola. Time of flight depends only on height; range depends on both horizontal speed and height. These results form a foundation for solving any projectile problem.

水平抛体运动由两个独立运动支配：水平方向的匀速直线运动与竖直方向的匀加速直线运动。核心公式为 x = u₀t、y = ½gt²、T = √(2h/g)、R = u₀√(2h/g)、v = √(u₀² + (gt)²)。轨迹为抛物线。飞行时间只由高度决定；射程由水平速度和高度共同决定。掌握这些结论，就能解决各种抛体运动问题。

Published by TutorHao | Mathematics Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）