📚 Horizontal Projection | 水平抛体运动
Horizontal projection is a core topic in Edexcel A-Level Mechanics. It describes the motion of an object launched with an initial velocity that is entirely horizontal, after which the only force acting is gravity. This means the object follows a parabolic path while its horizontal and vertical motions can be analysed independently. Understanding this separation of components is essential for solving projectile problems and forms the foundation for more complex two-dimensional motion.
水平抛体运动是Edexcel A-Level力学中的一个核心主题。它描述的是物体以完全水平的初速度抛出后，仅受重力作用的运动。物体将沿抛物线轨迹运动，其水平与竖直方向的运动可以独立分析。理解这种运动分解是解决抛体问题的基础，也是学习更复杂的二维运动的前提。
1. Introduction to Horizontal Projection | 水平抛体运动简介
In horizontal projection, the object is given an initial speed u horizontally from a certain height. From the moment of launch, gravity pulls it downwards with constant acceleration g. The horizontal velocity remains unchanged because no horizontal force acts, assuming negligible air resistance. This creates a curved trajectory which is a parabola. The study of this motion allows us to predict the time of flight, range, and impact velocity using simple kinematic equations.
在水平抛体运动中，物体从一定高度以水平初速度 u 抛出。自抛出瞬间起，重力以恒定加速度 g 将其向下拉。由于忽略空气阻力，没有水平力作用，水平速度保持不变。这就形成了抛物线轨迹。研究这一运动可以利用简单的运动学方程预测飞行时间、射程和着地速度。
This topic typically appears in the Mechanics component of Edexcel Mathematics. It tests your ability to break motion into perpendicular directions, apply suvat equations, and interpret real-world scenarios. Common exam questions include finding the range from a cliff or the velocity vector on impact.
该主题通常出现在Edexcel数学的力学部分，考查你将运动分解为互相垂直方向的能力、运用suvat方程的能力以及解释实际情景的能力。常见考题包括计算从悬崖抛出的射程或撞击时的速度矢量。
2. Modelling Assumptions | 模型假设
We make several simplifying assumptions to model horizontal projection. The object is treated as a particle, so its size and shape are ignored, and no rotational effects are considered. Air resistance is neglected, meaning the only force after launch is weight. The acceleration due to gravity, g, is assumed constant at 9.8 m s⁻² and acts vertically downwards. The motion takes place near the Earth's surface, so g does not vary with height.
我们建立模型时作了几项简化假设。物体被视为质点，忽略其大小和形状，也不考虑旋转效应。空气阻力忽略不计，因此抛出后唯一的作用力是重力。重力加速度 g 取常数 9.8 m s⁻²，方向竖直向下。运动发生在地表附近，所以 g 不随高度变化。
These assumptions allow us to treat the horizontal and vertical motions as completely independent. The horizontal motion undergoes zero acceleration, while the vertical motion is uniformly accelerated from rest with acceleration g. This independence makes the mathematics straightforward and is the key to solving any horizontal projection problem.
这些假设使得我们可以把水平和竖直运动看作完全独立。水平方向加速度为零，竖直方向从静止开始以加速度 g 匀加速。这种独立性使数学处理变得直接，是解决任何水平抛体运动问题的关键。
3. Coordinate System and Initial Conditions | 坐标系与初始条件
We set up a coordinate system with the origin at the point of projection. The x-axis is taken horizontally in the direction of the initial velocity, and the y-axis is chosen vertically downwards to make displacements positive as the object falls. This is the most convenient convention, although taking upwards as positive is also valid if signs are handled carefully. The initial velocity vector can then be written as (u, 0): horizontal component u, vertical component zero.
我们以抛出点为原点建立坐标系。x 轴沿初速度方向水平放置，y 轴竖直向下以使物体下落时位移为正。这是最方便的约定，当然若注意符号，取向上为正也有效。初始速度矢量可写为 (u, 0)：水平分量为 u，竖直分量为零。
ement when it hits the ground is y = h. The following table summarises the key quantities.
加速度恒定，矢量形式为 (0, g)。初始位置为 (0,0)。对于从离地高度 h 的悬崖水平抛出的物体，着地时的竖直位移为 y = h。下表总结了关键量。
| Quantity | Symbol | Value / Expression |
|---|---|---|
| Initial horizontal velocity | u | u (constant) |
| Initial vertical velocity | – | 0 |
| Horizontal acceleration | aₓ | 0 |
| Vertical acceleration | g | 9.8 m s⁻² (downwards) |
| Initial height | h | given |
4. Horizontal Motion: Constant Velocity | 水平运动：匀速直线运动
t velocity: distance = speed × time. In suvat terms, we use s = ut + ½at² with a = 0, giving the simple relation below.
由于没有水平力（忽略了空气阻力），水平方向加速度为零。因此，水平速度分量在整个飞行过程中保持为 u。时间 t 内的水平位移 x 直接由匀速运动的定义得到：距离 = 速度 × 时间。用 suvat 表达时，取 s = ut + ½at² 且 a = 0，即得到以下简单关系式。
x = u t
This equation tells us that the horizontal distance covered is proportional to time, with u as the constant of proportionality. No matter how far the object falls, the horizontal motion remains uniform. This is the reason why the range depends linearly on the initial speed and time of flight.
这个方程表明，经过的水平距离与时间成正比，比例系数为 u。无论物体竖直下落多远，水平运动始终保持匀速。这正是射程与初速度和时间成线性关系的原因。
5. Vertical Motion: Uniformly Accelerated Motion | 垂直运动：匀加速运动
½ g t². The vertical component of velocity at any instant is v = g t, directed downwards.
竖直运动完全由重力支配。由于初速度竖直分量为零且取向下为正，物体从静止开始以恒定的加速度 g 加速。时间 t 内的竖直位移 y 由 suvat 方程 s = ut + ½at² 令 u = 0 得出，即 y = ½ g t²。任意时刻的竖直速度分量为 v = g t，方向向下。
y = ½ g t²
v = g t
tly links the vertical speed to the vertical distance fallen. Since the vertical motion starts from rest, the speed gained depends only on the height lost, regardless of the horizontal speed. This reinforces the independence of the two components.
消去 t 还可得到 v² = 2 g y 这一有用关系，它直接联系了竖直速度与下落距离。由于竖直运动从静止开始，获得的速度仅取决于下落高度，与水平速度无关，这再次说明两个分量的独立性。
6. Time of Flight | 飞行时间
ing for T yields the expression below. Notice that the mass of the object does not appear, reflecting that gravity accelerates all objects equally in the absence of air resistance.
飞行时间 T 是从抛出到物体着地的时间长度。对于从高度 h 抛出的情况，着地时竖直位移为 y = h。代入 y = ½ g t² 得 h = ½ g T²，解出 T 得到下式。注意物体质量并未出现，反映出在没有空气阻力时，重力对一切物体的加速作用相同。
T = √2h / g
It is crucial to recognise that T depends only on the height from which the object is projected and on g. The horizontal speed u has no effect on how long the object stays in the air. This is a common source of misconception; a faster horizontal throw does not make the object fall more slowly – it simply travels further horizontally while falling.
重要的是认识到 T 仅取决于抛出高度和 g。水平速度 u 对物体在空中停留的时间没有影响。这是一个常见的误解：更快的水平抛出并不会让物体落得更慢，它只会在下落的同时前进得更远。
7. Horizontal Range | 水平射程
constant horizontal speed u by the time of flight T. Therefore, R = u T. Substituting T = √2h / g gives the range formula shown below. This shows that for a fixed height, doubling u doubles the range.
水平射程 R 是物体着地时经过的总水平距离。由于水平运动是匀速的，用恒定的水平速度 u 乘以飞行时间 T 即可：R = u T。代入 T =
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