Projectile Motion Core Formulas | 抛体运动核心公式总结

📚 Projectile Motion Core Formulas | 抛体运动核心公式总结

Projectile motion describes the motion of an object launched into the air, subject only to the acceleration due to gravity. Understanding the core formulas allows us to predict the object’s position, velocity, and time of flight without needing to know the exact path taken at every instant.

抛体运动描述的是一个物体被抛入空中后，仅在重力加速度作用下的运动。掌握核心公式，我们就能在不逐点了解运动路径的情况下，预测物体的位置、速度和飞行时间。

1. Key Assumptions & SUVAT Equations | 关键假设与 SUVAT 方程

Before applying the formulas, we must state the standard assumptions: air resistance is negligible, the gravitational field strength (g) is constant, and the object remains close to the Earth’s surface. Under these conditions, the motion can be analyzed using the SUVAT equations independently for horizontal and vertical components.

在套用公式前，我们必须明确标准假设：空气阻力可忽略不计，重力场强度 (g) 恒定，且物体始终接近地球表面。在这些条件下，我们可以利用 SUVAT 方程对水平与垂直方向的分运动分别进行分析。

Where ‘u’ is the initial velocity, ‘v’ is the final velocity, ‘a’ is the constant acceleration, ‘t’ is the time, and ‘s’ is the displacement.

其中 ‘u’ 是初速度，’v’ 是末速度，’a’ 是恒定加速度，’t’ 是时间，’s’ 是位移。

2. Resolving Initial Velocity | 初速度的分解

At the moment of launch, the initial velocity (u) must be resolved into two perpendicular components. Taking the angle of projection (θ) to the horizontal, we can define the horizontal component (uₓ) and the vertical component (uᵧ).

在发射瞬间，初速度 (u) 必须分解为两个相互垂直的分量。设发射角（与水平方向的夹角）为 (θ)，我们可以定义水平分量 (uₓ) 和垂直分量 (uᵧ)。

This separation is the foundation for analyzing the two independent motions later.

这一分解是后续分析两个独立运动的基础。

3. Horizontal Motion | 水平方向运动

Since no horizontal force acts on the projectile (ignoring air resistance), the horizontal acceleration is zero. Therefore, the horizontal velocity remains constant throughout the flight.

由于抛体在水平方向不受力（忽略空气阻力），水平加速度为零。因此，整个飞行过程中水平速度保持不变。

vₓ = uₓ = u cos(θ)

x = (u cos(θ)) × t

Here, ‘x’ represents the horizontal displacement at time ‘t’.

此处，’x’ 代表在时间 ‘t’ 时的水平位移。

4. Vertical Motion | 垂直方向运动

The vertical motion is affected by gravity, which acts downwards. Taking upwards as positive, the vertical acceleration is -g.

垂直方向运动受重力影响，重力方向向下。若取向上为正，则垂直加速度为 -g。

vᵧ = u sin(θ) – g × t

y = (u sin(θ)) × t – ½ × g × t²

Here, ‘y’ represents the vertical displacement at time ‘t’.

此处，’y’ 代表在时间 ‘t’ 时的垂直位移。

5. Time of Flight | 飞行时间

The total time of flight (T) is the duration the projectile remains in the air. This occurs when the vertical displacement y returns to zero (landing at the same height). Setting y = 0 in the vertical displacement equation gives:

总飞行时间 (T)

Published by TutorHao | Mathematics Revision Series | aleveler.com

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