Projectile Motion Formulae | 抛体运动公式

📚 Projectile Motion Formulae | 抛体运动公式

In Edexcel A-Level Mechanics, projectile motion is treated as two independent linear motions: constant horizontal velocity and constant vertical acceleration due to gravity. The formulae below connect initial speed, angle of projection, time, displacement and velocity.

在 Edexcel A-Level 力学中，抛体运动被分解为两个独立的直线运动：水平方向速度恒定，竖直方向受重力加速度影响。以下公式将初速度、抛射角、时间、位移和速度联系起来。

1. Modelling Assumptions for Projectile Motion | 抛体运动的基本假设

In projectile motion questions, the object is treated as a particle moving under gravity alone. This means air resistance is ignored, the horizontal acceleration is zero, and the vertical acceleration is constant with magnitude g = 9.8 m s⁻² directed downward.

在抛体运动题目中，物体被视为仅在重力作用下的质点。这意味着忽略空气阻力，水平加速度为零，竖直加速度恒为 g = 9.8 m s⁻²，方向向下。

These assumptions allow the horizontal and vertical motions to be analysed separately using SUVAT equations. The horizontal and vertical components are linked only by the time of flight t.

这些假设使我们能够使用匀加速运动公式分别分析水平与竖直运动。水平分量和竖直分量只通过飞行时间 t 相互联系。

2. Resolving Initial Velocity into Components | 初速度的正交分解

If the initial speed is u and the angle of projection is θ above the horizontal, resolve the initial velocity into two perpendicular components.

若初速度为 u，发射角为水平上方 θ，则将初速度分解为两个正交分量。

Horizontal component: u cos θ

Vertical component: u sin θ

The horizontal component is the adjacent side of the right-angled vector triangle, so it uses cos θ. The vertical component is opposite the angle, so it uses sin θ.

水平分量是速度矢量直角三角形中的邻边，因此使用 cos θ。竖直分量是该角的对边，因此使用 sin θ。

3. Horizontal Motion Formula | 水平方向运动公式

The horizontal component of velocity is constant because there is no horizontal force. Therefore the horizontal displacement after time t is found directly from speed × time.

由于没有水平力，水平速度分量保持不变。因此时间 t 后的水平位移可直接用速度 × 时间求得。

x = u cos θ × t

This equation is valid for the entire flight, regardless of the vertical motion. In suvat form, horizontal acceleration is zero and initial horizontal velocity is u cos θ.

该方程在整个飞行过程中都成立，与竖直运动无关。在匀加速公式中，水平加速度为零，初始水平速度为 u cos θ。

4. Vertical Motion Formula | 竖直方向运动公式

Taking upward as positive, vertical motion has constant acceleration a = −g. The two main vertical equations are the displacement equation and the velocity equation.

取向上为正方向，竖直运动具有恒定加速度 a = −g。两个主要的竖直方程是位移方程和速度方程。

y = u sin θ × t − ½ g t²

Vertical velocity: v_y = u sin θ − g t

In these equations, y is the vertical displacement above the launch point and v_y is the vertical component of velocity at time t. The term ½ g t² is subtracted because gravity acts downward while upward is positive.

在这些方程中，y 是相对于抛出点的竖直位移，v_y 是时间 t 时的竖直

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