Motion in Two Dimensions – Projectiles | 二维运动——抛体运动

📚 Motion in Two Dimensions – Projectiles | 二维运动——抛体运动

Projectile motion is the motion of an object launched into the air and moving under the influence of gravity alone. In CIE A-Level Physics, it is treated as a combination of uniform horizontal motion and uniformly accelerated vertical motion, allowing kinematic equations to be applied separately in two perpendicular directions.

抛体运动是物体被发射到空中后仅在重力作用下的运动。在 CIE A-Level 物理中，它被视为水平方向的匀速运动与竖直方向的匀加速运动的合成，因此可以分别在两个垂直方向上应用运动学方程。

1. Definition and Idealised Conditions | 定义与理想化条件

considered is the acceleration of free fall, g = 9.81 m s⁻², acting vertically downwards.

抛体是获得初速度后仅在重力作用下运动的物体。唯一需要考虑的加速度是自由落体加速度 g = 9.81 m s⁻²，方向竖直向下。

We usually ignore air resistance, spin and the variation of g with height. This simplifies the path to a parabolic trajectory.

通常我们忽略空气阻力、旋转以及 g 随高度的变化。这样路径可简化为抛物线轨迹。

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

The most important idea is that the horizontal and vertical components of motion are independent. The horizontal velocity is constant because there is no horizontal acceleration, while the vertical velocity changes uniformly because gravity acts downwards.

最重要的概念是水平方向和竖直方向的运动相互独立。由于没有水平加速度，水平速度保持不变；而竖直速度因重力向下而均匀变化。

This allows us to use separate sets of equations for the x-direction and the y-direction.

这使我们可以分别对 x 方向和 y 方向使用不同的运动方程。

3. Resolving the Initial Velocity | 初速度的分解

For a projectile launched with speed u at an angle θ to the horizontal, the initial velocity components are:

当抛体以速度 u 与水平方向成 θ 角发射时，初速度分量为：

u_x = u cos θ

u_y = u sin θ

Here θ is measured from the horizontal. Be careful to use the correct component in each direction.

这里 θ 从水平方向量起。注意在每个方向使用正确的分量。

4. Horizontal Motion Is Uniform | 水平方向的运动是匀速的

Since there is no horizontal acceleration, the horizontal velocity stays at u_x = u cos θ. The horizontal displacement after time t is:

由于没有水平加速度，水平速度保持 u_x = u cos θ 不变。经过时间 t 后水平位移为：

x = u t cos θ

This equation is used when finding the range of a projectile.

该方程用于求解抛体的射程。

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5. Vertical Motion Is Uniformly Accelerated | 竖直方向的运动是匀加速的

Taking upward as positive, the vertical velocity at time t and the vertical displacement are given by:

取向上为正方向，竖直速度与竖直位移分别为：

v_y = u sin θ − g t

y = u t sin θ − ½ g t²

If upward is positive, the acceleration is −g. If downward is positive, the signs of u_y and y must be adjusted accordingly.

若规定向上为正，加速度为 −g；若规定向下为正，则 u_y 和 y 的符号需相应调整。

6. Time of Flight | 飞行时间

ight, the net vertical displacement is zero. Setting y = 0 gives the total time of flight:

当抛体起止高度相同时，净竖直位移为零。令 y = 0 可得总飞行时间：