📚 Motion in Two Dimensions – Projectiles | 二维运动——抛体运动
Projectile motion is the motion of an object thrown or projected into the air, subject to only the acceleration of gravity. The path followed by a projectile is called its trajectory. In two-dimensional kinematics, we treat the horizontal and vertical components of motion independently, which greatly simplifies analysis. A deep understanding of projectiles is essential for A-Level Physics, as it bridges one-dimensional kinematics and vector resolution.
抛体运动是指物体被抛射到空中后，仅在重力加速度作用下的运动。抛体所经过的路径称为轨迹。在二维运动学中，我们可以将水平方向和竖直方向的运动分量独立处理，这大大简化了分析。深入理解抛体运动对 A-Level 物理至关重要，因为它将一维运动学与矢量分解联系在了一起。
1. Introduction to Projectile Motion | 抛体运动简介
A projectile is any object upon which the only significant force is gravity. We ignore air resistance for the basic model, making the horizontal acceleration zero and the vertical acceleration equal to g downwards. This simplified model predicts a parabolic trajectory, which is a key result in classical mechanics.
抛体是指仅受重力显著作用的物体。在基本模型中我们忽略空气阻力，因此水平加速度为零，竖直加速度等于向下的 g。这一简化模型预测出的轨迹是抛物线，这是经典力学中的一个重要结论。
In the standard model of projectile motion, the horizontal and vertical motions are completely independent of each other. The time taken for the vertical motion determines the total flight time, which is shared by both components. This independence is the foundation for solving any projectile problem.
在标准的抛体运动模型中，水平运动和竖直运动彼此完全独立。竖直运动所花费的时间决定了总飞行时间，这一时间同样适用于水平分量。这种独立性是解决任何抛体问题的基础。
2. Resolving the Initial Velocity | 分解初速度
If a projectile is launched with an initial speed v₀ at an angle θ to the horizontal, the initial velocity vector must be resolved. The horizontal component is v₀ₓ = v₀ cosθ, and the vertical component is v₀ᵧ = v₀ sinθ. These become the starting values for the two independent sets of equations.
如果抛体以初速率 v₀ 与水平方向成 θ 角发射，就必须分解初速度矢量。水平分量是 v₀ₓ = v₀ cosθ，竖直分量是 v₀ᵧ = v₀ sinθ。这两个值即作为两套独立运动方程的初始值。
It is crucial to assign a consistent sign convention, usually taking upward as positive and downward as negative. Thus the vertical acceleration is aᵧ = −g, where g = 9.81 m s⁻². For an object projected above the horizontal, v₀ᵧ is positive, while gravity acts to reduce this component.
建立一致的符号规则至关重要，通常取向上为正、向下为负。因此竖直加速度为 aᵧ = −g，其中 g = 9.81 m s⁻²。对于斜向上抛出的物体，v₀ᵧ 为正值，而重力会使该分量逐渐减小。
3. Independent Horizontal Motion | 独立的水平运动
Since there is no horizontal force (ignoring air resistance), the horizontal acceleration aₓ = 0. Consequently, the horizontal component of velocity remains constant throughout the flight: vₓ = v₀ cosθ. The horizontal displacement after time t is simply sₓ = (v₀ cosθ) t.
由于没有水平方向的力（忽略空气阻力），水平加速度 aₓ = 0。因此，整个飞行过程中速度的水平分量保持不变：vₓ = v₀ cosθ。经过时间 t 后的水平位移就是 sₓ = (v₀ cosθ) t。
This constant horizontal velocity means that the range of a projectile depends only on the horizontal component of the initial velocity and the total time of flight. Any change in the vertical motion that affects flight time will directly alter the horizontal range.
水平速度恒定意味着抛体的射程仅取决于初速度的水平分量和总飞行时间。任何影响飞行时间的竖直运动变化都会直接改变水平射程。
4. Vertical Motion under Gravity | 重力作用下的竖直运动
The vertical motion is governed by constant downward acceleration g. Using the initial vertical velocity v₀ᵧ = v₀ sinθ, the vertical velocity at any time t is vᵧ = v₀ sinθ − gt. The vertical displacement is sᵧ = (v₀ sinθ) t − ½ g t². These equations assume upward is positive.
竖直运动由向下的恒定加速度 g 支配。利用初始竖直速度 v₀ᵧ = v₀ sinθ，任意时刻 t 的竖直速度为 vᵧ = v₀ sinθ − gt。竖直位移为 sᵧ = (v₀ sinθ) t − ½ g t²。这些公式均假定向上为正方向。
At the highest point of the trajectory, the vertical velocity momentarily becomes zero. This condition, vᵧ = 0, allows us to find the time to reach maximum height: tₘₐₓ = (v₀ sinθ) / g. The symmetry of the parabolic path also means that the time to go up equals the time to come back down to the same vertical level.
在轨迹的最高点，竖直速度瞬时为零。利用这一条件 vᵧ = 0，可以求出到达最大高度的时间：tₘₐₓ = (v₀ sinθ) / g。抛物线路径的对称性还意味着从抛出点到最高点的时间等于从最高点落回同一水平面的时间。
5. Equations of Motion for Projectiles | 抛体运动方程组
The complete set of kinematic equations for a projectile can be summarised as follows:
Horizontal: aₓ = 0, vₓ = v₀ cosθ, sₓ = (v₀ cosθ) t.
Vertical: aᵧ = −g, vᵧ = v₀ sinθ − gt, sᵧ = (v₀ sinθ) t − ½ g t², and vᵧ² = (v₀ sinθ)² − 2g sᵧ.
抛体的完整运动学方程组可总结如下：
水平方向：aₓ = 0，vₓ = v₀ cosθ，sₓ = (v₀ cosθ) t。
竖直方向：aᵧ = −g，vᵧ = v₀ sinθ − gt，sᵧ = (v₀ sinθ) t − ½ g t²，以及 vᵧ² = (v₀ sinθ)² − 2g sᵧ。
These equations assume that the launch and landing points are at the same vertical height. When the heights differ, you must treat sᵧ as the net vertical displacement, which can be positive or negative, and solve the quadratic in t. The positive root gives the flight time.
上述方程假定抛射点和落地点处于同一高度。当高度不同时，必须将 sᵧ 视为净竖直位移，该值可正可负，然后求解关于 t 的二次方程。正根即为飞行时间。
6. Time of Flight | 飞行时间
For a projectile that lands at the same vertical level from which it was launched, the total time of flight T can be found by setting sᵧ = 0. Solving 0 = (v₀ sinθ) T − ½ g T² gives T = (2 v₀ sinθ) / g. This result highlights that a larger initial vertical speed or a smaller g increases air time.
对于落回与抛出点同一水平面的抛体，总飞行时间 T 可令 sᵧ = 0 求得。求解 0 = (v₀ sinθ) T − ½ g T² 可得 T = (2 v₀ sinθ) / g。这一结果表明，较大的初速竖直分量或较小的 g 会增加滞空时间。
If the projectile lands at a different height, say a vertical displacement h, then sᵧ = h and we solve the quadratic equation ½ g t² − (v₀ sinθ) t + h = 0. In such cases the motion loses its symmetry, and you must carefully choose the physically meaningful root for time.
如果抛体落在不同高度，例如竖直位移为 h，那么 sᵧ = h，我们需要求解二次方程 ½ g t² − (v₀ sinθ) t + h = 0。这种情况下运动失去对称性，必须谨慎选择物理上有意义的时间根。
7. Maximum Height | 最大高度
The maximum height H is reached when the vertical velocity becomes zero. Using vᵧ² = (v₀ sinθ)² − 2g sᵧ and setting vᵧ = 0 at sᵧ = H, we obtain H = (v₀ sinθ)² / (2g). This depends only on the vertical component of the launch velocity.
当竖直速度为零时达到最大高度 H。利用 vᵧ² = (v₀ sinθ)² − 2g sᵧ，并在 sᵧ = H 处令 vᵧ = 0，可得 H = (v₀ sinθ)² / (2g)。该值仅取决于发射速度的竖直分量。
Maximum height is proportional to the square of the initial vertical speed and inversely proportional to g. A projectile launched straight up (θ = 90°) achieves the greatest possible height for a given v₀, while for θ = 30° the height is one quarter of that at 90°.
最大高度与初速竖直分量的平方成正比，与 g 成反比。对于给定的 v₀，竖直向上发射（θ = 90°）时高度最大；而对于 θ = 30°，其高度仅为 90° 时的四分之一。
8. Range of a Projectile | 射程
The horizontal range R is the horizontal distance travelled during the time of flight. For level ground, R = (v₀ cosθ) × T = (v₀ cosθ) × (2 v₀ sinθ / g) = (v₀² sin 2θ) / g. This elegant result shows that maximum range is achieved when sin 2θ = 1, i.e. θ = 45°.
水平射程 R 是飞行时间内经过的水平距离。对于水平地面，R = (v₀ cosθ) × T = (v₀ cosθ) × (2 v₀ sinθ / g) = (v₀² sin 2θ) / g。这个简洁的表达式表明，当 sin 2θ = 1，即 θ = 45° 时可获得最大射程。
For a given initial speed, complementary angles such as 30° and 60° produce the same range, because sin(2×30°) = sin(2×60°). However, the trajectories are different: the larger angle gives a higher but shorter-range flight until the maximum, and a steeper descent.
对于给定的初速率，互余角（如 30° 和 60°）能产生相同的射程，因为 sin(2×30°) = sin(2×60°)。但轨迹不同：较大的发射角会使抛体飞得更高，上升和下降更陡，但在水平方向前进相同距离。
9. The Trajectory Equation | 轨迹方程
By eliminating time t from the equations for sₓ and sᵧ, we obtain the path equation: sᵧ = (tanθ) sₓ − (g / (2 v₀² cos²θ)) sₓ². This is of the form y = mx − kx², a parabola. The coefficient of sₓ² is always negative, confirming that the trajectory curves downward.
通过从 sₓ 和 sᵧ 的方程中消去时间 t，我们得到轨迹方程：sᵧ = (tanθ) sₓ − (g / (2 v₀² cos²θ)) sₓ²。该方程形如 y = mx − kx²，是一条抛物线。sₓ² 的系数恒为负值，这证实了轨迹是向下弯曲的。
The trajectory equation is particularly useful for determining whether a projectile will clear an obstacle or hit a target located at a specific point. You can substitute the horizontal coordinate of the target into the equation and check whether the calculated vertical position matches the target’s height.
轨迹方程对于判断抛体能否越过障碍物或命中特定位置的目标非常有用。可以将目标的水平坐标代入方程，检验计算出的竖直位置是否与目标高度一致。
10. Projection on an Inclined Plane | 斜面上的抛体运动
When a projectile is launched on an incline, the situation becomes more complex. The range along the slope, time of flight and maximum distance are found by resolving the motion parallel and perpendicular to the inclined surface, with g having components along both axes.
当抛体在斜面上发射时，情况变得更加复杂。沿斜面的射程、飞行时间和最大距离可以通过将运动分解为平行和垂直于斜面的两个分量来求得，此时重力加速度 g 在两个方向上都有分量。
Although the CIE A-Level syllabus focuses mainly on horizontal ground, the conceptual understanding of rotating axes and using components of g is good practice. Often you will see that the optimum angle for maximum range up an incline is not 45°, but depends on the slope angle.
尽管 CIE A-Level 考纲主要关注水平地面的情况，但理解旋转坐标轴和运用 g 的分量是一种很好的思维训练。通常你会发现，斜面上获得最大射程的最佳角度不再是 45°，而是取决于斜面倾角。
11. Effect of Air Resistance | 空气阻力的影响
In reality, air resistance cannot always be ignored. It acts opposite to the velocity, reducing both horizontal and vertical speeds. The trajectory becomes asymmetric: the ascending branch is steeper, the maximum height and range are reduced, and the descending branch is shorter and more curved near the end.
实际中，空气阻力不能总是忽略。它作用于速度的反方向，同时减小水平和竖直速率。轨迹变得不对称：上升段更陡，最大高度和射程减小，下降段更短并在末端更加弯曲。
Qualitatively, a projectile with air resistance will have a smaller range, a lower peak, and a steeper descent. Drag forces depend on speed, so the deceleration is not constant, making analytical solutions very difficult. At A-Level you are expected to describe these effects but not to calculate them quantitatively.
定性地看，受空气阻力影响的抛体射程更小、最高点更低、下落更陡。阻力取决于速度，因此减速度不是恒定的，解析求解非常困难。在 A-Level 阶段，你需要描述这些影响但无需定量计算。
12. Summary and Key Points | 总结与要点
Projectile motion is solved by separating it into constant-velocity horizontal motion and constant-acceleration vertical motion. The key quantities—time of flight, maximum height and range—follow from simple kinematics when the launch and landing heights are equal.
抛体运动是通过将其分解为匀速水平运动和匀加速竖直运动来求解的。当发射和落地高度相同时，飞行时间、最大高度和射程这些关键量都可从简单运动学公式得出。
Always define a sign convention, resolve the launch velocity into components, and apply the equations of motion independently. Remember that the parabolic trajectory and the 45° maximum range are idealised results; real projectiles experience air resistance, which alters the path. Mastery of these fundamentals will allow you to tackle any two-dimensional kinematics problem confidently.
务必定义符号规则，分解发射速度，然后独立地应用运动方程。记住抛物线轨迹和 45° 最大射程是理想化的结果；真实的抛体会受空气阻力而改变路径。掌握这些基本原理将让你自信地应对任何二维运动学问题。
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