HKDSE Physics Core
Chapter 6: Projectile Motion | 拋體運動
Billy Sir’s Smart Notes: Master 2D Motion, Trajectories, and the Monkey-Hunter Experiment.
由中大物理系碩士 Billy Sir 編寫，助你極速掌握二維運動、軌跡計算及馬騮與獵人實驗。
1. Definition of Projectile Motion | 拋體運動定義
Projectile motion is the motion of an object thrown or projected into the air, subject to only the acceleration of gravity. 拋體運動是指物體被拋入空中後，僅在重力加速度作用下的運動。
🔑 The Golden Rule of Projectiles | 拋體運動黃金法則
The horizontal and vertical motions are completely independent of each other! (水平和垂直方向的運動是完全獨立的！)
- Horizontal (x-axis): Uniform motion. Velocity is constant (\( a_x = 0 \)). (水平：勻速運動，加速度為零)
- Vertical (y-axis): Uniformly accelerated motion (Free fall). Acceleration is due to gravity (\( a_y = g \)). (垂直：勻加速運動/自由落體，加速度為重力加速度)
2. Horizontal Projection | 水平拋出
When an object is projected horizontally, its initial vertical velocity is zero (\( u_y = 0 \)). It is essentially a combination of a ball moving uniformly in the horizontal direction and a ball in vertical free fall. 當物體被水平拋出時，其初始垂直速度為零。它本質上是水平勻速運動和垂直自由落體的結合。
Illustration: Dropped Ball vs. Horizontally Projected Ball
Notice that both balls are at the same vertical height at any given time \( t \). (注意：在任何給定時間，兩球的垂直高度完全相同。)
3. Projection at an Angle | 傾斜拋出
When a ball is projected with an initial velocity \( u \) at an angle \( \theta \) to the horizontal, we resolve the velocity into two components: (當球以初速 \( u \) 及仰角 \( \theta \) 拋出時，我們將速度分解為兩個分量：)
- \( u_x = u \cos \theta \) (Constant throughout the flight)
- \( u_y = u \sin \theta \) (Changes due to gravity)
🧮 Key Calculations | 核心計算公式
1. Time of Flight (\( T \)): The total time the projectile is in the air. (飛行總時間)
2. Maximum Height (\( H \)): Reached when vertical velocity \( v_y = 0 \). (最大高度)
3. Horizontal Range (\( R \)): The total horizontal distance traveled. (水平射程)
🎯 Maximizing Range & Complementary Angles | 最大射程與互餘角
Which angle gives the farthest range?
The maximum horizontal range occurs when \( \sin 2\theta = 1 \), which means \( 2\theta = 90^\circ \). Therefore, the optimal angle is \( \theta = 45^\circ \). (當角度為 45° 時，水平射程最遠。)
Which two angles land at the same point?
Any two complementary angles (angles that add up to \( 90^\circ \), such as \( 30^\circ \) and \( 60^\circ \)) will result in the exact same horizontal range, assuming the same initial speed. (任何兩個互餘的角度，如 30° 和 60°，將落在同一點。)
4. Energy Changes (PE & KE) | 能量轉換 (勢能與動能)
Assuming air resistance is negligible, the total mechanical energy is conserved throughout the flight. (假設空氣阻力可忽略，整個飛行過程中的總機械能守恆。)
| Position (位置) | Kinetic Energy (KE) | Potential Energy (PE) |
|---|---|---|
| Going Up (上升時) | Decreases (減少) | Increases (增加) |
| Maximum Height (最高點) | Minimum (Not Zero!) Because \( v_x \) still exists. \( KE = \frac{1}{2}m(u \cos \theta)^2 \) |
Maximum |
| Going Down (下降時) | Increases (增加) | Decreases (減少) |
5. The Monkey and Hunter Experiment | 馬騮與獵人實驗
A classic physics thought experiment: A hunter aims a dart gun directly at a monkey hanging from a tree branch. The exact moment the gun is fired, the monkey lets go and falls freely. Will the dart hit the monkey? (獵人瞄準樹上的馬騮開槍，開槍的瞬間馬騮鬆手自由下落。飛鏢會擊中馬騮嗎？)
🐒 The Result | 結果
Yes, the dart will always hit the monkey! (Assuming the dart reaches the monkey’s horizontal position before hitting the ground). (是的，飛鏢一定會擊中馬騮！)
🧠 The Physics Explanation | 物理原理解釋
Gravity acts on both the dart and the monkey equally. If there were no gravity, the dart would travel in a straight line and hit the monkey (who would remain stationary in the air). Because of gravity, both the dart and the monkey fall at the exact same rate (\( \frac{1}{2}gt^2 \)) relative to their gravity-free positions. Since they fall by the same vertical distance in the same amount of time, their paths will intersect! (重力對飛鏢和馬騮的作用是相同的。兩者在相同時間內下落的垂直距離完全相等，因此必定相撞！)
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