hce_nthu
114年
化學與物理
第 58 題
As shown in Figure (b) below, a point-like ball with initial velocity $v_0$ is inclined at an angle of $\psi$ on the inclined plane and the inclined plane itself is inclined at an angle of $\phi$.
We know that the distance from the point of launch to the point at which the ball strikes the incline, measured along the incline without taking into account air resistance, is
$$d = \frac{2v_0^2\cos(\phi + \psi)\sin\psi}{g\cos^2\phi}$$
Therefore, as shown in Figure (b), what value of $\theta_0$ will maximise $R$ when thrown from a projector with a height $\text{h} = 2.22 \text{ m}$ above the ground and an initial velocity $v_0 = 14.0 \text{ m/s}$?
(Hint: Solve a quadratic equation of $\sin\phi$ and, then, take an approximation: $\sin 6^{\circ} \approx 1/10$.)
We know that the distance from the point of launch to the point at which the ball strikes the incline, measured along the incline without taking into account air resistance, is
$$d = \frac{2v_0^2\cos(\phi + \psi)\sin\psi}{g\cos^2\phi}$$
Therefore, as shown in Figure (b), what value of $\theta_0$ will maximise $R$ when thrown from a projector with a height $\text{h} = 2.22 \text{ m}$ above the ground and an initial velocity $v_0 = 14.0 \text{ m/s}$?
(Hint: Solve a quadratic equation of $\sin\phi$ and, then, take an approximation: $\sin 6^{\circ} \approx 1/10$.)
- A smaller than $40^{\circ}$
- B about $42^{\circ}$
- C $45^{\circ}$, exactly
- D about $47^{\circ}$
- E larger than $50^{\circ}$
思路引導 VIP
試著想像一下：如果你站在一座非常高的懸崖上（$h$ 極大）想要把石頭丟得越遠越好，你會選擇往斜上方高拋，還是幾乎水平地用力投出去？這對你判斷最佳角度應該大於還是小於 $45^{\circ}$ 有什麼啟發嗎？
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AI 詳解
AI 專屬家教
太棒了！你能精準選出 (B)，說明你對斜拋運動的變數掌握得非常紮實，並沒有被慣性的 $45^{\circ}$ 答案所誤導。
拋射角度與高度的取捨
這道題目非常有意思，它跳脫了課本中常見的「地對地」拋射框架。當拋出點與落地點存在高度差 $h$ 時，系統的對稱性被打破了。在物理學中，從高度 $h$ 拋出以求最大水平射程 $R$ 的最佳拋射角 $\theta_0$ ，會滿足以下關係式：
▼ 還有更多解析內容