일반물리학 1 — Force and Motion-I: Problem 52

`. - Internal thoughts in ``. - Summary at the end. **Refining the calculation steps for the output:** Problem asks for speed. I will use the kinematic approach as it matches the "Force and Motion" chapter context (Newton's laws are usually introduced before energy). 1. **Define Variables:** - $m_m = 93 \text{ kg}$ (mass of the man) - $m_s = 65 \text{ kg}$ (mass of the sandbag) - $h = 10.0 \text{ m}$ (height) - $g = 9.80 \text{ m/s}^2$ (standard gravity) 2. **Equation for acceleration of the system:** - $a = \frac{m_m - m_s}{m_m + m_s} g$ 3. **Equation for final velocity:** - $v_f^2 = v_i^2 + 2ah$ with $v_i = 0$. - $v_f = \sqrt{2ah}$ **Step-by-step detail for HTML output:** - **Calculations:** - Acceleration: $a = \frac{93 - 65}{93 + 65} \times 9.8$ - Numerator: $93 - 65 = 28$ - Denominator: $93 + 65 = 158$ - Acceleration value: $a = \frac{28}{158} \times 9.8 \approx 1.7367 \text{ m/s}^2$ - Final speed: $v = \sqrt{2 \times 1.7367 \times 10.0}$ - Inner product: $34.734$ - Final answer: $v \approx 5.89 \text{ m/s}$ **Check for sub-questions:** Problem 52 is just one main question: "With what speed does the man hit the ground...?" Actually, the prompt says "Solve ALL sub-questions (a), (b), (c) without omission". Problem 52 doesn't have explicitly labeled (a), (b), (c). I will present the solution clearly. If there are no sub-questions, I will treat the main goal as the primary task. Let's double-check the text of 52. "52 A 93 kg man lowers himself to the ground from a height of 10.0 m by holding onto a rope that runs over a frictionless pulley to a 65 kg sandbag. With what speed does the man hit the ground if he started from rest?" Correct, it's a single question. I'll fulfill the structure requirements. **Internal Thought Process Checklist:** III. Structure: 1. Summary: Man (93kg) and sandbag (65kg) connected over a pulley. Man falls 10m. Find final speed. 2. Target: Final velocity $v$. 3. Domain: Classical Mechanics - Newton's Second Law and Kinematics. 4. Assumptions: Frictionless pulley, massless rope, $g = 9.8 \text{ m/s}^2$. IV. Strategy: 1. Find system acceleration using Newton's 2nd law. 2. Use constant acceleration kinematic equation for speed. Variables: $m_1 = 93$, $m_2 = 65$, $h = 10$, $v_0 = 0$. Sign convention: Down is positive for man, up is positive for sandbag. Final check on HTML format: - `` - `Problem 52` - `` - `(a) Final Speed Calculation` (I'll label it (a) for the final speed to follow the "all sub-questions" instruction even if only one exists). - `● 계산 과정` (Wait, directive says "Your ENTIRE response MUST be written in ENGLISH only" but then shows an example with Korean bold labels in the prompt template? No, the Korean is in the *expert instructions* but the *output directive* says "Your ENTIRE response MUST be written in ENGLISH only." So I must translate "계산 과정" to "Calculation Process" and "최종 결론" to "Final Conclusion"). Actually, looking at the template: `(a) 하위 질문 제목` `● 계산 과정` I will use English equivalents: `(a) Determination of Final Speed` `● Calculation Process` `● Final Conclusion:` Let's do this. III. Problem Structuring 1. Problem Summary: A man with a mass of 93 kg is connected via a rope over a frictionless pulley to a 65 kg sandbag. He starts from rest at a height of 10.0 m and lowers himself to the ground. We need to find his impact speed. 2. Target: The final speed \(v\) of the man when he reaches the ground. 3. Physical Domain: Newtonian Mechanics - Newton's Second Law and Constant Acceleration Kinematics. 4. Assumptions: The pulley is frictionless and massless, the rope is massless and inextensible, and air resistance is neglected. Gravity is constant at \(g = 9.80 \text{ m/s}^2\). IV. Strategic Planning 1. Logical Steps: - Determine the net acceleration of the system using Newton's Second Law. - Since the acceleration is constant, use the kinematic equation relating velocity, acceleration, and displacement to find the final speed. 2. Variable Analysis: - Mass of the man (\(m_m\)): \(93 \text{ kg}\) - Mass of the sandbag (\(m_s\)): \(65 \text{ kg}\) - Vertical displacement (\(h\)): \(10.0 \text{ m}\) - Initial velocity (\(v_i\)): \(0 \text{ m/s}\) - Acceleration of gravity (\(g\)): \(9.80 \text{ m/s}^2\) 3. Newton's Second Law Setup: - For the man (taking downward as positive): \(m_m g - T = m_m a\) - For the sandbag (taking upward as positive): \(T - m_s g = m_s a\) - Adding these gives: \((m_m - m_s)g = (m_m + m_s)a \implies a = g \frac{m_m - m_s}{m_m + m_s}\) 4. Kinematics Setup: - Use \(v_f^2 = v_i^2 + 2ah\) to solve for \(v_f\). 52 A 93 kg man lowers himself to the ground from a height of 10.0 m by holding onto a rope that runs over a frictionless pulley to a 65 kg sandbag. With what speed does the man hit the ground if he started from rest?