Question

Tarzan, in one tree, sights Jane in another tree. He grabs the end of a vine with length 2020 m that makes an angle of 45°45°  with the vertical, steps off his tree limb, and swings down and then up to Jane's open arms. When he arrives, his vine makes an angle of 30°30° with the vertical. Determine whether he gives her a tender embrace or knocks her off her limb by calculating Tarzan's speed just before he reaches Jane. Ignore air resistance and the mass of the vine.

Accepted Answer

Step 1: Identify the energy conservation principle. Since air resistance and the mass of the vine are ignored, we can use the conservation of mechanical energy. Tarzan's potential energy at the starting point is converted into kinetic energy and potential energy at the final point. Step 2: Write the expression for Tarzan's potential energy at the starting point. The height difference between Tarzan's initial position and the lowest point of the swing can be calculated using the vine length and the angle with the vertical. Use the formula for height: $$ h = L(1 - \cos	heta) $$, where $$ L  is $$the vine length and $$ 	heta  is $$the angle with the vertical. Step 3: Write the expression for Tarzan's potential energy at the final point. Similarly, calculate the height difference between Tarzan's final position and the lowest point of the swing using $$ h = L(1 - \cos	heta) $$, where $$ 	heta  is $$the final angle with the vertical. Step 4: Apply the conservation of mechanical energy. The total mechanical energy at the starting point (potential energy) is equal to the total mechanical energy at the final point (kinetic energy + potential energy). Use the equation: $$ m g h_{initial} = \frac{1}{2} m v^2 + m g h_{final} $$, where $$ m  is $$Tarzan's mass, $$ g  is $$the acceleration due to gravity, $$ h_{initial}  is $$the initial height, $$ h_{final}  is $$the final height, and $$ v  is $$Tarzan's speed just before reaching Jane. Step 5: Solve for Tarzan's speed $$ v . $$Rearrange the equation to isolate $$ v $$: $$ v = \sqrt{2 g (h_{initial} - h_{final})} . $$Substitute the values for $$ g $$, $$ h_{initial} $$, and $$ h_{final}  to $$calculate Tarzan's speed. Ensure all units are consistent.

비슷한 문제에 대한 검증된 영상 답변:

주요 개념

Conservation of Energy

Kinematics of Circular Motion

Forces and Acceleration