Problem 49 - College Physics

Question

\(\bullet\) Tarzan and Jane. Tarzan, in one tree, sights Jane in another tree. He grabs the end of a vine with length 20 \(\mathrm{m}\) that makes an angle of \(45^{\circ}\) 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 \(30^{\circ}\) with the vertical. Deter- mine whether he gives her a tender embrace or knocks her off her limb by calculating Tarzan's speed just before he reaches Jane. You can ignore air resistance and the mass of the vine.

Step-by-Step Solution

Answer

Tarzan's speed is approximately 7.9 m/s, indicating he may knock Jane off her limb.

Step 1: Identify Variables

We have a vine with length \( L = 20 \, \mathrm{m} \). Initially, the vine makes an angle of \( \theta_1 = 45^{\circ} \) with the vertical, and at the final point, it makes \( \theta_2 = 30^{\circ} \). Tarzan is moving under gravity, ignoring other resistive forces.

Step 2: Understand Potential Energy Changes

At the initial position, the height \( h_1 \) above the lowest point can be calculated using trigonometry: \[h_1 = L(1 - \cos \theta_1) = 20(1 - \cos 45^{\circ}).\]Similarly, the height \( h_2 \) at the final position is: \[h_2 = L(1 - \cos \theta_2) = 20(1 - \cos 30^{\circ}).\]

Step 3: Calculate Initial and Final Heights

Substituting the cosine values, calculate:\[h_1 = 20(1 - \frac{\sqrt{2}}{2}) = 20(1 - 0.7071) \approx 5.86 \, \mathrm{m},\]\[h_2 = 20(1 - \frac{\sqrt{3}}{2}) = 20(1 - 0.8660) \approx 2.68 \, \mathrm{m}.\]

Step 4: Apply Conservation of Energy

Since there are no non-conservative forces doing work, mechanical energy is conserved: \[mgh_1 = \frac{1}{2}mv^2 + mgh_2.\]Canceling the mass \( m \) from both sides and solving for the speed \( v \) of Tarzan just before he reaches Jane, we have:\[gh_1 = \frac{1}{2}v^2 + gh_2,\]so,\[v^2 = 2g(h_1 - h_2).\]

Step 5: Calculate Tarzan's Speed

Using the heights calculated:\[v^2 = 2 \times 9.81 \, \mathrm{m/s^2} \times (5.86 - 2.68) \, \mathrm{m},\]\[v^2 = 2 \times 9.81 \times 3.18 \approx 62.3844,\]\[v = \sqrt{62.3844} \approx 7.9 \, \mathrm{m/s}.\]

Step 6: Conclusion on Tarzan's Action

Tarzan's speed just before he reaches Jane is \( 7.9 \, \mathrm{m/s} \). This is quite fast, suggesting that Tarzan will likely knock Jane off her limb rather than give her a tender embrace.

Key Concepts

Kinematics

Kinematics is the branch of physics that deals with the motion of objects without considering the forces that cause the motion. In our scenario with Tarzan swinging on a vine, kinematics helps us understand his path and the velocities involved. Tarzan swings like a pendulum, beginning at an angle and descending along the arc of a circle.
Key aspects of kinematics that are relevant include:

• The description of Tarzan's motion along the vine as he moves from an initial height to a lower height.
• The calculation of his speed as he reaches Jane, applying formulae that relate path traveled and change in heights.

These fundamental points allow us to calculate and predict his speed just before reaching Jane, which is crucial for determining whether his arrival will be gentle or forceful. Understanding kinematics, therefore, forms a significant base for analyzing the problem at hand.

Potential Energy

Potential energy is energy stored due to an object's position or height. In the Tarzan exercise, potential energy plays a central role in calculating how Tarzan’s energy transforms throughout his swing.
Initial potential energy is calculated when Tarzan is at the top of his swing. This energy is greatest when he hangs at an angle with the vertical:

• Initial height,\(h_1 = L(1 - \cos \theta_1)\)

At the final point before reaching Jane:

• Potential energy is less as he lowers his height,\(h_2 = L(1 - \cos \theta_2)\)

As Tarzan swings downward, gravitational potential energy converts into kinetic energy. This conservation of energy allows us to use the difference in heights to calculate his speed. Mastering potential energy concepts helps us understand how initial potential transforms to speed which determines his speed.

Trigonometry

Trigonometry involves the study of angles and the relationships between side lengths in triangles. It is crucial for handling scenarios like Tarzan's swing, where understanding angular displacement and its impact on height is essential.
In this exercise, we apply trigonometry to determine the changes in height by considering the angles formed by the vine with the vertical:

• Using\( \cos \theta \), we calculate the vertical distance below the initial branch height.

This trigonometric application allows us to assess the initial versus final heights (

h_1

and

h_2

), giving a clearer picture of energy changes.

Understanding trigonometry helps to resolve problems involving angle variations, such as determining accurate positions and predicting movements based on angles.