Problem 12 - University Physics with Modern Physics

Question

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. Determine, 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 can can can ignore air resistance and the mass of the vine.

Step-by-Step Solution

Answer

Tarzan's speed is approximately 4.9 m/s just before he reaches Jane.

Step 1: Identify the Problem

We need to determine Tarzan's speed just before he reaches Jane as he swings from one tree to another using a vine. His motion can be analyzed using the concept of energy conservation because air resistance and the vine's mass are negligible.

Step 2: Use Energy Conservation Principle

The principle of energy conservation states that the total mechanical energy (potential + kinetic) of the system remains constant if there are no non-conservative forces (e.g., friction) acting on it. Thus, the initial potential energy (PE) when the vine makes an angle of \( 45^\circ \) equals the kinetic energy (KE) plus the potential energy when it makes an angle of \( 30^\circ \).

Step 3: Initial Height Calculation

Tarzan's initial height (relative to the lowest point of the swing) can be calculated using his initial angle with the vertical. The vertical height can be determined by the formula: \[ h_i = L - L \cos \theta_1 \] where \( \theta_1 = 45^\circ \) and \( L = 20 \) m. Thus, \( h_i = 20 - 20 \cos 45^\circ = 20(1 - \frac{\sqrt{2}}{2}) \).

Step 4: Final Height Calculation

Similarly, the final height (when the vine makes an angle \( 30^\circ \) with the vertical) is given by: \[ h_f = L - L \cos \theta_2 \] where \( \theta_2 = 30^\circ \). Hence, \( h_f = 20 - 20 \cos 30^\circ = 20(1 - \frac{\sqrt{3}}{2}) \).

Step 5: Calculate Change in Potential Energy

The change in potential energy is \( PE = mgh \), so the difference in height from initial to final is needed. This change is \( \Delta h = h_i - h_f \), substituting the height values from previous steps gives: \[ \Delta h = 20(1 - \frac{\sqrt{2}}{2}) - 20(1 - \frac{\sqrt{3}}{2}) \].

Step 6: Apply Energy Conservation Equation

Using the energy conservation formula: \[ mgh_i = \frac{1}{2}mv^2 + mgh_f \], solve for the velocity \( v \) right before reaching Jane: \[ \frac{1}{2}mv^2 = mg(h_i - h_f) \], thus \[ v = \sqrt{2g(h_i - h_f)} \].

Step 7: Calculate Tarzan's Speed

Let \( g = 9.8 \text{ m/s}^2 \), and using \( h_i \) and \( h_f \) calculated earlier, \( v = \sqrt{2 \times 9.8 \times [(20 - 20\frac{\sqrt{2}}{2}) - (20 - 20\frac{\sqrt{3}}{2})] }\). Fill in the values to get Tarzan's speed before reaching Jane.

Key Concepts

Mechanical Energy

Mechanical energy in physics refers to the sum of potential energy (PE) and kinetic energy (KE) in a system. It's important to understand that mechanical energy is a critical concept in dynamics and describes the energy due to both position and motion.
When studying problems involving swing dynamics, like with Tarzan swinging on a vine, we often rely on the conservation of mechanical energy. This states that if no external forces like air resistance or friction do work, the total mechanical energy of a system remains constant.

In the case of Tarzan's swing, his mechanical energy at the starting point consists entirely of potential energy because he begins at rest. As Tarzan swings down, potential energy is converted into kinetic energy. Mechanical energy helps us predict his speed before reaching Jane.

Potential Energy

Potential energy is the energy an object has due to its position or configuration. In physics, potential energy is often related to the height of an object above a reference point. The higher Tarzan starts, the more potential energy he has.
In our scenario, Tarzan starts at a specific height when the vine makes a 45-degree angle with the vertical. We calculate this height to determine his initial potential energy.

• The initial height is given by the formula: \[ h_i = L - L \cos \theta_1 \] where the vine length \( L \) is 20 meters and the initial angle \( \theta_1 \) is 45 degrees.

As Tarzan swings and descends, his potential energy decreases as it is converted into kinetic energy.

Kinetic Energy

Kinetic energy is the energy of motion. When Tarzan swings down on the vine, his speed increases, and thus his kinetic energy increases. This energy conversion from potential to kinetic allows us to calculate his speed just before reaching Jane.
It's important to see how kinetic energy is determined mathematically by the formula:
\[ KE = \frac{1}{2}mv^2 \]
where \( m \) is mass and \( v \) is velocity.
This formula shows that kinetic energy depends on the square of Tarzan's speed. As he approaches Jane, his increase in kinetic energy comes from the loss of potential energy during his descent.

Swing Dynamics

Swing dynamics is a fascinating area in physics that involves motion, gravity, and energy transformation. When we talk about Tarzan swinging on a vine, particularly focusing on angles and speeds, we have a perfect example of swing dynamics.
As Tarzan swings towards Jane, several forces act upon him:

• Gravity pulls him downwards, which helps convert potential energy into kinetic energy.
• The tension in the vine prevents him from falling straight down, directing his motion along an arc.

Understanding these forces and energy conversions allows us to solve for his speed accurately by applying conservation of energy principles.
This motion can be complex but analyzing it using physics makes it simpler.

Problem-Solving in Physics

Problem-solving in physics often involves breaking down complex scenarios into manageable parts. Analyzing energy transformations, like in the Tarzan swing problem, is an excellent example of practical physics problem-solving.
When tackling these problems, it's essential to:

• Identify relevant principles, such as energy conservation.
• Calculate required starting conditions, like initial height for potential energy.
• Use formulas correctly to solve for desired quantities, like speed.

By following these steps, we can solve for Tarzan’s speed before he reaches Jane effectively. Practicing these strategies improves our ability to approach and solve various real-world physics problems.