Problem 12 - Physics for Scientists and Engineers with Modern Physics

Question

(I) Jane, looking for Tarzan, is running at top speed \((5.0 \mathrm{~m} / \mathrm{s})\) and grabs a vine hanging vertically from a tall tree in the jungle. How high can she swing upward? Does the length of the vine affect your answer?

Step-by-Step Solution

Answer

Jane can swing upward to a height of 1.275 meters. The vine length does not affect the height.

Step 1: Understand the Energy Conservation Principle

In this problem, we can use the conservation of mechanical energy. When Jane grabs the vine, her kinetic energy is converted into potential energy as she swings upwards and comes to a momentary stop at her highest point.

Step 2: Write Down the Kinetic Energy Equation

The initial kinetic energy (KE) when Jane is running is given by the equation \( KE = \frac{1}{2} m v^2 \), where \( m \) is her mass and \( v = 5.0 \mathrm{~m/s} \) is her speed. We do not need to know \( m \) because it cancels out later.

Step 3: Write Down the Potential Energy Equation

At the highest point in her swing, all kinetic energy is converted to gravitational potential energy \( PE = mgh \), where \( h \) is the height and \( g = 9.8 \mathrm{~m/s^2} \) is the acceleration due to gravity.

Step 4: Set the Kinetic and Potential Energy Equations Equal

Since energy is conserved, set the initial kinetic energy equal to the potential energy at the highest point: \( \frac{1}{2} m v^2 = mgh \).

Step 5: Solve for Height \( h \)

Cancel the mass \( m \) from both sides in \( \frac{1}{2} v^2 = gh \). Solve for \( h \) by dividing both sides by \( g \): \[ h = \frac{v^2}{2g} = \frac{(5.0)^2}{2 \times 9.8} \approx 1.275 \mathrm{~m} \].

Step 6: Determine the Effect of Vine Length

The length of the vine does not affect the height she can swing upwards. The height is purely dependent on her initial speed and the acceleration due to gravity.

Key Concepts

Kinetic Energy

Kinetic energy is the energy possessed by an object due to its motion. In physics, it's a fundamental type of mechanical energy. The formula to calculate kinetic energy is:\[ KE = \frac{1}{2} m v^2 \]Here, \( m \) represents mass and \( v \) represents velocity. Even if the mass seems like a missing part of our problem, it effectively cancels out as you progress through the solution steps, so you may not need its exact value. This allows us to focus primarily on the velocity to understand kinetic energy.Kinetic energy is proportional to the square of the velocity. This means if you double the speed of an object, its kinetic energy increases fourfold. Thus, the speed is a crucial factor when calculating kinetic energy. Jane, in our scenario, reaching a speed of \( 5.0 \, \mathrm{m/s} \), gains enough kinetic energy to subsequently convert it into potential energy as she swings upwards using the vine.

Potential Energy

Potential energy is primarily the energy an object possesses due to its position relative to other objects. Gravitational potential energy is one of the common types, and it applies in scenarios where height and gravity come into play, like Jane swinging on a vine.To calculate potential energy, use the formula:\[ PE = mgh \]where \( m \) is the mass, \( g \) is the acceleration due to gravity \( (9.8 \, \mathrm{m/s^2}) \), and \( h \) is the height above the reference point. When Jane swings upward, the kinetic energy gets transferred into potential energy, reaching her highest point where all kinetic energy becomes potential energy.At this point, her speed momentarily becomes zero, meaning all her energy is stored due to her elevated position. For our problem, solving for height \( h \) using the principle of energy conservation reveals how high she can swing without needing the vine length.

Energy Conversion

Energy conversion refers to the process where energy changes from one form to another. In Jane's scenario, it involves the exchange between kinetic energy and potential energy. As Jane grabs the vine and swings upwards, the kinetic energy she has while running is gradually converted into potential energy. This conversion is best exemplified at her highest swing point—where all her kinetic energy has transformed into potential energy. The entire process perfectly illustrates the conservation of mechanical energy, where the total energy in the system remains constant, just switching forms. The kinetic energy she initially possesses because of her running transfers into potential energy due to her increased height. This exchange showcases the beauty and simplicity of mechanical energy transformations.

Physics Problem Solving

Solving physics problems like Jane swinging on a vine requires a good understanding of key principles and clear logic. Here are some steps to make problems like these simpler:

• Start by identifying the type of problem you're solving, which in this case is energy conservation.
• Clearly write down known variables and assign equations accordingly, such as kinetic and potential energy formulas.
• Look for quantities that might cancel out, such as mass in both sides of the equation when dealing with energy equations.
• Equalize the energy equations directly to find unknowns, using algebraic manipulation when necessary.
• Conclude by interpreting results contextually. For Jane's swing, the length of the vine doesn't change the outcome as it primarily depends on her speed and gravitational pull.

Each step in physics is crucial, shedding light on why it's important to understand the underlying concepts rather than just crunching numbers. This fundamental understanding helps tackle a variety of problems effectively.