Problem 13 - Applied Physics

Question

A 250 -kg part falls from a plane and hits the ground at \(150 \mathrm{~km} / \mathrm{h}\). Find its kinetic energy.

Step-by-Step Solution

Answer

The kinetic energy of the part is 217,013.875 Joules.

Step 1: Convert Velocity

First, we need to convert the velocity from kilometers per hour to meters per second to use the kinetic energy formula. We know that 1 km/h = 0.27778 m/s. Thus, the velocity in meters per second is: \[ 150 \, \text{km/h} \times 0.27778 \, \text{m/s per km/h} = 41.667 \, \text{m/s} \]

Step 2: Kinetic Energy Formula

The formula for kinetic energy (KE) is: \[ KE = \frac{1}{2} m v^2 \] where \( m \) is the mass in kilograms and \( v \) is the velocity in meters per second.

Step 3: Substitute the Values

Substitute the known values into the kinetic energy formula: Mass \( m = 250 \, \text{kg} \) and velocity \( v = 41.667 \, \text{m/s} \).\[ KE = \frac{1}{2} \times 250 \, \text{kg} \times (41.667 \, \text{m/s})^2 \]

Step 4: Calculate Kinetic Energy

Calculate the result of the expression:\[ KE = \frac{1}{2} \times 250 \times 1736.11 = 217013.875 \, \text{Joules} \]So the kinetic energy is 217,013.875 Joules.

Key Concepts

Velocity Conversion

Velocity conversion is a crucial step in tackling physics problems, especially when dealing with different units. To accurately calculate kinetic energy, we need the velocity in meters per second (m/s) rather than kilometers per hour (km/h). This is because the standard unit of speed in the International System of Units (SI) is m/s.

• To convert km/h to m/s, multiply the velocity by a factor of 0.27778. This conversion factor stems from the relationship between kilometers and meters, as well as hours and seconds.
• For example, the velocity given is 150 km/h. By multiplying 150 by 0.27778, we achieve a converted velocity of approximately 41.667 m/s. This ensures the velocity aligns with other SI units in the kinetic energy formula.

Converting units is an essential skill in physics that allows you to solve problems correctly and consistently.

Kinetic Energy Formula

The kinetic energy formula is a fundamental equation used to calculate the energy an object has due to its motion. This formula is given by: \[ KE = \frac{1}{2} m v^2 \] where:

• \( KE \) is the kinetic energy
• \( m \) is the mass of the object, in kilograms
• \( v \) is the velocity of the object, in meters per second

The formula indicates that kinetic energy relates to both mass and the square of the velocity.

• This means that even small changes in velocity can lead to significant changes in kinetic energy, emphasizing the velocity's quadratic impact.
• The factor \( \frac{1}{2} \) in the formula accounts for the uniform distribution of energy during acceleration.

By understanding this formula, one can appreciate how both mass and velocity confer kinetic energy.

Physics Problem Solving

Physics problem solving often involves breaking down a problem into understandable parts and addressing each step systematically. This exercise demonstrates that:

• Defined steps, like converting velocity and utilizing the kinetic energy formula, guide the process.
• Identifying the correct units and formulas is key to solving problems accurately.
• Analyzing the components given, such as mass and velocity, ensures correct input into equations.

Each step builds on the previous one, leading to a final answer. Practicing these structured approaches enhances your problem-solving skills in physics, boosting confidence and accuracy.
A strategic problem-solving approach can turn complex physics problems into manageable tasks, ultimately making the subject more accessible.

Unit Conversion in Physics

Unit conversion is a frequent necessity in physics problems due to the diverse units used in different contexts. Recognizing the need to convert units allows for accurate calculations and results. In physics, common conversions include:

• Kilometers to meters, to align with the SI system
• Hours to seconds, ensuring time calculations adhere to standard units

Converting unit measurements bridges the gap between everyday language and scientific language.

• A misunderstanding in unit conversion can have significant impacts on the outcomes of a calculation, leading to incorrect answers.
• Promptly identifying when and how to convert units ensures fluid problem-solving and avoids errors.

Developing a habit of checking units can greatly enhance one’s ability to handle challenging physics problems with confidence.