Kinetic energy can be defined as (1/2)mv^2 or as (3/2)PV. Show that the derived SI units of each of these terms are those of energy. (Pressure is force/area and force is mass * acceleration.)

Verified step by step guidance

1. Start by analyzing the first expression for kinetic energy: \( \frac{1}{2}mv^2 \). Here, \( m \) is mass with SI units of kilograms (kg), and \( v \) is velocity with SI units of meters per second (m/s).
2. Calculate the units for \( v^2 \): since velocity \( v \) is in m/s, \( v^2 \) will be in \((\text{m/s})^2 = \text{m}^2/\text{s}^2\).
3. Combine the units for mass and velocity squared: \( \text{kg} \times \text{m}^2/\text{s}^2 = \text{kg} \cdot \text{m}^2/\text{s}^2 \), which are the units of energy, known as a joule (J).
4. Now, consider the second expression for kinetic energy: \( \frac{3}{2}PV \). Here, \( P \) is pressure with units of force/area, and \( V \) is volume with units of cubic meters (m^3).
5. Express pressure \( P \) in terms of its fundamental units: force is mass times acceleration (\( \text{kg} \cdot \text{m/s}^2 \)), and area is \( \text{m}^2 \), so pressure \( P \) is \( \text{kg} \cdot \text{m/s}^2/\text{m}^2 = \text{kg}/(\text{m} \cdot \text{s}^2) \). Multiply by volume \( V \) to get \( \text{kg} \cdot \text{m}^2/\text{s}^2 \), which are again the units of energy, a joule (J).

Key Concepts

Kinetic Energy Formula

Kinetic energy (KE) is defined as the energy an object possesses due to its motion, mathematically expressed as KE = (1/2)mv^2, where m is mass and v is velocity. This formula indicates that kinetic energy is directly proportional to the mass of the object and the square of its velocity, highlighting how changes in either variable significantly affect the energy.

Pressure and Its Units

Pressure (P) is defined as force (F) applied per unit area (A), expressed as P = F/A. The SI unit of pressure is the pascal (Pa), which is equivalent to one newton per square meter (N/m²). Understanding pressure is crucial for deriving energy expressions, as it relates to the force exerted by gas molecules in a given volume.

Dimensional Analysis

Dimensional analysis is a mathematical technique used to convert units and verify the consistency of equations by ensuring that both sides have the same dimensions. In the context of energy, it involves checking that the derived units from different expressions (like (1/2)mv^2 and (3/2)PV) yield the same unit of energy, which is the joule (J) in the SI system.