Kinetic energy can be defined as (1/2)mv^2 or as (3/2)PV. Show th...

Question

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.)

Accepted Answer

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).. Calculate the units for $ v^2 $: since velocity $ v $ is in m/s, $ v^2 $ will be in $(\text{m/s})^2 = \text{m}^2/\text{s}^2$.. 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).. 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).. 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).