📂Classical Mechanics

Kinetic and Potential Energy Definitions in Physics

Introduction¹

When a force acts on a body, the body’s state of motion changes. The cumulative effect of force during the body’s displacement is called work.

$$ W_{\mathbf{a}\mathbf{b}} = \int_{\mathbf{a}}^{\mathbf{b}} \mathbf{F} \cdot \mathrm{d}\mathbf{r} $$

Using Newton’s second law and the chain rule, the equation can be expanded as follows.

$$ \begin{align*} W_{\mathbf{a}\mathbf{b}} &= \int_{\mathbf{a}}^{\mathbf{b}} \mathbf{F} \cdot \mathrm{d}\mathbf{r} \\ &= \int m \mathbf{a} \cdot \dfrac{\mathrm{d}\mathbf{r}}{\mathrm{d}t} \mathrm{d}t \\ &= m \int \dfrac{\mathrm{d}\mathbf{v}}{\mathrm{d}t} \cdot \mathbf{v} \mathrm{d}t \\ &\overset{4\text{th}}{=} \dfrac{m}{2} \int \dfrac{\mathrm{d}(\mathbf{v} \cdot \mathbf{v})}{\mathrm{d}t} \mathrm{d}t = \dfrac{m}{2} \int \mathrm{d} v^{2} \\[1em] &= \dfrac{m}{2} \left[ v^{2} \right]_{\mathbf{a}}^{\mathbf{b}} \\[1em] &= \dfrac{m}{2} \left( v_{\mathbf{b}}^{2} - v_{\mathbf{a}}^{2} \right) \end{align*} $$

The fourth equality holds because $\dfrac{\mathrm{d}(\mathbf{v} \cdot \mathbf{v})}{\mathrm{d}t} = 2\mathbf{v} \cdot \dfrac{\mathrm{d}\mathbf{v}}{\mathrm{d}t}$. $v = \sqrt{\mathbf{v} \cdot \mathbf{v}}$ is the speed of the object. Let us look at the obtained expression again.

$$ W_{\mathbf{a}\mathbf{b}} = \dfrac{1}{2} m v_{\mathbf{b}}^{2} - \dfrac{1}{2} m v_{\mathbf{a}}^{2} \tag{1} $$

The work done in moving the object from point $\mathbf{a}$ to point $\mathbf{b}$ is expressed in terms of the states of motion at those points, $\mathbf{v}_{\mathbf{a}}$ and $\mathbf{v}_{\mathbf{b}}$. Therefore, we define this physical quantity as follows.

Definition

For a body undergoing translational motion at speed $\mathbf{v}$, the kinetic energy of the body is defined as follows.

$$ T = \dfrac{1}{2} m v^{2}, \quad v^{2} = \mathbf{v} \cdot \mathbf{v} $$

If the momentum is written in the form $\mathbf{p} = m \mathbf{v}$,

$$ T = \dfrac{p^{2}}{2m} , \quad p^{2} = \mathbf{p} \cdot \mathbf{p} $$

Explanation

The symbol for kinetic energy is often taken from the first letter of “kinetic” and is written as $K$ or $E_{K}$. From $(1)$, one can see that kinetic energy is the amount of work required to accelerate a body at rest to that speed. Also, from the definition of kinetic energy we obtain the fact that the work done on a body equals the change in its kinetic energy.

Meanwhile, from the equations developed in the Introduction, the following holds. It is also the differential form of the work–energy theorem.

$$ \mathrm{d}T = \mathbf{F} \cdot \mathrm{d}\mathbf{r} \quad \text{or} \quad \dfrac{\mathrm{d}T}{\mathrm{d}t} = \mathbf{F} \cdot \mathbf{v} $$