Kinetic Energy

Definition: Kinetic Energy

The kinetic energy of a physical object is the work needed to bring the object from rest to its current state of motion.

NOTATION

$$E_{\text{kin}}{E}_{\text{k}}K$$

Theorem: Non-Relativistic Kinetic Energy of a Point Mass

The kinetic energy of a point mass $m$ moving with speed $v$ is given by the formula

$$E_{\text{kin}}=\frac{1}{2}m{v}^2$$

provided that $v$ is a lot smaller than the speed of light.

PROOF

At time $t_1$, the resultant force $\mathbf{F}$ begins to accelerates the point mass from 0 m/s and at time $t_2$ the speed reaches $v$ m/s, i.e. $||\mathbf{v}(t_1)||=0$ and $||\mathbf{v}(t_2)||=v$. During this time $\mathbf{F}$ does work

$$W={\int }_{t_1}^{t_2}\mathbf{F}\cdot \mathbf{v}\mathrm{dt}$$

According to Newton’s second laws of motion, $\mathbf{F}=m\mathbf{a}$.

$$W={\int }_{t_1}^{t_2}m\mathbf{a}\cdot \mathbf{v}\mathrm{dt}=m{\int }_{t_1}^{t_2}\mathbf{a}\cdot \mathbf{v}\mathrm{dt}$$

And since acceleration is the temporal derivative of velocity, we have

$$\frac{\mathrm{d}\mathbf{v}}{\mathrm{d}t}=\mathbf{a}$$ $$\mathbf{a}\mathrm{dt}=\mathrm{dv}$$ $$\begin{array}{rl}W=m{\int }_{t_1}^{t_2}\mathbf{a}\cdot \mathbf{v}\mathrm{dt}& =m{\int }_{t_1}^{t_2}(\mathbf{a}\mathrm{dt})\cdot \mathbf{v}\\ & =m{\int }_{t_1}^{t_2}\mathbf{v}\cdot \mathrm{dv}\end{array}$$

Let’s see what the temporal derivative of $\mathbf{v}\cdot \mathbf{v}$ is. Using the product rule, we obtain

$$\frac{\mathrm{d}}{\mathrm{d}t}(\mathbf{v}\cdot \mathbf{v})=\frac{\mathrm{d}\mathbf{v}}{\mathrm{d}t}\cdot \mathbf{v}+\mathbf{v}\cdot \frac{\mathrm{d}\mathbf{v}}{\mathrm{d}t}=2\left(\mathbf{v}\cdot \frac{\mathrm{d}\mathbf{v}}{\mathrm{d}t}\right)$$ $$\mathbf{v}\cdot \mathrm{dv}=\frac{1}{2}\mathrm{d}(\mathbf{v}\cdot \mathbf{v})$$

We just substitute this into the above formula for the work $W$.

$$W=m{\int }_{t_1}^{t_2}\mathbf{v}\cdot \mathrm{dv}=m{\int }_{t_1}^{t_2}\frac{1}{2}\mathrm{d}(\mathbf{v}\cdot \mathbf{v})=\frac{1}{2}m{\int }_{t_1}^{t_2}\mathrm{d}(\mathbf{v}\cdot \mathbf{v})$$

Solve the integral:

$$W=\frac{1}{2}m{\int }_{t_1}^{t_2}\mathrm{d}(\mathbf{v}\cdot \mathbf{v})=\frac{1}{2}m[\mathbf{v}(t_2)\cdot \mathbf{v}(t_2)-\mathbf{v}(t_1)\cdot \mathbf{v}(t_1)]$$

Using the properties of the dot product, we obtain

$$\mathbf{v}(t_2)\cdot \mathbf{v}(t_2)=||\mathbf{v}(t_2)|{|}^2=v^2$$ $$\mathbf{v}(t_1)\cdot \mathbf{v}(t_1)=||\mathbf{v}(t_1)|{|}^2=0^2$$

Thus

$$W=\frac{1}{2}m(v^2-0^2)=\frac{1}{2}m{v}^2$$