Kinetic Energy

• The kinetic energy of an object is defined as its energy of motion.
• If an object with a mass $m$ moves with a speed $v$, the kinetic energy can be calculated according to

$$E_\mathrm{kin} = \frac{1}{2}mv^2$$ Kinetic energy is defined as the ability of an object to carry out work due to its mechanical motion. Every moving object, such as a falling stone, a driving car, or a planet orbiting around a star, has a certain amount of kinetic energy. This form of energy is, therefore, called energy of motion. Kinetic energy is stored in a moving object and can be converted into other energy forms like heat through friction or transferred to other objects during collisions. In the first step, we want to derive a formula for the kinetic energy for a constant acceleration $a$. In this case, the energy can simply be calculated according to $$E = Fs$$ where $F$ is the driving force and $s$ is the distance. In the next step, we can replace the force with Newton's second law $F=ma$ which leads to $$E = mas$$ For constant acceleration, we can use the previously derived quadratic relation between the distance and time which is given as $s = \frac{1}{2}at^2$. The result is then given as $$E = \frac{1}{2}ma^2t^2$$ If the body is accelerated with $a$, it reaches the speed $v$ after the time $t = v/a$. After inserting this relation into the equation above, the acceleration cancels out and the following equation remains: $$\boxed{E = \frac{1}{2}mv^2}$$ This equation is the famous formula for calculating the kinetic energy of an object after a certain acceleration takes place. Now we want to show that we can obtain the same formula when we drop the condition of a constant acceleration, but rather consider any arbitrary trajectory. If an object undergoes an acceleration $\vec{a}$, the definition of the energy can be rewritten as follows: $$E = \int_0^\vec{r} \vec{F}\cdot\mathrm{d}\vec{r} = \int_0^\vec{r} m\vec{a}\cdot\mathrm{d}\vec{r}$$ In the next step, the acceleration $\vec{a}$ can be replaced with its definition $\mathrm{d}\vec{v}/\mathrm{d}t$ which leads to the following equation: $$E = m\int_0^t \frac{\mathrm{d}\vec{v}}{\mathrm{d}t}\cdot\mathrm{d}\vec{r}$$ The mass $m$ is a constant and can therefore be drawn in front of the integral. If the derivation of the velocity with respect to time is treated as an ordinary fraction, the denominator $\mathrm{d}t$ can be shifted to the right. Together with the replacement $\vec{v} = \mathrm{d}\vec{r}/\mathrm{d}t$, the integrand becomes $$E = \int_0^\vec{v} m \vec{v}\cdot\mathrm{d}\vec{v}$$ Now, the product rule can be applied. Therefore, only the absolute value of the velocity remains that can easily be integrated to $$\boxed{E_\mathrm{kin} = \frac{1}{2}mv^2}$$ This formula describes the energy of motion of a body that has been accelerated speed until it reaches the velocity $v$ and is therefore called kinetic energy. This formula can be applied for a speed $v$ which is much smaller than the speed of light. If $v$ gets too large (around 10% of c), then the approximation gets worse and formulas from the theory of relativity have to be used. This subject will be discussed at a later stage.