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# Kinetic energy

## Kinetic energy definition

Kinetic energy $(E_{\text{k}})$ is the energy that an object possesses due to its movement (its speed) and its mass.

Doing work on an object can give it kinetic energy.

To accelerate a stationary object so that it has $100 \text{ J}$ of kinetic energy, you need to do $100 \text{ J}$ of work.

Objects that are heavier or moving faster are harder to stop. This means they have more kinetic energy.

A car travelling at $30 \ukmph$ has less kinetic energy than a car travelling at $60 \ukmph.$

An object's kinetic energy can be calculated using the formula: $$\Torange{\text{kinetic energy}}=\frac{1}{2}\times\Tblue{\text{mass}} \times \Tred{\text{speed}^{2}}$$

Kinetic energy is also often abbreviated as $KE$ or $T$.

A car of mass of $1000 \text{ kg}$ travelling at a speed of $20 \umps$ has a kinetic energy of $200,000 \text{ J}.$

## Derivation of the kinetic energy formula

The formula for kinetic energy can be derived by first finding the displacement of the object in terms of the magnitude of acceleration $a$ when it is accelerated by a force of magnitude $F$ from rest to a speed $v$: \begin{align*} v^{2}&=u^{2}+2as\quad\quad u=0\\ \Rightarrow s&=\frac{1}{2}\left(\frac{v^{2}}{a}\right) \end{align*} We then find the work done by the force in accelerating the body from rest to that speed (which is equivalent to the kinetic energy possessed by the body): \begin{align*} W&=Fs\\ &=ma\cdot s\\ &=ma\times\frac{1}{2}\left(\frac{v^{2}}{a}\right)\\ &=\frac{1}{2}mv^{2}\\ W &=E_{k} \end{align*}