## What is Kinetic Energy

**The kinetic energy, K**, is defined as the energy stored in an object because of its motion. An object in motion has the ability to do work and thus can be said to have energy. It is called kinetic energy, from the Greek word kinetikos, meaning “motion.”

The **kinetic energy** depends on the speed of an object and is the ability of a moving object to do work on other objects when it collides with them. On the other hand, the kinetic energy of an object represents the amount of energy required to increase the velocity of the object from rest (v = 0) to its final velocity. The kinetic energy also depends linearly on the mass, which is a numerical measure of object’s **inertia **and the measure of an object’s resistance to acceleration when a force is applied.

We define the quantity:

**K = ½ mv ^{2}**

to be the **translational kinetic energy** of the object. It must be added, it is called the “translational” kinetic energy to distinguish it from rotational kinetic energy.

## Examples of Kinetic Energy

## Block sliding down a frictionless incline slope

The 1 kg block starts out a height H (let say 1 m) above the ground, with **potential energy** **mgH** and** kinetic energy** that is equal to 0. It slides to the ground (without friction) and arrives with no potential energy and kinetic energy **K = ½ mv ^{2}**. Calculate the velocity of the block on the ground and its kinetic energy.

*E*

_{mech}

*= U + K = const**=> ½ mv*^{2}* = mgH*

*=> v = √2gH = 4.43 m/s*

*=> K*_{2}* = ½ x 1 kg x (4.43 m/s)*^{2}* = 19.62 kg.m*^{2}*.s*^{-2}* = 19.62 J*

## Pendulum

Assume a **pendulum** (ball of mass m suspended on a string of length **L** that we have pulled up so that the ball is a height **H < L** above its lowest point on the arc of its stretched string motion. The pendulum is subjected to the **conservative gravitational force** where frictional forces like air drag and friction at the pivot are negligible.

We release it from rest. **How fast is it going at the bottom?**

The pendulum reaches **greatest kinetic energy** and **least potential energy** when in the **vertical position**, because it will have the greatest speed and be nearest the Earth at this point. On the other hand, it will have its **least kinetic energy** and **greatest potential energy** at the **extreme positions** of its swing, because it has zero speed and is farthest from Earth at these points.

If the amplitude is limited to small swings, the period *T* of a simple pendulum, the time taken for a complete cycle, is:

where ** L** is the length of the pendulum and

**is the local acceleration of gravity. For small swings the period of swing is approximately the same for different size swings. That is,**

*g***the period is independent of amplitude**.

## Example: Proton’s kinetic energy

A proton (**m = 1.67 x 10 ^{-27} kg**) travels at a speed

**v = 0.9900c = 2.968 x 10**. What is its

^{8}m/s**kinetic energy**?

According to a classical calculation, which is not correct, we would obtain:

K = 1/2mv^{2} = ½ x (1.67 x 10^{-27} kg) x (2.968 x 10^{8}m/s)^{2} = **7.355 x 10 ^{-11} J**

With relativistic correction the **relativistic kinetic energy** is equal to:

**K = (ɣ – 1)mc ^{2}**

where the Lorentz factor

ɣ = 7.089

therefore

K = 6.089 x (1.67 x 10^{-27} kg) x (2.9979 x 10^{8}m/s)^{2} = **9.139 x 10 ^{-10} J = 5.701 GeV**

This is about** 12 times higher** energy as in the classical calculation. According to this relationship, an acceleration of a proton beam to 5.7 GeV requires energies that are in the order different.

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