# Kinetic Energy

## Kinetic energy of a rigid body is the energy possessed by its motion

Work must be done to set any object in motion, and any moving object can do work. Energy is the ability to do work and kinetic energy is the energy of motion. There are several forms of kinetic energy

- vibration - the energy due to vibration motion
- rotational - the energy due to rotational motion
- translational - the energy due to motion from one location to another

Energy has the same units as work and work is *force times distance*. *One Joule* is *one Newton* of force acting through *one meter* - *Nm* or *Joule* in SI-units. The Imperial units are *foot-pound*.

*1 ft lb = 1.356 N m (Joule)*

### Translational Kinetic Energy

Translational kinetic energy can be expressed as

E_{t}= 1/2 m v^{2 }(1)

where

E_{t}= kinetic translation energy (Joule, ft lb)

m = mass (kg, slugs)

v = velocity (m/s, ft/s)

*one slug = 32.1740 pounds (as mass) - lb*_{m}

### Rotational Kinetic Energy

Rotational kinetic energy can be expressed as

E_{r}= 1/2 I ω^{2 }(2)

where

E_{m}= kinetic rotation energy (Joule, ft lb)

I = moment of inertia - an object's resistance to changes in rotation direction (kg m^{2}, slug ft^{2})

ω = angular velocity (rad/s)

### Example - Kinetic Energy in a Car

The kinetic energy of a car with mass of *1000 kg* at speed *70 km/h* can be calculated as

E_{t}= 1/2 (1000 kg) ((70 km/h) (1000 m/km) / (3600 s/h))^{2}

= 189043 Joule

The kinetic energy of the same car at speed *90 km/h* can be expressed as

E_{t}= 1/2 (1000 kg) ((90 km/h) (1000 m/km) / (3600 s/h))^{2}

= 312500 Joule

**Note!** - when the speed of a car is increased with *28%* (from* 70* to *90 km/h*) - the kinetic energy of the car is increased with *65% *(from *189043* to *312500 J*). This **huge rise** in kinetic energy must be absorbed by the safety construction of the car to provide the same protection in a crash - which is very hard to achieve. In a modern car it is possible to survive a crash at *70 km/h*. A crash at *90 km/h* is more likely fatal.

### Example - Kinetic Energy in a Steel Cube moving on a Conveyor Belt

A steel cube with weight *500 lb _{}* is moved on a conveyor belt with a speed of

*9 ft/s*. The steel cube mass can be calculated as

*m = (500 lb _{}) / (32.1740 ft/s^{2})*

* = 15.54 slugs *

The kinetic energy of the steel cube can be calculated as

E_{t}= 1/2 (15.54 slugs) (9 ft/s)^{2}

= 629 ft lbs

### Example - Kinetic Energy in a Flywheel

A flywheel with Moment of Inertia *I = 0.15 kg m ^{2} *is rotating with

*1000 rpm (revolutions/min)*. The angular velocity can be calculated as

*ω = (1000 revolutions/min) (0.01667 min/s) (2 π rad/revolution)*

* = 104 rad/s*

The flywheel kinetic energy can be calculated

*E _{r} = 1/2 (0.15 kg m^{2}) (104 rad/s)^{2 }*

* = 821 J*