# Kinetic energy

## Explanation

Kinetic energy is the energy possessed by a system due to its state of being in a continuous motion. A good example of explaining this would be using a bow and arrow. When the bowstring is stressed back along with the arrow, the bowstring would have a potential energy stored in it in the form of elastic potential energy. When the archer releases the bowstring, the potential energy is converted to the kinetic energy to the arrow released, which accelerates to its maximum velocity. Kinetic energy is measured in SI units as joule, J. 1 J is 1 kg m2 s−2.

### How to calculate the kinetic energy of an object?

The kinetic energy can be simplified as the work required to be done on an object like an arrow to accelerate to its maximum speed, or to bring that same arrow from its maximum speed to the state of rest. So for a point mass, the kinetic energy can be calculated as the work.

Work done = Force × Distance

Kinetic Energy = ma × vavgt

Where m is the mass, a is the acceleration, vavg is the average velocity of the final and initial velocity ((vfinal - vinitial)/2) of the object and t is the time taken for the acceleration or deceleration. Therefore, the kinetic energy of a point mass can be calculated as below.

 $KE_\text{translational}=\tfrac{1}{2} mv^2$

### How to calculate the kinetic energy of a rotating object?

The kinetic energy of a rotating body is similar to its linear motion part, except that it uses the rotational counterparts like angular velocity ω, torque τ and moment of inertia I. The rotational kinetic energy of an object can be written as follows.

 $KE_\text{rotational}=\tfrac{1}{2} I\omega^2$

### How is the equation of rotational kinetic energy obtained?

Rotational kinetic energy is simply the work done on an object by exerting a torque τ, which makes it rotate at a certain acceleration to attain the maximum angular velocity ω about an angle θ. Following the same derivation, as seen above for linear translational kinetic energy, an equation can be obtained by substituting for rotational motion.

Rotational work done = Torque × Angle

Applying Newton’s second law to the rotational system,

Kinetic Energy = Ia × θ

Where I is the moment of inertia and a is the angular acceleration. As angular acceleration is rate of change of angular velocity ω over time t and angular velocity is simply the rate of change of the rotational angle θ over the same time t, we can rewrite the equation as below.

Kinetic Energy = I(ω/t) × ωavgt

Where ωavg is the average angular velocity of the final and initial angular velocity (final - ωinitial)/2) of the object. Therefore, we obtain the below equation for the rotational kinetic energy.

 $KE_\text{rotational}=\tfrac{1}{2} I\omega^2$