Kinetic energy is the energy in motion. An object that has motion – whether it is vertical or horizontal motion – has kinetic energy. These are many forms of kinetic energy – vibrational (the energy due to vibrational motion), rotational (due to rotational energy), and translational (due to motion from one location to another). Focusing on translational, the following equation is used to represent the kinetic energy (KE) of an object:
Where, m = mass of object
v = speed of the object
This equation reveals that the kinetic energy of an object is already proportional to the square of its speed. That means, for a twofold increase in speed, the kinetic energy will increase by a factor of four. For a threefold increase in speed, the kinetic energy will increase by a factor of nine. And for a fourfold increase speed, the kinetic energy will increase by a factor of sixteen. The kinetic energy is dependent upon the square of the speed.
Kinetic energy is a scalar quantity; it does not have a direction. The kinetic energy of an object is completely described by magnitude alone. The standard metric unit of measurement for kinetic energy is the Joule – one Joule is equivalent to 1kg*(m/s) ^2.
Equations:
Linear motion:
Rotational motion:
where, cor means center of rotation, and com is the center of the mass.
Conservation of energy:
Linear motion (relativistic):
where the final approximate equality holds for v << c.
Extended Explanation:
When a (non-relativistic) particle of mass moves with velocity, the particle’s kinetic energy is given by
The relationship between the momentum p and velocity is p = mv equation (1) can also be written
For a collection of particles (labeled by index) the total kinetic energy is given by
where m(i) is the mass of the ith particle and v_{(i)} is the magnitude of the ith particle’s velocity.
For the case of a continuous distribution of particles
For the case of the rigid body v(r) = ω x r for constant ω, and the above equation reduces to
where
In many cases (e.g., cubic, spherical) the system is symmetric enough that
in which case the above equation for kinetic energy reduces to
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