kinetic energy (idea)

Kinetic energy is the energy of motion. The motion itself may involve the movement of a body from one position to another (translational kinetic energy), the vibration of, for example, a spring (vibrational kinetic energy) or the rotation of a body about its own axis (rotational kinetic energy).

The strict definition of kinetic energy is that it is the work required to get an object to move (in the translational case) with a velocity v. Given a force F exerted on a body of mass m over a distance ds, the work done is given by

W= ∫F·ds

Noting Newtons second law (F=m dv/dt) and the definition of velocity (v=ds/dt) the equation may be rewritten.

W=m∫ dv/dt·(vdt)

The nature of differentiation allows one to write

dv/dt.v=(1/2)d(v.v)/dt=(1/2)dv²/dt

Substituting this into the above yields a simple expression for the translational kinetic energy

(1/2)mv²

Similarly, the rotational kinetic energy of a body with moment of inertia I and angular velocity ω is given by

(1/2)Iω²

When driving at 30 m.p.h an increase of speed to 33 m.p.h leads to an over 20% increase in kinetic energy and a corresponding increase in the chance of serious injury or death if there were an accident at the increased speed (not 10% as might be supposed by someone who supposes a linear relationship between speed and energy). Note, a similar relationship applies between the stopping distance of a car and its speed (thanks trikyguy).

The temperature T of a body is proportional to the average translational kinetic energy of its atoms or molecules.

<(1/2)mv²>_average=(3/2)kT

where k is the Boltzmann constant