Deriving Kinetic EnergyKinetic energy is closely linked with the concept of work, which is the scalar product (or dot product) of force and the displacement vector over which the force is applied.
Using some basic kinematics equations, we obtain an equation for the acceleration of an object which changes speed. (In the following equation, the term x - x0 has been replaced by s, a term which represents the total distance of displacement.)
v2 = v02 + 2asThe kinetic energy, K (or sometimes Ek) is, therefore, defined as:
a = ( v2 - v02 ) / 2s
Applying Newton's Second Law of Motion, F = ma, we get:
F = ma = m ( v2 - v02 ) / 2s
and, multiplying by the distance s (for work) and breaking it apart, we get:
W = Fs = 0.5mv2 - 0.5mv02
K = 0.5mv2It should be noted that, as mentioned before, this quantity will always be a non-zero scalar quantity. If the object has mass and is moving, it will always be positive. It will be zero in the case of a massless object or an object at rest (zero velocity). The kinetic energy equation, therefore, gives us no information about the direction of the motion, only about the speed.
Work-Energy TheoremThe work-energy theorem comes from the above derivation, and indicates that the work done by an external force on a particle is equal to the change in kinetic energy of the particle. Mathematically, then, you get:
Wtot = K2 - K1 = delta-K