Kinetic Energy

Kinetic Energy = \dfrac{1}{2}mv^{2}

The good student is cautious not to confuse kinetic energy with momentum.

Check it out for yourself–the above expression has units that are the units of energy.

When we multiply mass times velocity the units are

\frac{kg m}{s}

For systems considered to be conservative (i.e. no friction, etc.) the total energy is the sum of kinetic energy and potential energy.

The “one half m v squared” form of kinetic energy is what is commonly taught in early chemistry and physics courses, but another form is used in Quantum Mechanics, so work will be shown below to prove that the two are equivalent.

\dfrac{1}{2}mv^{2}

Change the expression so that everything is either in the numerator or the denominator:

\dfrac{mv^2}{2}

Multiply numerator and denominator by m:

\dfrac {m}{m} \dfrac{mv^2}{2} = \dfrac {m^2 v^2} {2m}

Recall that momentum, p, is mass times velocity: p=mv \rightarrow p^2=m^2v^2

\dfrac {m^2 v^2} {2m} = \dfrac {p^2} {2m}