Skip to main content

Mathematics for Physics

184 Newtonian Kinetic Energy Formula Derivation

A derivation of the Newtonian kinetic energy formula that is used to obtain \(K = \dfrac{1}{2}mv^{2}\) from the work-energy theorem.

Note: Also called \(T\) in some sources.

The Newtonian kinetic energy is

\[ K = \dfrac{1}{2}mv^{2} \]

The work-energy theorem is

\[ K_{2} - K_{1} = W \]

where

  • \(K\) is the kinetic energy.
  • \(m\) is the mass.
  • \(v\) is the speed.
  • \(K_{1}\) and \(K_{2}\) are the initial and final kinetic energies.
  • \(W\) is the work done by the force.

184.1 References

  1. Shankar, R. Fundamentals of Physics I. Yale University Press, 2019. — from \(v_{2}^{2}=v_{1}^{2}+2\left(\dfrac{F}{m}\right)d\) to \(K_{2}-K_{1}=W\) with \(K=\dfrac{1}{2}mv^{2}\).
  2. Logan, J. D. A First Course in Differential Equations. Springer, 2015. — multiply \(m\dfrac{d^{2}x}{dt^{2}}+\gamma\dfrac{dx}{dt}+kx=0\) by \(\dfrac{dx}{dt}\) to identify \(T=\dfrac{1}{2}m\left(\dfrac{dx}{dt}\right)^{2}\).