The momentum of a particle having mass m is p. Its kinetic energy will be

Kinetic energy of a particle can be expressed either in terms of its velocity or, equivalently, in terms of its momentum and mass. For a particle of mass m moving with velocity v, momentum is p = mv and kinetic energy is K = (1/2)mv2; eliminating v between these two definitions gives kinetic energy purely in terms of p and m.

1. From the definition of momentum, p = mv, so the velocity is v = p/m.

2. The kinetic energy is defined as K = (1/2)mv2.

3. Substituting v = p/m into this expression gives K = (1/2)m(p/m)2 = (1/2)·p2/m.

4. Simplifying, K = p2/2m.

As an independent check, verify the units: p2/m has units of (kg·m/s)2/kg = kg·m2/s2, which is the joule, the correct SI unit for energy — confirming the expression is dimensionally consistent with kinetic energy.

Hence the particle's kinetic energy in terms of its momentum p and mass m is p2/2m.